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BT 227.982 767.476 Td /F1 9.8 Tf [(Emergence: Complexity and Organization. Emergence: Complexity and Organization.)] TJ ET
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BT 15.000 745.137 Td /F2 21.0 Tf [(The logistic map, functional iteration, and complexity)] TJ ET
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BT 15.000 723.567 Td /F2 13.5 Tf [(An introduction to Robert M. May's "Simple mathematical models with very complicated )] TJ ET
BT 15.000 707.529 Td /F2 13.5 Tf [(dynamics")] TJ ET
Q
BT 15.000 689.055 Td /F3 9.8 Tf [(June 30, 2015)] TJ ET
BT 74.846 689.055 Td /F3 9.8 Tf [()] TJ ET
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BT 79.721 689.055 Td /F3 9.8 Tf [(Classic Paper)] TJ ET
BT 26.250 677.214 Td /F1 9.8 Tf [(Professor Jeffrey Goldstein)] TJ ET
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BT 26.250 666.042 Td /F4 9.0 Tf [(1)] TJ ET
BT 31.254 666.042 Td /F1 9.0 Tf [( Adelphi University)] TJ ET
BT 26.250 654.321 Td /F1 9.8 Tf [(Goldstein J. The logistic map, functional iteration, and complexity: An introduction to Robert M. May's "Simple mathematical )] TJ ET
BT 26.250 642.416 Td /F1 9.8 Tf [(models with very complicated dynamics". Emergence: Complexity and Organization. 2015 Jun 30 [last modified: 2015 Oct 4]. )] TJ ET
BT 26.250 630.511 Td /F1 9.8 Tf [(Edition 1. doi: 10.emerg/10.17357.abddc31159b71e5a63f3b8bfddcd064a.)] TJ ET
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BT 26.250 601.409 Td /F4 12.0 Tf [(Abstract)] TJ ET
BT 26.250 581.454 Td /F1 9.8 Tf [(In this and the next issue of )] TJ ET
BT 147.657 581.454 Td /F5 9.8 Tf [(E:CO)] TJ ET
BT 171.496 581.454 Td /F1 9.8 Tf [(, we are reprinting two classic papers emanating from and even generating much of the)] TJ ET
BT 26.250 569.550 Td /F1 9.8 Tf [(ground swell of burgeoning enthusiasm for chaos theory forty years ago. Yes, its been that long! These papers were among the )] TJ ET
BT 26.250 557.645 Td /F1 9.8 Tf [(first to explore the nature of bifurcation, attractors, and chaos as discovered in complex systems through developing new )] TJ ET
BT 26.250 545.740 Td /F1 9.8 Tf [(mathematical tools and conceptual frameworks. These papers are truly foundational in their ramifications and implications and )] TJ ET
BT 26.250 533.835 Td /F1 9.8 Tf [(yet come across as fresh as if they were written yesterday. )] TJ ET
BT 26.250 497.233 Td /F4 12.0 Tf [(Introduction)] TJ ET
BT 26.250 477.279 Td /F1 9.8 Tf [(The routes by which its aficionados have come to complexity science are many and varied but several prototypes of complex )] TJ ET
BT 26.250 465.374 Td /F1 9.8 Tf [(systems stand out as salient attractors of interest \(for more on these prototypes, see )] TJ ET
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BT 398.534 466.881 Td /F4 8.7 Tf [(1)] TJ ET
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BT 403.353 465.374 Td /F1 9.8 Tf [(\):)] TJ ET
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48.078 437.480 48.522 437.664 48.838 437.980 c
49.154 438.296 49.337 438.740 49.337 439.187 c f
BT 54.750 436.219 Td /F1 9.8 Tf [(Self-organizing systems and the schools devoted to studying them, e.g., Hakens synergetics and Prigogines far-from-)] TJ ET
BT 54.750 424.314 Td /F1 9.8 Tf [(equilibrium thermodynamics;)] TJ ET
49.337 411.627 m
49.337 412.074 49.154 412.518 48.838 412.834 c
48.522 413.150 48.078 413.334 47.631 413.334 c
47.185 413.334 46.741 413.150 46.425 412.834 c
46.109 412.518 45.925 412.074 45.925 411.627 c
45.925 411.181 46.109 410.737 46.425 410.421 c
46.741 410.105 47.185 409.921 47.631 409.921 c
48.078 409.921 48.522 410.105 48.838 410.421 c
49.154 410.737 49.337 411.181 49.337 411.627 c f
BT 54.750 408.660 Td /F1 9.8 Tf [(Computational emergence observed in agent-based modeling, artificial life, and other computational simulations;)] TJ ET
49.337 395.973 m
49.337 396.419 49.154 396.863 48.838 397.179 c
48.522 397.495 48.078 397.679 47.631 397.679 c
47.185 397.679 46.741 397.495 46.425 397.179 c
46.109 396.863 45.925 396.419 45.925 395.973 c
45.925 395.526 46.109 395.082 46.425 394.766 c
46.741 394.450 47.185 394.266 47.631 394.266 c
48.078 394.266 48.522 394.450 48.838 394.766 c
49.154 395.082 49.337 395.526 49.337 395.973 c f
BT 54.750 393.005 Td /F1 9.8 Tf [(Networks of various kinds including social/organizational networks, electronic networks, communication networks, and )] TJ ET
BT 54.750 381.100 Td /F1 9.8 Tf [(so forth;)] TJ ET
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48.078 366.707 48.522 366.891 48.838 367.206 c
49.154 367.522 49.337 367.966 49.337 368.413 c f
BT 54.750 365.445 Td /F1 9.8 Tf [(Phase transitions and collective condensed matter phenomena such as Quantum Protectorates;)] TJ ET
49.337 352.758 m
49.337 353.205 49.154 353.649 48.838 353.965 c
48.522 354.281 48.078 354.465 47.631 354.465 c
47.185 354.465 46.741 354.281 46.425 353.965 c
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48.078 351.052 48.522 351.236 48.838 351.552 c
49.154 351.868 49.337 352.312 49.337 352.758 c f
BT 54.750 349.791 Td /F1 9.8 Tf [(Biological emergence including symbiogenesis, lateral gene transfer, novel speciations, etc.;)] TJ ET
49.337 337.104 m
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48.078 335.397 48.522 335.581 48.838 335.897 c
49.154 336.213 49.337 336.657 49.337 337.104 c f
BT 54.750 334.136 Td /F1 9.8 Tf [(Bifrucation scenarios in nonlinear dynamical systems \(NDS; chaos theory\).)] TJ ET
BT 26.250 299.731 Td /F1 9.8 Tf [(The last entry in this list is the only one that is primarily mathematical in nature although the others rely upon and are deeply )] TJ ET
BT 26.250 287.826 Td /F1 9.8 Tf [(invested in mathematical frameworks. It would be hard put not to find elements of NDS, in one form or another, in the research )] TJ ET
BT 26.250 275.922 Td /F1 9.8 Tf [(and theorizing that finds its way into )] TJ ET
BT 183.410 275.922 Td /F5 9.8 Tf [(E:CO)] TJ ET
BT 207.249 275.922 Td /F1 9.8 Tf [(. The ongoing vitality of NDS is attested by its predominance in the work of this years )] TJ ET
BT 26.250 264.017 Td /F1 9.8 Tf [(Fields Medalists winners, particularly in the research of the Brazilian savant Arthur Avila who has found surprising connections )] TJ ET
BT 26.250 252.112 Td /F1 9.8 Tf [(between dynamical systems and other branches of math.)] TJ ET
BT 26.250 232.707 Td /F1 9.8 Tf [(Nonlinear dynamical systems goes back at least as far as Poincares glimpses of phenomena mathematically so bizarre, he )] TJ ET
BT 26.250 220.803 Td /F1 9.8 Tf [(declared he couldnt dare contemplate them! What he was seeing was a type of instability in his formulations of differential )] TJ ET
BT 26.250 208.898 Td /F1 9.8 Tf [(equations hithertofore unprecedented and conceptually unexpected. This glimpse of chaos threatened the stability of nothing )] TJ ET
BT 26.250 196.993 Td /F1 9.8 Tf [(less than the cosmos itself and moreover his work was taking place during an age when religious solace in such matters was )] TJ ET
BT 26.250 185.088 Td /F1 9.8 Tf [(considerably on the wane. Poincares quest and the methods he developed were taken up as subjects of inquiry in their own )] TJ ET
BT 26.250 173.184 Td /F1 9.8 Tf [(own right as well prompting the development of topology and other rich mathematical fields.)] TJ ET
BT 26.250 153.779 Td /F1 9.8 Tf [(The two classic papers that are being reprinted in the current and next issues of )] TJ ET
BT 372.541 153.779 Td /F5 9.8 Tf [(E:CO)] TJ ET
BT 396.380 153.779 Td /F1 9.8 Tf [( represent milestones, forty some years )] TJ ET
BT 26.250 141.874 Td /F1 9.8 Tf [(ago, in the burgeoning growth of chaos theory. These papers exhibit incisive probings into the inner structure of one of the main )] TJ ET
BT 26.250 129.969 Td /F1 9.8 Tf [(mathematical processes at work in complex systems, namely, that of )] TJ ET
BT 325.351 129.969 Td /F5 9.8 Tf [(functional iteration)] TJ ET
BT 404.472 129.969 Td /F1 9.8 Tf [( especially as exhibited in the logistic )] TJ ET
BT 26.250 118.065 Td /F1 9.8 Tf [(map, a nonlinear difference equation that has been found of great usefulness in several arenas of applications including, e.g., )] TJ ET
BT 26.250 106.160 Td /F1 9.8 Tf [(population dynamics. It is important to note, however, that neither of these two papers focuses on the applications of the logistic )] TJ ET
BT 26.250 94.255 Td /F1 9.8 Tf [(map but instead go right into its mathematical core in order to lay bare the unforeseen role of parameters in prompting )] TJ ET
BT 26.250 82.350 Td /F1 9.8 Tf [(bifurcation of new attractor regimes.)] TJ ET
BT 26.250 62.946 Td /F1 9.8 Tf [(Over many years of teaching complexity science, I have come to appreciate the instructional utility of thematic images or )] TJ ET
BT 26.250 51.041 Td /F1 9.8 Tf [(pictures of complexity imbuing each prototype of complexity. In this I have followed the preeminent philosopher of science Mary )] TJ ET
BT 26.250 39.136 Td /F1 9.8 Tf [(Hesses)] TJ ET
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BT 65.740 39.136 Td /F1 9.8 Tf [( cogent account of the heart of scientific creativity as resting on metaphoric/analogical visualizations of the natural )] TJ ET
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BT 15.000 745.137 Td /F2 21.0 Tf [(The logistic map, functional iteration, and complexity)] TJ ET
0.271 0.267 0.267 rg
BT 15.000 723.567 Td /F2 13.5 Tf [(An introduction to Robert M. May's "Simple mathematical models with very complicated )] TJ ET
BT 15.000 707.529 Td /F2 13.5 Tf [(dynamics")] TJ ET
Q
BT 15.000 689.055 Td /F3 9.8 Tf [(June 30, 2015)] TJ ET
BT 74.846 689.055 Td /F3 9.8 Tf [()] TJ ET
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BT 79.721 689.055 Td /F3 9.8 Tf [(Classic Paper)] TJ ET
BT 26.250 677.214 Td /F1 9.8 Tf [(Professor Jeffrey Goldstein)] TJ ET
0.271 0.267 0.267 rg
BT 143.835 681.103 Td /F1 8.7 Tf [(1)] TJ ET
BT 26.250 666.042 Td /F4 9.0 Tf [(1)] TJ ET
BT 31.254 666.042 Td /F1 9.0 Tf [( Adelphi University)] TJ ET
BT 26.250 654.321 Td /F1 9.8 Tf [(Goldstein J. The logistic map, functional iteration, and complexity: An introduction to Robert M. May's "Simple mathematical )] TJ ET
BT 26.250 642.416 Td /F1 9.8 Tf [(models with very complicated dynamics". Emergence: Complexity and Organization. 2015 Jun 30 [last modified: 2015 Oct 4]. )] TJ ET
BT 26.250 630.511 Td /F1 9.8 Tf [(Edition 1. doi: 10.emerg/10.17357.abddc31159b71e5a63f3b8bfddcd064a.)] TJ ET
q
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0.271 0.267 0.267 rg
BT 26.250 601.409 Td /F4 12.0 Tf [(Abstract)] TJ ET
BT 26.250 581.454 Td /F1 9.8 Tf [(In this and the next issue of )] TJ ET
BT 147.657 581.454 Td /F5 9.8 Tf [(E:CO)] TJ ET
BT 171.496 581.454 Td /F1 9.8 Tf [(, we are reprinting two classic papers emanating from and even generating much of the)] TJ ET
BT 26.250 569.550 Td /F1 9.8 Tf [(ground swell of burgeoning enthusiasm for chaos theory forty years ago. Yes, its been that long! These papers were among the )] TJ ET
BT 26.250 557.645 Td /F1 9.8 Tf [(first to explore the nature of bifurcation, attractors, and chaos as discovered in complex systems through developing new )] TJ ET
BT 26.250 545.740 Td /F1 9.8 Tf [(mathematical tools and conceptual frameworks. These papers are truly foundational in their ramifications and implications and )] TJ ET
BT 26.250 533.835 Td /F1 9.8 Tf [(yet come across as fresh as if they were written yesterday. )] TJ ET
BT 26.250 497.233 Td /F4 12.0 Tf [(Introduction)] TJ ET
BT 26.250 477.279 Td /F1 9.8 Tf [(The routes by which its aficionados have come to complexity science are many and varied but several prototypes of complex )] TJ ET
BT 26.250 465.374 Td /F1 9.8 Tf [(systems stand out as salient attractors of interest \(for more on these prototypes, see )] TJ ET
0.267 0.267 0.267 rg
BT 398.534 466.881 Td /F4 8.7 Tf [(1)] TJ ET
0.271 0.267 0.267 rg
BT 403.353 465.374 Td /F1 9.8 Tf [(\):)] TJ ET
0.965 0.965 0.965 rg
26.250 316.755 555.000 138.738 re f
0.267 0.267 0.267 rg
0.267 0.267 0.267 RG
26.250 455.493 m 581.250 455.493 l 581.250 454.743 l 26.250 454.743 l f
26.250 316.755 m 581.250 316.755 l 581.250 317.505 l 26.250 317.505 l f
0.271 0.267 0.267 rg
0.271 0.267 0.267 RG
49.337 439.187 m
49.337 439.633 49.154 440.077 48.838 440.393 c
48.522 440.709 48.078 440.893 47.631 440.893 c
47.185 440.893 46.741 440.709 46.425 440.393 c
46.109 440.077 45.925 439.633 45.925 439.187 c
45.925 438.740 46.109 438.296 46.425 437.980 c
46.741 437.664 47.185 437.480 47.631 437.480 c
48.078 437.480 48.522 437.664 48.838 437.980 c
49.154 438.296 49.337 438.740 49.337 439.187 c f
BT 54.750 436.219 Td /F1 9.8 Tf [(Self-organizing systems and the schools devoted to studying them, e.g., Hakens synergetics and Prigogines far-from-)] TJ ET
BT 54.750 424.314 Td /F1 9.8 Tf [(equilibrium thermodynamics;)] TJ ET
49.337 411.627 m
49.337 412.074 49.154 412.518 48.838 412.834 c
48.522 413.150 48.078 413.334 47.631 413.334 c
47.185 413.334 46.741 413.150 46.425 412.834 c
46.109 412.518 45.925 412.074 45.925 411.627 c
45.925 411.181 46.109 410.737 46.425 410.421 c
46.741 410.105 47.185 409.921 47.631 409.921 c
48.078 409.921 48.522 410.105 48.838 410.421 c
49.154 410.737 49.337 411.181 49.337 411.627 c f
BT 54.750 408.660 Td /F1 9.8 Tf [(Computational emergence observed in agent-based modeling, artificial life, and other computational simulations;)] TJ ET
49.337 395.973 m
49.337 396.419 49.154 396.863 48.838 397.179 c
48.522 397.495 48.078 397.679 47.631 397.679 c
47.185 397.679 46.741 397.495 46.425 397.179 c
46.109 396.863 45.925 396.419 45.925 395.973 c
45.925 395.526 46.109 395.082 46.425 394.766 c
46.741 394.450 47.185 394.266 47.631 394.266 c
48.078 394.266 48.522 394.450 48.838 394.766 c
49.154 395.082 49.337 395.526 49.337 395.973 c f
BT 54.750 393.005 Td /F1 9.8 Tf [(Networks of various kinds including social/organizational networks, electronic networks, communication networks, and )] TJ ET
BT 54.750 381.100 Td /F1 9.8 Tf [(so forth;)] TJ ET
49.337 368.413 m
49.337 368.860 49.154 369.304 48.838 369.620 c
48.522 369.935 48.078 370.119 47.631 370.119 c
47.185 370.119 46.741 369.935 46.425 369.620 c
46.109 369.304 45.925 368.860 45.925 368.413 c
45.925 367.966 46.109 367.522 46.425 367.206 c
46.741 366.891 47.185 366.707 47.631 366.707 c
48.078 366.707 48.522 366.891 48.838 367.206 c
49.154 367.522 49.337 367.966 49.337 368.413 c f
BT 54.750 365.445 Td /F1 9.8 Tf [(Phase transitions and collective condensed matter phenomena such as Quantum Protectorates;)] TJ ET
49.337 352.758 m
49.337 353.205 49.154 353.649 48.838 353.965 c
48.522 354.281 48.078 354.465 47.631 354.465 c
47.185 354.465 46.741 354.281 46.425 353.965 c
46.109 353.649 45.925 353.205 45.925 352.758 c
45.925 352.312 46.109 351.868 46.425 351.552 c
46.741 351.236 47.185 351.052 47.631 351.052 c
48.078 351.052 48.522 351.236 48.838 351.552 c
49.154 351.868 49.337 352.312 49.337 352.758 c f
BT 54.750 349.791 Td /F1 9.8 Tf [(Biological emergence including symbiogenesis, lateral gene transfer, novel speciations, etc.;)] TJ ET
49.337 337.104 m
49.337 337.550 49.154 337.994 48.838 338.310 c
48.522 338.626 48.078 338.810 47.631 338.810 c
47.185 338.810 46.741 338.626 46.425 338.310 c
46.109 337.994 45.925 337.550 45.925 337.104 c
45.925 336.657 46.109 336.213 46.425 335.897 c
46.741 335.581 47.185 335.397 47.631 335.397 c
48.078 335.397 48.522 335.581 48.838 335.897 c
49.154 336.213 49.337 336.657 49.337 337.104 c f
BT 54.750 334.136 Td /F1 9.8 Tf [(Bifrucation scenarios in nonlinear dynamical systems \(NDS; chaos theory\).)] TJ ET
BT 26.250 299.731 Td /F1 9.8 Tf [(The last entry in this list is the only one that is primarily mathematical in nature although the others rely upon and are deeply )] TJ ET
BT 26.250 287.826 Td /F1 9.8 Tf [(invested in mathematical frameworks. It would be hard put not to find elements of NDS, in one form or another, in the research )] TJ ET
BT 26.250 275.922 Td /F1 9.8 Tf [(and theorizing that finds its way into )] TJ ET
BT 183.410 275.922 Td /F5 9.8 Tf [(E:CO)] TJ ET
BT 207.249 275.922 Td /F1 9.8 Tf [(. The ongoing vitality of NDS is attested by its predominance in the work of this years )] TJ ET
BT 26.250 264.017 Td /F1 9.8 Tf [(Fields Medalists winners, particularly in the research of the Brazilian savant Arthur Avila who has found surprising connections )] TJ ET
BT 26.250 252.112 Td /F1 9.8 Tf [(between dynamical systems and other branches of math.)] TJ ET
BT 26.250 232.707 Td /F1 9.8 Tf [(Nonlinear dynamical systems goes back at least as far as Poincares glimpses of phenomena mathematically so bizarre, he )] TJ ET
BT 26.250 220.803 Td /F1 9.8 Tf [(declared he couldnt dare contemplate them! What he was seeing was a type of instability in his formulations of differential )] TJ ET
BT 26.250 208.898 Td /F1 9.8 Tf [(equations hithertofore unprecedented and conceptually unexpected. This glimpse of chaos threatened the stability of nothing )] TJ ET
BT 26.250 196.993 Td /F1 9.8 Tf [(less than the cosmos itself and moreover his work was taking place during an age when religious solace in such matters was )] TJ ET
BT 26.250 185.088 Td /F1 9.8 Tf [(considerably on the wane. Poincares quest and the methods he developed were taken up as subjects of inquiry in their own )] TJ ET
BT 26.250 173.184 Td /F1 9.8 Tf [(own right as well prompting the development of topology and other rich mathematical fields.)] TJ ET
BT 26.250 153.779 Td /F1 9.8 Tf [(The two classic papers that are being reprinted in the current and next issues of )] TJ ET
BT 372.541 153.779 Td /F5 9.8 Tf [(E:CO)] TJ ET
BT 396.380 153.779 Td /F1 9.8 Tf [( represent milestones, forty some years )] TJ ET
BT 26.250 141.874 Td /F1 9.8 Tf [(ago, in the burgeoning growth of chaos theory. These papers exhibit incisive probings into the inner structure of one of the main )] TJ ET
BT 26.250 129.969 Td /F1 9.8 Tf [(mathematical processes at work in complex systems, namely, that of )] TJ ET
BT 325.351 129.969 Td /F5 9.8 Tf [(functional iteration)] TJ ET
BT 404.472 129.969 Td /F1 9.8 Tf [( especially as exhibited in the logistic )] TJ ET
BT 26.250 118.065 Td /F1 9.8 Tf [(map, a nonlinear difference equation that has been found of great usefulness in several arenas of applications including, e.g., )] TJ ET
BT 26.250 106.160 Td /F1 9.8 Tf [(population dynamics. It is important to note, however, that neither of these two papers focuses on the applications of the logistic )] TJ ET
BT 26.250 94.255 Td /F1 9.8 Tf [(map but instead go right into its mathematical core in order to lay bare the unforeseen role of parameters in prompting )] TJ ET
BT 26.250 82.350 Td /F1 9.8 Tf [(bifurcation of new attractor regimes.)] TJ ET
BT 26.250 62.946 Td /F1 9.8 Tf [(Over many years of teaching complexity science, I have come to appreciate the instructional utility of thematic images or )] TJ ET
BT 26.250 51.041 Td /F1 9.8 Tf [(pictures of complexity imbuing each prototype of complexity. In this I have followed the preeminent philosopher of science Mary )] TJ ET
BT 26.250 39.136 Td /F1 9.8 Tf [(Hesses)] TJ ET
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BT 60.921 40.643 Td /F4 8.7 Tf [(2)] TJ ET
0.271 0.267 0.267 rg
BT 65.740 39.136 Td /F1 9.8 Tf [( cogent account of the heart of scientific creativity as resting on metaphoric/analogical visualizations of the natural )] TJ ET
Q
q
15.000 698.321 577.500 66.774 re W n
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BT 15.000 745.137 Td /F2 21.0 Tf [(The logistic map, functional iteration, and complexity)] TJ ET
0.271 0.267 0.267 rg
BT 15.000 723.567 Td /F2 13.5 Tf [(An introduction to Robert M. May's "Simple mathematical models with very complicated )] TJ ET
BT 15.000 707.529 Td /F2 13.5 Tf [(dynamics")] TJ ET
Q
BT 15.000 689.055 Td /F3 9.8 Tf [(June 30, 2015)] TJ ET
BT 74.846 689.055 Td /F3 9.8 Tf [()] TJ ET
0.267 0.267 0.267 rg
BT 79.721 689.055 Td /F3 9.8 Tf [(Classic Paper)] TJ ET
BT 26.250 677.214 Td /F1 9.8 Tf [(Professor Jeffrey Goldstein)] TJ ET
0.271 0.267 0.267 rg
BT 143.835 681.103 Td /F1 8.7 Tf [(1)] TJ ET
BT 26.250 666.042 Td /F4 9.0 Tf [(1)] TJ ET
BT 31.254 666.042 Td /F1 9.0 Tf [( Adelphi University)] TJ ET
BT 26.250 654.321 Td /F1 9.8 Tf [(Goldstein J. The logistic map, functional iteration, and complexity: An introduction to Robert M. May's "Simple mathematical )] TJ ET
BT 26.250 642.416 Td /F1 9.8 Tf [(models with very complicated dynamics". Emergence: Complexity and Organization. 2015 Jun 30 [last modified: 2015 Oct 4]. )] TJ ET
BT 26.250 630.511 Td /F1 9.8 Tf [(Edition 1. doi: 10.emerg/10.17357.abddc31159b71e5a63f3b8bfddcd064a.)] TJ ET
q
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BT 26.250 601.409 Td /F4 12.0 Tf [(Abstract)] TJ ET
BT 26.250 581.454 Td /F1 9.8 Tf [(In this and the next issue of )] TJ ET
BT 147.657 581.454 Td /F5 9.8 Tf [(E:CO)] TJ ET
BT 171.496 581.454 Td /F1 9.8 Tf [(, we are reprinting two classic papers emanating from and even generating much of the)] TJ ET
BT 26.250 569.550 Td /F1 9.8 Tf [(ground swell of burgeoning enthusiasm for chaos theory forty years ago. Yes, its been that long! These papers were among the )] TJ ET
BT 26.250 557.645 Td /F1 9.8 Tf [(first to explore the nature of bifurcation, attractors, and chaos as discovered in complex systems through developing new )] TJ ET
BT 26.250 545.740 Td /F1 9.8 Tf [(mathematical tools and conceptual frameworks. These papers are truly foundational in their ramifications and implications and )] TJ ET
BT 26.250 533.835 Td /F1 9.8 Tf [(yet come across as fresh as if they were written yesterday. )] TJ ET
BT 26.250 497.233 Td /F4 12.0 Tf [(Introduction)] TJ ET
BT 26.250 477.279 Td /F1 9.8 Tf [(The routes by which its aficionados have come to complexity science are many and varied but several prototypes of complex )] TJ ET
BT 26.250 465.374 Td /F1 9.8 Tf [(systems stand out as salient attractors of interest \(for more on these prototypes, see )] TJ ET
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BT 398.534 466.881 Td /F4 8.7 Tf [(1)] TJ ET
0.271 0.267 0.267 rg
BT 403.353 465.374 Td /F1 9.8 Tf [(\):)] TJ ET
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49.337 439.187 m
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48.078 437.480 48.522 437.664 48.838 437.980 c
49.154 438.296 49.337 438.740 49.337 439.187 c f
BT 54.750 436.219 Td /F1 9.8 Tf [(Self-organizing systems and the schools devoted to studying them, e.g., Hakens synergetics and Prigogines far-from-)] TJ ET
BT 54.750 424.314 Td /F1 9.8 Tf [(equilibrium thermodynamics;)] TJ ET
49.337 411.627 m
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48.522 413.150 48.078 413.334 47.631 413.334 c
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48.078 409.921 48.522 410.105 48.838 410.421 c
49.154 410.737 49.337 411.181 49.337 411.627 c f
BT 54.750 408.660 Td /F1 9.8 Tf [(Computational emergence observed in agent-based modeling, artificial life, and other computational simulations;)] TJ ET
49.337 395.973 m
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48.078 394.266 48.522 394.450 48.838 394.766 c
49.154 395.082 49.337 395.526 49.337 395.973 c f
BT 54.750 393.005 Td /F1 9.8 Tf [(Networks of various kinds including social/organizational networks, electronic networks, communication networks, and )] TJ ET
BT 54.750 381.100 Td /F1 9.8 Tf [(so forth;)] TJ ET
49.337 368.413 m
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48.078 366.707 48.522 366.891 48.838 367.206 c
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BT 54.750 365.445 Td /F1 9.8 Tf [(Phase transitions and collective condensed matter phenomena such as Quantum Protectorates;)] TJ ET
49.337 352.758 m
49.337 353.205 49.154 353.649 48.838 353.965 c
48.522 354.281 48.078 354.465 47.631 354.465 c
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48.078 351.052 48.522 351.236 48.838 351.552 c
49.154 351.868 49.337 352.312 49.337 352.758 c f
BT 54.750 349.791 Td /F1 9.8 Tf [(Biological emergence including symbiogenesis, lateral gene transfer, novel speciations, etc.;)] TJ ET
49.337 337.104 m
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BT 54.750 334.136 Td /F1 9.8 Tf [(Bifrucation scenarios in nonlinear dynamical systems \(NDS; chaos theory\).)] TJ ET
BT 26.250 299.731 Td /F1 9.8 Tf [(The last entry in this list is the only one that is primarily mathematical in nature although the others rely upon and are deeply )] TJ ET
BT 26.250 287.826 Td /F1 9.8 Tf [(invested in mathematical frameworks. It would be hard put not to find elements of NDS, in one form or another, in the research )] TJ ET
BT 26.250 275.922 Td /F1 9.8 Tf [(and theorizing that finds its way into )] TJ ET
BT 183.410 275.922 Td /F5 9.8 Tf [(E:CO)] TJ ET
BT 207.249 275.922 Td /F1 9.8 Tf [(. The ongoing vitality of NDS is attested by its predominance in the work of this years )] TJ ET
BT 26.250 264.017 Td /F1 9.8 Tf [(Fields Medalists winners, particularly in the research of the Brazilian savant Arthur Avila who has found surprising connections )] TJ ET
BT 26.250 252.112 Td /F1 9.8 Tf [(between dynamical systems and other branches of math.)] TJ ET
BT 26.250 232.707 Td /F1 9.8 Tf [(Nonlinear dynamical systems goes back at least as far as Poincares glimpses of phenomena mathematically so bizarre, he )] TJ ET
BT 26.250 220.803 Td /F1 9.8 Tf [(declared he couldnt dare contemplate them! What he was seeing was a type of instability in his formulations of differential )] TJ ET
BT 26.250 208.898 Td /F1 9.8 Tf [(equations hithertofore unprecedented and conceptually unexpected. This glimpse of chaos threatened the stability of nothing )] TJ ET
BT 26.250 196.993 Td /F1 9.8 Tf [(less than the cosmos itself and moreover his work was taking place during an age when religious solace in such matters was )] TJ ET
BT 26.250 185.088 Td /F1 9.8 Tf [(considerably on the wane. Poincares quest and the methods he developed were taken up as subjects of inquiry in their own )] TJ ET
BT 26.250 173.184 Td /F1 9.8 Tf [(own right as well prompting the development of topology and other rich mathematical fields.)] TJ ET
BT 26.250 153.779 Td /F1 9.8 Tf [(The two classic papers that are being reprinted in the current and next issues of )] TJ ET
BT 372.541 153.779 Td /F5 9.8 Tf [(E:CO)] TJ ET
BT 396.380 153.779 Td /F1 9.8 Tf [( represent milestones, forty some years )] TJ ET
BT 26.250 141.874 Td /F1 9.8 Tf [(ago, in the burgeoning growth of chaos theory. These papers exhibit incisive probings into the inner structure of one of the main )] TJ ET
BT 26.250 129.969 Td /F1 9.8 Tf [(mathematical processes at work in complex systems, namely, that of )] TJ ET
BT 325.351 129.969 Td /F5 9.8 Tf [(functional iteration)] TJ ET
BT 404.472 129.969 Td /F1 9.8 Tf [( especially as exhibited in the logistic )] TJ ET
BT 26.250 118.065 Td /F1 9.8 Tf [(map, a nonlinear difference equation that has been found of great usefulness in several arenas of applications including, e.g., )] TJ ET
BT 26.250 106.160 Td /F1 9.8 Tf [(population dynamics. It is important to note, however, that neither of these two papers focuses on the applications of the logistic )] TJ ET
BT 26.250 94.255 Td /F1 9.8 Tf [(map but instead go right into its mathematical core in order to lay bare the unforeseen role of parameters in prompting )] TJ ET
BT 26.250 82.350 Td /F1 9.8 Tf [(bifurcation of new attractor regimes.)] TJ ET
BT 26.250 62.946 Td /F1 9.8 Tf [(Over many years of teaching complexity science, I have come to appreciate the instructional utility of thematic images or )] TJ ET
BT 26.250 51.041 Td /F1 9.8 Tf [(pictures of complexity imbuing each prototype of complexity. In this I have followed the preeminent philosopher of science Mary )] TJ ET
BT 26.250 39.136 Td /F1 9.8 Tf [(Hesses)] TJ ET
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BT 65.740 39.136 Td /F1 9.8 Tf [( cogent account of the heart of scientific creativity as resting on metaphoric/analogical visualizations of the natural )] TJ ET
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BT 291.710 19.825 Td /F1 11.0 Tf [(1)] TJ ET
BT 25.000 19.825 Td /F1 11.0 Tf [(Emergence: Complexity and Organization)] TJ ET
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BT 26.250 767.476 Td /F1 9.8 Tf [(objects being studied. Mathematics can also be expounded by recognizing and probing the imagistic and pictorial associations )] TJ ET
BT 26.250 755.571 Td /F1 9.8 Tf [(of equations which employ figurative notations and codes of the processes, operations, parameters, and other means, that is, )] TJ ET
BT 26.250 743.667 Td /F1 9.8 Tf [(a kind of metaphoric short-hand that delineates relationships among different mathematical objects)] TJ ET
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BT 451.652 745.174 Td /F4 8.7 Tf [(3)] TJ ET
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BT 456.471 743.667 Td /F1 9.8 Tf [(.)] TJ ET
BT 26.250 724.262 Td /F1 9.8 Tf [(By invoking the notion of a picture or image, I am not suggesting a dumbing down of the math. That this is not the case will be )] TJ ET
BT 26.250 712.357 Td /F1 9.8 Tf [(obvious when we encounter: first, the mathematical images of the logistic map which, as a nonlinear difference equation, )] TJ ET
BT 26.250 700.452 Td /F1 9.8 Tf [(portrays the changes in the values of variable\(s\) at discrete time intervals; second, , bifurcation diagrams showing the critical )] TJ ET
BT 26.250 688.548 Td /F1 9.8 Tf [(points of the bifurcation or control variables when attractors; third, parameter values ratio equations \(e.g., revealing the amazing )] TJ ET
BT 26.250 676.643 Td /F1 9.8 Tf [(constant Feigenbaum discovered\);, and so on. These specific thematic images follow from the fact that the centerpiece of both )] TJ ET
BT 26.250 664.738 Td /F1 9.8 Tf [(Mays and Feigenbaums inquiries is the sequence of bifurcations that occur during the operation of functional iteration in the )] TJ ET
BT 26.250 652.833 Td /F1 9.8 Tf [(now notorious logistic map which functions as the apotheosis of thematic imagery in their explorations of chaos. Without )] TJ ET
BT 26.250 640.929 Td /F1 9.8 Tf [(needing to dumb it down, however, a few preliminary remarks on what May explicitly focused on mathematically can be helpful.)] TJ ET
BT 26.250 604.326 Td /F4 12.0 Tf [(Population dynamics and the logistic map)] TJ ET
BT 26.250 584.372 Td /F1 9.8 Tf [(The author of the first classic paper, Robert May, stands out as a true polymath. An Australian trained as a theoretical physicist, )] TJ ET
BT 26.250 572.467 Td /F1 9.8 Tf [(serving stints at premier universities like Princeton, Harvard, Oxford \(only the beginning of a long list\); he has made )] TJ ET
BT 26.250 560.562 Td /F1 9.8 Tf [(groundbreaking discoveries in theoretical ecology and biology, population dynamics, pure and applied mathematics and many )] TJ ET
BT 26.250 548.658 Td /F1 9.8 Tf [(other fields, being knighted along the way for his persistently outstanding work.)] TJ ET
BT 26.250 529.253 Td /F1 9.8 Tf [(In his forays in population dynamics and theoretical ecology, May inevitably came into contact with the so-called logistic map, a )] TJ ET
BT 26.250 517.348 Td /F1 9.8 Tf [(type of discrete \(a difference not a differential\) function fashioned in order to capture certain features of how populations )] TJ ET
BT 26.250 505.443 Td /F1 9.8 Tf [(change, for instance, the manner by which a predator/prey model effectuates an increase, then decrease of population:)] TJ ET
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BT 54.750 476.289 Td /F1 9.8 Tf [(First, the number of predators increases since the amount of prey is high because they have not as yet been eaten;)] TJ ET
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BT 54.750 460.634 Td /F1 9.8 Tf [(After a peak in the population of predators, there are less prey to bear consumed resulting to a decline in the number )] TJ ET
BT 560.385 460.634 Td /F1 9.8 Tf [(of )] TJ ET
BT 54.750 448.729 Td /F1 9.8 Tf [(predators \(they starve to death\);)] TJ ET
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BT 54.750 433.074 Td /F1 9.8 Tf [(As the population of predators decline, the population of prey rises since there are less predators to eat them.)] TJ ET
BT 26.250 398.670 Td /F1 9.8 Tf [(It is useful to keep this parabolic \(when graphed\) dynamic in mind since the logistic map \(see Eqn. 1 below\), to which May and )] TJ ET
BT 26.250 386.765 Td /F1 9.8 Tf [(Feigenbaum devote so much of their attention, displays this rise and fall in terms of discrete changes in the population, e.g., )] TJ ET
BT 26.250 374.860 Td /F1 9.8 Tf [(from one year to the next. The logistic map discretizes the population variable by repeating or iterating the functional operation )] TJ ET
BT 26.250 362.955 Td /F1 9.8 Tf [(at each temporal interval of time \(stipulated such as weekly, or monthly, or yearly\). This repetition is called functional iteration )] TJ ET
BT 26.250 351.051 Td /F1 9.8 Tf [(since each newly-arrived-at value of the variable is repeatedly pumped-back into the function and the functional operation is )] TJ ET
BT 26.250 339.146 Td /F1 9.8 Tf [(iterated to arrive at the next value.)] TJ ET
BT 26.250 319.741 Td /F1 9.8 Tf [(Here is the logistic map in the usual form found in population dynamics:)] TJ ET
BT 26.250 296.660 Td /F1 9.8 Tf [(Eqn 1. )] TJ ET
q
111.000 0 0 16.500 57.148 293.360 cm /I2 Do
Q
BT 26.250 276.336 Td /F6 9.8 Tf [(P)] TJ ET
BT 32.753 274.272 Td /F6 8.7 Tf [(t)] TJ ET
BT 35.639 276.336 Td /F1 9.8 Tf [( is the population at some arbitrarily chosen initial year )] TJ ET
BT 272.983 276.336 Td /F6 9.8 Tf [(t)] TJ ET
BT 276.230 276.336 Td /F1 9.8 Tf [(, )] TJ ET
BT 284.362 276.336 Td /F6 9.8 Tf [(P)] TJ ET
BT 290.865 276.336 Td /F6 8.7 Tf [(t)] TJ ET
BT 293.751 274.272 Td /F4 8.7 Tf [(+1)] TJ ET
BT 303.631 276.336 Td /F1 9.8 Tf [(is the population at the next year )] TJ ET
BT 449.969 276.336 Td /F6 9.8 Tf [(t)] TJ ET
BT 453.216 276.336 Td /F4 9.8 Tf [(+1)] TJ ET
BT 464.331 276.336 Td /F1 9.8 Tf [(, and)] TJ ET
BT 486.015 276.336 Td /F6 9.8 Tf [( a)] TJ ET
BT 494.146 276.336 Td /F1 9.8 Tf [( is the birth/death )] TJ ET
BT 26.250 264.432 Td /F1 9.8 Tf [(rate \(after suitable adjustments such as setting at a constant some such factor comprising the system/environment, e.g., )] TJ ET
BT 26.250 252.527 Td /F1 9.8 Tf [(available space, food, etc.\) This Eqn. 1a is nonlinear because the variable )] TJ ET
BT 347.620 252.527 Td /F6 9.8 Tf [(P)] TJ ET
BT 354.123 250.463 Td /F6 8.7 Tf [(t)] TJ ET
BT 357.009 252.527 Td /F1 9.8 Tf [( is multiplied by itself:)] TJ ET
BT 26.250 204.246 Td /F1 9.8 Tf [(Eqn 1a. )] TJ ET
q
48.000 0 0 48.000 62.569 194.646 cm /I3 Do
Q
0.500 0.500 0.500 rg
BT 62.569 235.246 Td /F1 8.0 Tf [(Image not found)] TJ ET
BT 62.569 225.246 Td /F1 8.0 Tf [(http://bit.ly/1JraLBy)] TJ ET
0.271 0.267 0.267 rg
BT 26.250 177.622 Td /F1 9.8 Tf [(It is also useful to keep in mind that although a big deal is usually made about the nonlinearity of the nonlinear dynamical )] TJ ET
BT 26.250 165.717 Td /F1 9.8 Tf [(systems aspect of complexity science, by itself, this doesnt imply much more than the simple fact that multiplication in some )] TJ ET
BT 26.250 153.813 Td /F1 9.8 Tf [(form is involved and that when displayed graphically, a curvilinear outcome is drawn. Accordingly, what May is getting at with )] TJ ET
BT 26.250 141.908 Td /F1 9.8 Tf [(the complicatedness in his papers title, is not this implication of nonlinearity. Rather, it is whats been called the folded )] TJ ET
BT 26.250 130.003 Td /F1 9.8 Tf [(nonlinearity of the logistic maps operation which results in the startling results of bifurcation \(even to the point of chaos )] TJ ET
BT 26.250 118.098 Td /F1 9.8 Tf [(emerging from what appears so simple\). The logistic map as a relatively simple one-variable/one-parameter difference equation )] TJ ET
BT 26.250 106.194 Td /F1 9.8 Tf [(exhibiting bifurcation is an apt way to represent jumps in variable values in a host of different fields besides population growth, )] TJ ET
BT 26.250 94.289 Td /F1 9.8 Tf [(for instance, in epidemiology the fraction of population infected with a pathogen; in economics the relation between the quantity )] TJ ET
BT 26.250 82.384 Td /F1 9.8 Tf [(of a commodity and its price as well as in theories of business cycles; in learning theory how many bits of information can be )] TJ ET
BT 26.250 70.479 Td /F1 9.8 Tf [(retained in some time period; in social science, the propagation of rumors in differently configured social systems.)] TJ ET
BT 26.250 51.075 Td /F1 9.8 Tf [(In portraying the function graphically for the purpose of representing nontrivial values of the population and thus the )] TJ ET
BT 26.250 39.170 Td /F1 9.8 Tf [(complicated/complex dynamics hidden within it, )] TJ ET
BT 242.983 39.170 Td /F6 9.8 Tf [(P)] TJ ET
BT 249.486 39.170 Td /F6 8.7 Tf [(t)] TJ ET
BT 252.372 39.170 Td /F1 9.8 Tf [(is plotted against )] TJ ET
BT 328.247 39.170 Td /F6 9.8 Tf [(P)] TJ ET
BT 334.750 39.170 Td /F6 8.7 Tf [(t)] TJ ET
BT 337.636 37.106 Td /F4 8.7 Tf [(+1)] TJ ET
BT 347.516 39.170 Td /F1 9.8 Tf [(only when the parameter )] TJ ET
BT 457.525 39.170 Td /F6 9.8 Tf [(a)] TJ ET
BT 462.946 39.170 Td /F1 9.8 Tf [( is less than 4 which )] TJ ET
Q
q
15.000 24.884 577.500 752.116 re W n
0.271 0.267 0.267 rg
BT 26.250 767.476 Td /F1 9.8 Tf [(objects being studied. Mathematics can also be expounded by recognizing and probing the imagistic and pictorial associations )] TJ ET
BT 26.250 755.571 Td /F1 9.8 Tf [(of equations which employ figurative notations and codes of the processes, operations, parameters, and other means, that is, )] TJ ET
BT 26.250 743.667 Td /F1 9.8 Tf [(a kind of metaphoric short-hand that delineates relationships among different mathematical objects)] TJ ET
0.267 0.267 0.267 rg
BT 451.652 745.174 Td /F4 8.7 Tf [(3)] TJ ET
0.271 0.267 0.267 rg
BT 456.471 743.667 Td /F1 9.8 Tf [(.)] TJ ET
BT 26.250 724.262 Td /F1 9.8 Tf [(By invoking the notion of a picture or image, I am not suggesting a dumbing down of the math. That this is not the case will be )] TJ ET
BT 26.250 712.357 Td /F1 9.8 Tf [(obvious when we encounter: first, the mathematical images of the logistic map which, as a nonlinear difference equation, )] TJ ET
BT 26.250 700.452 Td /F1 9.8 Tf [(portrays the changes in the values of variable\(s\) at discrete time intervals; second, , bifurcation diagrams showing the critical )] TJ ET
BT 26.250 688.548 Td /F1 9.8 Tf [(points of the bifurcation or control variables when attractors; third, parameter values ratio equations \(e.g., revealing the amazing )] TJ ET
BT 26.250 676.643 Td /F1 9.8 Tf [(constant Feigenbaum discovered\);, and so on. These specific thematic images follow from the fact that the centerpiece of both )] TJ ET
BT 26.250 664.738 Td /F1 9.8 Tf [(Mays and Feigenbaums inquiries is the sequence of bifurcations that occur during the operation of functional iteration in the )] TJ ET
BT 26.250 652.833 Td /F1 9.8 Tf [(now notorious logistic map which functions as the apotheosis of thematic imagery in their explorations of chaos. Without )] TJ ET
BT 26.250 640.929 Td /F1 9.8 Tf [(needing to dumb it down, however, a few preliminary remarks on what May explicitly focused on mathematically can be helpful.)] TJ ET
BT 26.250 604.326 Td /F4 12.0 Tf [(Population dynamics and the logistic map)] TJ ET
BT 26.250 584.372 Td /F1 9.8 Tf [(The author of the first classic paper, Robert May, stands out as a true polymath. An Australian trained as a theoretical physicist, )] TJ ET
BT 26.250 572.467 Td /F1 9.8 Tf [(serving stints at premier universities like Princeton, Harvard, Oxford \(only the beginning of a long list\); he has made )] TJ ET
BT 26.250 560.562 Td /F1 9.8 Tf [(groundbreaking discoveries in theoretical ecology and biology, population dynamics, pure and applied mathematics and many )] TJ ET
BT 26.250 548.658 Td /F1 9.8 Tf [(other fields, being knighted along the way for his persistently outstanding work.)] TJ ET
BT 26.250 529.253 Td /F1 9.8 Tf [(In his forays in population dynamics and theoretical ecology, May inevitably came into contact with the so-called logistic map, a )] TJ ET
BT 26.250 517.348 Td /F1 9.8 Tf [(type of discrete \(a difference not a differential\) function fashioned in order to capture certain features of how populations )] TJ ET
BT 26.250 505.443 Td /F1 9.8 Tf [(change, for instance, the manner by which a predator/prey model effectuates an increase, then decrease of population:)] TJ ET
0.965 0.965 0.965 rg
26.250 415.694 555.000 79.869 re f
0.267 0.267 0.267 rg
0.267 0.267 0.267 RG
26.250 495.562 m 581.250 495.562 l 581.250 494.812 l 26.250 494.812 l f
26.250 415.694 m 581.250 415.694 l 581.250 416.444 l 26.250 416.444 l f
0.271 0.267 0.267 rg
0.271 0.267 0.267 RG
49.337 479.256 m
49.337 479.703 49.154 480.147 48.838 480.463 c
48.522 480.779 48.078 480.962 47.631 480.962 c
47.185 480.962 46.741 480.779 46.425 480.463 c
46.109 480.147 45.925 479.703 45.925 479.256 c
45.925 478.810 46.109 478.366 46.425 478.050 c
46.741 477.734 47.185 477.550 47.631 477.550 c
48.078 477.550 48.522 477.734 48.838 478.050 c
49.154 478.366 49.337 478.810 49.337 479.256 c f
BT 54.750 476.289 Td /F1 9.8 Tf [(First, the number of predators increases since the amount of prey is high because they have not as yet been eaten;)] TJ ET
49.337 463.601 m
49.337 464.048 49.154 464.492 48.838 464.808 c
48.522 465.124 48.078 465.308 47.631 465.308 c
47.185 465.308 46.741 465.124 46.425 464.808 c
46.109 464.492 45.925 464.048 45.925 463.601 c
45.925 463.155 46.109 462.711 46.425 462.395 c
46.741 462.079 47.185 461.895 47.631 461.895 c
48.078 461.895 48.522 462.079 48.838 462.395 c
49.154 462.711 49.337 463.155 49.337 463.601 c f
BT 54.750 460.634 Td /F1 9.8 Tf [(After a peak in the population of predators, there are less prey to bear consumed resulting to a decline in the number )] TJ ET
BT 560.385 460.634 Td /F1 9.8 Tf [(of )] TJ ET
BT 54.750 448.729 Td /F1 9.8 Tf [(predators \(they starve to death\);)] TJ ET
49.337 436.042 m
49.337 436.489 49.154 436.933 48.838 437.249 c
48.522 437.564 48.078 437.748 47.631 437.748 c
47.185 437.748 46.741 437.564 46.425 437.249 c
46.109 436.933 45.925 436.489 45.925 436.042 c
45.925 435.595 46.109 435.151 46.425 434.835 c
46.741 434.520 47.185 434.336 47.631 434.336 c
48.078 434.336 48.522 434.520 48.838 434.835 c
49.154 435.151 49.337 435.595 49.337 436.042 c f
BT 54.750 433.074 Td /F1 9.8 Tf [(As the population of predators decline, the population of prey rises since there are less predators to eat them.)] TJ ET
BT 26.250 398.670 Td /F1 9.8 Tf [(It is useful to keep this parabolic \(when graphed\) dynamic in mind since the logistic map \(see Eqn. 1 below\), to which May and )] TJ ET
BT 26.250 386.765 Td /F1 9.8 Tf [(Feigenbaum devote so much of their attention, displays this rise and fall in terms of discrete changes in the population, e.g., )] TJ ET
BT 26.250 374.860 Td /F1 9.8 Tf [(from one year to the next. The logistic map discretizes the population variable by repeating or iterating the functional operation )] TJ ET
BT 26.250 362.955 Td /F1 9.8 Tf [(at each temporal interval of time \(stipulated such as weekly, or monthly, or yearly\). This repetition is called functional iteration )] TJ ET
BT 26.250 351.051 Td /F1 9.8 Tf [(since each newly-arrived-at value of the variable is repeatedly pumped-back into the function and the functional operation is )] TJ ET
BT 26.250 339.146 Td /F1 9.8 Tf [(iterated to arrive at the next value.)] TJ ET
BT 26.250 319.741 Td /F1 9.8 Tf [(Here is the logistic map in the usual form found in population dynamics:)] TJ ET
BT 26.250 296.660 Td /F1 9.8 Tf [(Eqn 1. )] TJ ET
q
111.000 0 0 16.500 57.148 293.360 cm /I5 Do
Q
BT 26.250 276.336 Td /F6 9.8 Tf [(P)] TJ ET
BT 32.753 274.272 Td /F6 8.7 Tf [(t)] TJ ET
BT 35.639 276.336 Td /F1 9.8 Tf [( is the population at some arbitrarily chosen initial year )] TJ ET
BT 272.983 276.336 Td /F6 9.8 Tf [(t)] TJ ET
BT 276.230 276.336 Td /F1 9.8 Tf [(, )] TJ ET
BT 284.362 276.336 Td /F6 9.8 Tf [(P)] TJ ET
BT 290.865 276.336 Td /F6 8.7 Tf [(t)] TJ ET
BT 293.751 274.272 Td /F4 8.7 Tf [(+1)] TJ ET
BT 303.631 276.336 Td /F1 9.8 Tf [(is the population at the next year )] TJ ET
BT 449.969 276.336 Td /F6 9.8 Tf [(t)] TJ ET
BT 453.216 276.336 Td /F4 9.8 Tf [(+1)] TJ ET
BT 464.331 276.336 Td /F1 9.8 Tf [(, and)] TJ ET
BT 486.015 276.336 Td /F6 9.8 Tf [( a)] TJ ET
BT 494.146 276.336 Td /F1 9.8 Tf [( is the birth/death )] TJ ET
BT 26.250 264.432 Td /F1 9.8 Tf [(rate \(after suitable adjustments such as setting at a constant some such factor comprising the system/environment, e.g., )] TJ ET
BT 26.250 252.527 Td /F1 9.8 Tf [(available space, food, etc.\) This Eqn. 1a is nonlinear because the variable )] TJ ET
BT 347.620 252.527 Td /F6 9.8 Tf [(P)] TJ ET
BT 354.123 250.463 Td /F6 8.7 Tf [(t)] TJ ET
BT 357.009 252.527 Td /F1 9.8 Tf [( is multiplied by itself:)] TJ ET
BT 26.250 204.246 Td /F1 9.8 Tf [(Eqn 1a. )] TJ ET
q
48.000 0 0 48.000 62.569 194.646 cm /I3 Do
Q
0.500 0.500 0.500 rg
BT 62.569 235.246 Td /F1 8.0 Tf [(Image not found)] TJ ET
BT 62.569 225.246 Td /F1 8.0 Tf [(http://bit.ly/1JraLBy)] TJ ET
0.271 0.267 0.267 rg
BT 26.250 177.622 Td /F1 9.8 Tf [(It is also useful to keep in mind that although a big deal is usually made about the nonlinearity of the nonlinear dynamical )] TJ ET
BT 26.250 165.717 Td /F1 9.8 Tf [(systems aspect of complexity science, by itself, this doesnt imply much more than the simple fact that multiplication in some )] TJ ET
BT 26.250 153.813 Td /F1 9.8 Tf [(form is involved and that when displayed graphically, a curvilinear outcome is drawn. Accordingly, what May is getting at with )] TJ ET
BT 26.250 141.908 Td /F1 9.8 Tf [(the complicatedness in his papers title, is not this implication of nonlinearity. Rather, it is whats been called the folded )] TJ ET
BT 26.250 130.003 Td /F1 9.8 Tf [(nonlinearity of the logistic maps operation which results in the startling results of bifurcation \(even to the point of chaos )] TJ ET
BT 26.250 118.098 Td /F1 9.8 Tf [(emerging from what appears so simple\). The logistic map as a relatively simple one-variable/one-parameter difference equation )] TJ ET
BT 26.250 106.194 Td /F1 9.8 Tf [(exhibiting bifurcation is an apt way to represent jumps in variable values in a host of different fields besides population growth, )] TJ ET
BT 26.250 94.289 Td /F1 9.8 Tf [(for instance, in epidemiology the fraction of population infected with a pathogen; in economics the relation between the quantity )] TJ ET
BT 26.250 82.384 Td /F1 9.8 Tf [(of a commodity and its price as well as in theories of business cycles; in learning theory how many bits of information can be )] TJ ET
BT 26.250 70.479 Td /F1 9.8 Tf [(retained in some time period; in social science, the propagation of rumors in differently configured social systems.)] TJ ET
BT 26.250 51.075 Td /F1 9.8 Tf [(In portraying the function graphically for the purpose of representing nontrivial values of the population and thus the )] TJ ET
BT 26.250 39.170 Td /F1 9.8 Tf [(complicated/complex dynamics hidden within it, )] TJ ET
BT 242.983 39.170 Td /F6 9.8 Tf [(P)] TJ ET
BT 249.486 39.170 Td /F6 8.7 Tf [(t)] TJ ET
BT 252.372 39.170 Td /F1 9.8 Tf [(is plotted against )] TJ ET
BT 328.247 39.170 Td /F6 9.8 Tf [(P)] TJ ET
BT 334.750 39.170 Td /F6 8.7 Tf [(t)] TJ ET
BT 337.636 37.106 Td /F4 8.7 Tf [(+1)] TJ ET
BT 347.516 39.170 Td /F1 9.8 Tf [(only when the parameter )] TJ ET
BT 457.525 39.170 Td /F6 9.8 Tf [(a)] TJ ET
BT 462.946 39.170 Td /F1 9.8 Tf [( is less than 4 which )] TJ ET
Q
q
15.000 24.884 577.500 752.116 re W n
0.271 0.267 0.267 rg
BT 26.250 767.476 Td /F1 9.8 Tf [(objects being studied. Mathematics can also be expounded by recognizing and probing the imagistic and pictorial associations )] TJ ET
BT 26.250 755.571 Td /F1 9.8 Tf [(of equations which employ figurative notations and codes of the processes, operations, parameters, and other means, that is, )] TJ ET
BT 26.250 743.667 Td /F1 9.8 Tf [(a kind of metaphoric short-hand that delineates relationships among different mathematical objects)] TJ ET
0.267 0.267 0.267 rg
BT 451.652 745.174 Td /F4 8.7 Tf [(3)] TJ ET
0.271 0.267 0.267 rg
BT 456.471 743.667 Td /F1 9.8 Tf [(.)] TJ ET
BT 26.250 724.262 Td /F1 9.8 Tf [(By invoking the notion of a picture or image, I am not suggesting a dumbing down of the math. That this is not the case will be )] TJ ET
BT 26.250 712.357 Td /F1 9.8 Tf [(obvious when we encounter: first, the mathematical images of the logistic map which, as a nonlinear difference equation, )] TJ ET
BT 26.250 700.452 Td /F1 9.8 Tf [(portrays the changes in the values of variable\(s\) at discrete time intervals; second, , bifurcation diagrams showing the critical )] TJ ET
BT 26.250 688.548 Td /F1 9.8 Tf [(points of the bifurcation or control variables when attractors; third, parameter values ratio equations \(e.g., revealing the amazing )] TJ ET
BT 26.250 676.643 Td /F1 9.8 Tf [(constant Feigenbaum discovered\);, and so on. These specific thematic images follow from the fact that the centerpiece of both )] TJ ET
BT 26.250 664.738 Td /F1 9.8 Tf [(Mays and Feigenbaums inquiries is the sequence of bifurcations that occur during the operation of functional iteration in the )] TJ ET
BT 26.250 652.833 Td /F1 9.8 Tf [(now notorious logistic map which functions as the apotheosis of thematic imagery in their explorations of chaos. Without )] TJ ET
BT 26.250 640.929 Td /F1 9.8 Tf [(needing to dumb it down, however, a few preliminary remarks on what May explicitly focused on mathematically can be helpful.)] TJ ET
BT 26.250 604.326 Td /F4 12.0 Tf [(Population dynamics and the logistic map)] TJ ET
BT 26.250 584.372 Td /F1 9.8 Tf [(The author of the first classic paper, Robert May, stands out as a true polymath. An Australian trained as a theoretical physicist, )] TJ ET
BT 26.250 572.467 Td /F1 9.8 Tf [(serving stints at premier universities like Princeton, Harvard, Oxford \(only the beginning of a long list\); he has made )] TJ ET
BT 26.250 560.562 Td /F1 9.8 Tf [(groundbreaking discoveries in theoretical ecology and biology, population dynamics, pure and applied mathematics and many )] TJ ET
BT 26.250 548.658 Td /F1 9.8 Tf [(other fields, being knighted along the way for his persistently outstanding work.)] TJ ET
BT 26.250 529.253 Td /F1 9.8 Tf [(In his forays in population dynamics and theoretical ecology, May inevitably came into contact with the so-called logistic map, a )] TJ ET
BT 26.250 517.348 Td /F1 9.8 Tf [(type of discrete \(a difference not a differential\) function fashioned in order to capture certain features of how populations )] TJ ET
BT 26.250 505.443 Td /F1 9.8 Tf [(change, for instance, the manner by which a predator/prey model effectuates an increase, then decrease of population:)] TJ ET
0.965 0.965 0.965 rg
26.250 415.694 555.000 79.869 re f
0.267 0.267 0.267 rg
0.267 0.267 0.267 RG
26.250 495.562 m 581.250 495.562 l 581.250 494.812 l 26.250 494.812 l f
26.250 415.694 m 581.250 415.694 l 581.250 416.444 l 26.250 416.444 l f
0.271 0.267 0.267 rg
0.271 0.267 0.267 RG
49.337 479.256 m
49.337 479.703 49.154 480.147 48.838 480.463 c
48.522 480.779 48.078 480.962 47.631 480.962 c
47.185 480.962 46.741 480.779 46.425 480.463 c
46.109 480.147 45.925 479.703 45.925 479.256 c
45.925 478.810 46.109 478.366 46.425 478.050 c
46.741 477.734 47.185 477.550 47.631 477.550 c
48.078 477.550 48.522 477.734 48.838 478.050 c
49.154 478.366 49.337 478.810 49.337 479.256 c f
BT 54.750 476.289 Td /F1 9.8 Tf [(First, the number of predators increases since the amount of prey is high because they have not as yet been eaten;)] TJ ET
49.337 463.601 m
49.337 464.048 49.154 464.492 48.838 464.808 c
48.522 465.124 48.078 465.308 47.631 465.308 c
47.185 465.308 46.741 465.124 46.425 464.808 c
46.109 464.492 45.925 464.048 45.925 463.601 c
45.925 463.155 46.109 462.711 46.425 462.395 c
46.741 462.079 47.185 461.895 47.631 461.895 c
48.078 461.895 48.522 462.079 48.838 462.395 c
49.154 462.711 49.337 463.155 49.337 463.601 c f
BT 54.750 460.634 Td /F1 9.8 Tf [(After a peak in the population of predators, there are less prey to bear consumed resulting to a decline in the number )] TJ ET
BT 560.385 460.634 Td /F1 9.8 Tf [(of )] TJ ET
BT 54.750 448.729 Td /F1 9.8 Tf [(predators \(they starve to death\);)] TJ ET
49.337 436.042 m
49.337 436.489 49.154 436.933 48.838 437.249 c
48.522 437.564 48.078 437.748 47.631 437.748 c
47.185 437.748 46.741 437.564 46.425 437.249 c
46.109 436.933 45.925 436.489 45.925 436.042 c
45.925 435.595 46.109 435.151 46.425 434.835 c
46.741 434.520 47.185 434.336 47.631 434.336 c
48.078 434.336 48.522 434.520 48.838 434.835 c
49.154 435.151 49.337 435.595 49.337 436.042 c f
BT 54.750 433.074 Td /F1 9.8 Tf [(As the population of predators decline, the population of prey rises since there are less predators to eat them.)] TJ ET
BT 26.250 398.670 Td /F1 9.8 Tf [(It is useful to keep this parabolic \(when graphed\) dynamic in mind since the logistic map \(see Eqn. 1 below\), to which May and )] TJ ET
BT 26.250 386.765 Td /F1 9.8 Tf [(Feigenbaum devote so much of their attention, displays this rise and fall in terms of discrete changes in the population, e.g., )] TJ ET
BT 26.250 374.860 Td /F1 9.8 Tf [(from one year to the next. The logistic map discretizes the population variable by repeating or iterating the functional operation )] TJ ET
BT 26.250 362.955 Td /F1 9.8 Tf [(at each temporal interval of time \(stipulated such as weekly, or monthly, or yearly\). This repetition is called functional iteration )] TJ ET
BT 26.250 351.051 Td /F1 9.8 Tf [(since each newly-arrived-at value of the variable is repeatedly pumped-back into the function and the functional operation is )] TJ ET
BT 26.250 339.146 Td /F1 9.8 Tf [(iterated to arrive at the next value.)] TJ ET
BT 26.250 319.741 Td /F1 9.8 Tf [(Here is the logistic map in the usual form found in population dynamics:)] TJ ET
BT 26.250 296.660 Td /F1 9.8 Tf [(Eqn 1. )] TJ ET
q
111.000 0 0 16.500 57.148 293.360 cm /I7 Do
Q
BT 26.250 276.336 Td /F6 9.8 Tf [(P)] TJ ET
BT 32.753 274.272 Td /F6 8.7 Tf [(t)] TJ ET
BT 35.639 276.336 Td /F1 9.8 Tf [( is the population at some arbitrarily chosen initial year )] TJ ET
BT 272.983 276.336 Td /F6 9.8 Tf [(t)] TJ ET
BT 276.230 276.336 Td /F1 9.8 Tf [(, )] TJ ET
BT 284.362 276.336 Td /F6 9.8 Tf [(P)] TJ ET
BT 290.865 276.336 Td /F6 8.7 Tf [(t)] TJ ET
BT 293.751 274.272 Td /F4 8.7 Tf [(+1)] TJ ET
BT 303.631 276.336 Td /F1 9.8 Tf [(is the population at the next year )] TJ ET
BT 449.969 276.336 Td /F6 9.8 Tf [(t)] TJ ET
BT 453.216 276.336 Td /F4 9.8 Tf [(+1)] TJ ET
BT 464.331 276.336 Td /F1 9.8 Tf [(, and)] TJ ET
BT 486.015 276.336 Td /F6 9.8 Tf [( a)] TJ ET
BT 494.146 276.336 Td /F1 9.8 Tf [( is the birth/death )] TJ ET
BT 26.250 264.432 Td /F1 9.8 Tf [(rate \(after suitable adjustments such as setting at a constant some such factor comprising the system/environment, e.g., )] TJ ET
BT 26.250 252.527 Td /F1 9.8 Tf [(available space, food, etc.\) This Eqn. 1a is nonlinear because the variable )] TJ ET
BT 347.620 252.527 Td /F6 9.8 Tf [(P)] TJ ET
BT 354.123 250.463 Td /F6 8.7 Tf [(t)] TJ ET
BT 357.009 252.527 Td /F1 9.8 Tf [( is multiplied by itself:)] TJ ET
BT 26.250 204.246 Td /F1 9.8 Tf [(Eqn 1a. )] TJ ET
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BT 62.569 235.246 Td /F1 8.0 Tf [(Image not found)] TJ ET
BT 62.569 225.246 Td /F1 8.0 Tf [(http://bit.ly/1JraLBy)] TJ ET
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BT 26.250 177.622 Td /F1 9.8 Tf [(It is also useful to keep in mind that although a big deal is usually made about the nonlinearity of the nonlinear dynamical )] TJ ET
BT 26.250 165.717 Td /F1 9.8 Tf [(systems aspect of complexity science, by itself, this doesnt imply much more than the simple fact that multiplication in some )] TJ ET
BT 26.250 153.813 Td /F1 9.8 Tf [(form is involved and that when displayed graphically, a curvilinear outcome is drawn. Accordingly, what May is getting at with )] TJ ET
BT 26.250 141.908 Td /F1 9.8 Tf [(the complicatedness in his papers title, is not this implication of nonlinearity. Rather, it is whats been called the folded )] TJ ET
BT 26.250 130.003 Td /F1 9.8 Tf [(nonlinearity of the logistic maps operation which results in the startling results of bifurcation \(even to the point of chaos )] TJ ET
BT 26.250 118.098 Td /F1 9.8 Tf [(emerging from what appears so simple\). The logistic map as a relatively simple one-variable/one-parameter difference equation )] TJ ET
BT 26.250 106.194 Td /F1 9.8 Tf [(exhibiting bifurcation is an apt way to represent jumps in variable values in a host of different fields besides population growth, )] TJ ET
BT 26.250 94.289 Td /F1 9.8 Tf [(for instance, in epidemiology the fraction of population infected with a pathogen; in economics the relation between the quantity )] TJ ET
BT 26.250 82.384 Td /F1 9.8 Tf [(of a commodity and its price as well as in theories of business cycles; in learning theory how many bits of information can be )] TJ ET
BT 26.250 70.479 Td /F1 9.8 Tf [(retained in some time period; in social science, the propagation of rumors in differently configured social systems.)] TJ ET
BT 26.250 51.075 Td /F1 9.8 Tf [(In portraying the function graphically for the purpose of representing nontrivial values of the population and thus the )] TJ ET
BT 26.250 39.170 Td /F1 9.8 Tf [(complicated/complex dynamics hidden within it, )] TJ ET
BT 242.983 39.170 Td /F6 9.8 Tf [(P)] TJ ET
BT 249.486 39.170 Td /F6 8.7 Tf [(t)] TJ ET
BT 252.372 39.170 Td /F1 9.8 Tf [(is plotted against )] TJ ET
BT 328.247 39.170 Td /F6 9.8 Tf [(P)] TJ ET
BT 334.750 39.170 Td /F6 8.7 Tf [(t)] TJ ET
BT 337.636 37.106 Td /F4 8.7 Tf [(+1)] TJ ET
BT 347.516 39.170 Td /F1 9.8 Tf [(only when the parameter )] TJ ET
BT 457.525 39.170 Td /F6 9.8 Tf [(a)] TJ ET
BT 462.946 39.170 Td /F1 9.8 Tf [( is less than 4 which )] TJ ET
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BT 291.710 19.825 Td /F1 11.0 Tf [(2)] TJ ET
BT 25.000 19.825 Td /F1 11.0 Tf [(Emergence: Complexity and Organization)] TJ ET
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BT 26.250 767.476 Td /F1 9.8 Tf [(keeps the population from exploding beyond any sensible interpretation. The plot is called a cobweb diagram since the way the )] TJ ET
BT 26.250 755.571 Td /F1 9.8 Tf [(solutions change discretely looks like a cobweb \(see the parabolic curves in Mays paper, Figure 1 on p. 2\). The birth/death )] TJ ET
BT 26.250 743.667 Td /F1 9.8 Tf [(parameter )] TJ ET
BT 73.391 743.667 Td /F6 9.8 Tf [(a)] TJ ET
BT 78.812 743.667 Td /F1 9.8 Tf [( shows up as the slope of the tangent line, when )] TJ ET
BT 289.646 743.667 Td /F6 9.8 Tf [(P)] TJ ET
BT 296.149 741.602 Td /F6 8.7 Tf [(t)] TJ ET
BT 299.036 743.667 Td /F1 9.8 Tf [(= )] TJ ET
BT 307.440 743.667 Td /F6 9.8 Tf [(P)] TJ ET
BT 313.943 743.667 Td /F6 8.7 Tf [(t)] TJ ET
BT 316.829 743.667 Td /F4 8.7 Tf [(+1)] TJ ET
BT 329.119 743.667 Td /F4 9.8 Tf [(.)] TJ ET
BT 26.250 724.262 Td /F1 9.8 Tf [(I remember from math courses that such parabolic equations were studied in calculus but usually in the form of differential )] TJ ET
BT 26.250 712.357 Td /F1 9.8 Tf [(equations that were not particularly interesting nor revealing but instead consisted of just finding solutions, graphically displaying )] TJ ET
BT 26.250 700.452 Td /F1 9.8 Tf [(the equations, the mechanisms of differentiation, and so forth \(unless we were fortunate enough to have taken classes in )] TJ ET
BT 26.250 688.548 Td /F1 9.8 Tf [(dynamical systems but they were few at that time, for undergraduates at least\). What May discovered and labeled complicated )] TJ ET
BT 26.250 676.643 Td /F1 9.8 Tf [(dynamics in the title of his classic paper was in radical contrast to what most of us encountered in calculus. His findings were )] TJ ET
BT 26.250 664.738 Td /F1 9.8 Tf [(due not to the usual exploration of changes of the values of the population but instead arose from playing around with the )] TJ ET
BT 26.250 652.833 Td /F1 9.8 Tf [(values of the parameter )] TJ ET
BT 131.384 652.833 Td /F4 9.8 Tf [(a)] TJ ET
BT 136.805 652.833 Td /F1 9.8 Tf [(. He refers to this manipulation of the parameter )] TJ ET
BT 345.982 652.833 Td /F6 9.8 Tf [(a)] TJ ET
BT 351.403 652.833 Td /F1 9.8 Tf [( tuning the nonlinearity of the logistic map since )] TJ ET
BT 26.250 640.929 Td /F1 9.8 Tf [(the steepness of the tangent line indicates how nonlinear, i.e., how complicated \(in his terminology\) things are going to get )] TJ ET
BT 26.250 629.024 Td /F1 9.8 Tf [(when bifurcation occurs.)] TJ ET
BT 26.250 609.619 Td /F1 9.8 Tf [(By tuning the nonlinearity of the logistic map, May found, against expectations, that as the values of the parameter )] TJ ET
BT 530.793 609.619 Td /F6 9.8 Tf [(a)] TJ ET
BT 26.250 597.714 Td /F1 9.8 Tf [(increased, the values of )] TJ ET
BT 134.641 597.714 Td /F6 9.8 Tf [(P)] TJ ET
BT 141.144 597.714 Td /F6 8.7 Tf [(t)] TJ ET
BT 144.030 595.650 Td /F4 8.7 Tf [(+1)] TJ ET
BT 153.910 597.714 Td /F1 9.8 Tf [(did not consistently change in the usual manner but instead could get trapped at some particular )] TJ ET
BT 26.250 585.810 Td /F1 9.8 Tf [(value\(s\) or circumscribed set of values and stay there no matter how many times the iteration involved took place, or they could )] TJ ET
BT 26.250 573.905 Td /F1 9.8 Tf [(in contrast display sudden jumps in values These trapped and jumped values, as May found out, hinged on the values of the )] TJ ET
BT 26.250 562.000 Td /F1 9.8 Tf [(parameters and whether they kept the system inside an attractor or instead prompted a bifurcation or jump out of the attractor. )] TJ ET
BT 26.250 550.095 Td /F1 9.8 Tf [(are constrained by attractors because they metaphorically attract the values of the variables in the long-run. It is only when )] TJ ET
BT 26.250 538.191 Td /F1 9.8 Tf [(the parameter values reach a critical threshold that they can prompt bifurcation or the emergence or disappearance of new )] TJ ET
BT 26.250 526.286 Td /F1 9.8 Tf [(attractors.)] TJ ET
BT 26.250 489.683 Td /F4 12.0 Tf [(Complexity from simplicity?)] TJ ET
BT 26.250 469.729 Td /F1 9.8 Tf [(It is generally thought that one of the most significant findings stemming from chaos theory is the idea that highly complex )] TJ ET
BT 26.250 457.824 Td /F1 9.8 Tf [(outcomes can emerge from very simple beginnings, a concept finding expression in the title of Mays classic paper and in the )] TJ ET
BT 26.250 445.920 Td /F1 9.8 Tf [(notion of deterministic chaos \(in the sense that whatever is determined by a causal linkagefunctional iteration consisting of )] TJ ET
BT 26.250 434.015 Td /F1 9.8 Tf [(such a causal linkageshould only generate results proportionate to whatever rudimentary elements are at work in the )] TJ ET
BT 26.250 422.110 Td /F1 9.8 Tf [(causation\). Yet, this is not what happens. Rather, deterministic chaos brings forth something as complex as a random seeming )] TJ ET
BT 26.250 410.205 Td /F1 9.8 Tf [(pattern, i.e., chaos. In Mays terminology, functional iteration engenders this complicatedness out of simple although )] TJ ET
BT 26.250 398.301 Td /F1 9.8 Tf [(complicated for May was not so much chaos per se as the operation of functional iteration, indeed, the entire bifurcation )] TJ ET
BT 26.250 386.396 Td /F1 9.8 Tf [(scenario generated by changes in the bifurcation parameter. Should we then, not attempt to understand what May )] TJ ET
BT 26.250 374.491 Td /F1 9.8 Tf [(accomplished as falling in line behind the slogan of complexity from simplicity.)] TJ ET
BT 26.250 355.086 Td /F1 9.8 Tf [(It is not only in chaos theory and its associated fields that this implication has found a foothold. We can hear it in other )] TJ ET
BT 26.250 343.182 Td /F1 9.8 Tf [(complexity science branches as well. For instance, it is very frequently heard in emergentist circles that emergence, in many of )] TJ ET
BT 26.250 331.277 Td /F1 9.8 Tf [(its different contexts, can only occur when a certain threshold of complexity \(this threshold measured by different metrics )] TJ ET
BT 26.250 319.372 Td /F1 9.8 Tf [(according to what kind of emergence is being queried\). Perhaps the most extreme, recent statement of such a viewpoint is the )] TJ ET
BT 26.250 307.467 Td /F1 9.8 Tf [(claim that consciousness is an emergent phenomena that can only come about when evolution has produced some sufficiently )] TJ ET
BT 26.250 295.563 Td /F1 9.8 Tf [(critical level of complexity, e.g., some supposed metric of neural connectivity or neural networking \(see a very recent version of )] TJ ET
BT 26.250 283.658 Td /F1 9.8 Tf [(this thesis formulated in terms of integrated information in the work of Giulio Tononi and Cristof Koch)] TJ ET
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BT 465.712 285.165 Td /F4 8.7 Tf [(4)] TJ ET
0.271 0.267 0.267 rg
BT 470.530 283.658 Td /F1 9.8 Tf [(\).)] TJ ET
BT 26.250 264.253 Td /F1 9.8 Tf [(The two classic papers being republished in )] TJ ET
BT 218.081 264.253 Td /F5 9.8 Tf [(E:CO)] TJ ET
BT 241.920 264.253 Td /F1 9.8 Tf [( should be of aid for us only only in grasping how functional iteration in )] TJ ET
BT 26.250 252.348 Td /F1 9.8 Tf [(nonlinear maps takes placethey should also help give the lie to any such conclusions of complexity coming from simplicity. )] TJ ET
BT 26.250 240.444 Td /F1 9.8 Tf [(That is, these two classic papers can be read as showing how the logistic map and similar complexity images in actuality )] TJ ET
BT 26.250 228.539 Td /F1 9.8 Tf [(contain a huge reservoir of complex dynamics that is made manifest under the right conditions. In other words, what May and )] TJ ET
BT 26.250 216.634 Td /F1 9.8 Tf [(Feigenbaum have done is to further the project of discerning how complex patterns emerge from antecedent patterns or as )] TJ ET
BT 26.250 204.729 Td /F1 9.8 Tf [(Turing described it in the arising of novel morphogenic patterns: developing from one pattern into another, rather than from )] TJ ET
BT 26.250 192.825 Td /F1 9.8 Tf [(homogeneity into a pattern.)] TJ ET
0.267 0.267 0.267 rg
BT 147.647 194.332 Td /F4 8.7 Tf [(5)] TJ ET
0.271 0.267 0.267 rg
BT 26.250 173.420 Td /F1 9.8 Tf [(This may seem like a semantical issue but I am contending it is much deeper since the popularization of the sloganeering of )] TJ ET
BT 26.250 161.515 Td /F1 9.8 Tf [(complexity coming out of simplicity can get in the way of appreciating just how much complexity, complicatedness, )] TJ ET
BT 26.250 149.610 Td /F1 9.8 Tf [(complexification are going on already or have been going on. For instance, consider functional iteration: it is fundamentally a )] TJ ET
BT 26.250 137.706 Td /F1 9.8 Tf [(complexifying operation in the sense likened to kneading dough to make bread, an operation that aims at optimal mixing of )] TJ ET
BT 26.250 125.801 Td /F1 9.8 Tf [(ingredients, especially diffusing yeast throughout the flour where the yeast can do its work. This can be seen vividly in the )] TJ ET
BT 26.250 113.896 Td /F1 9.8 Tf [(closely related horseshoe map of Steve Smale which illustrates the folding nonlinearity of functional iteration by way of a )] TJ ET
BT 26.250 101.991 Td /F1 9.8 Tf [(horseshoe which is elongated, then folded over, the result being elongated and then folded over again and so on.)] TJ ET
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BT 26.250 767.476 Td /F1 9.8 Tf [(keeps the population from exploding beyond any sensible interpretation. The plot is called a cobweb diagram since the way the )] TJ ET
BT 26.250 755.571 Td /F1 9.8 Tf [(solutions change discretely looks like a cobweb \(see the parabolic curves in Mays paper, Figure 1 on p. 2\). The birth/death )] TJ ET
BT 26.250 743.667 Td /F1 9.8 Tf [(parameter )] TJ ET
BT 73.391 743.667 Td /F6 9.8 Tf [(a)] TJ ET
BT 78.812 743.667 Td /F1 9.8 Tf [( shows up as the slope of the tangent line, when )] TJ ET
BT 289.646 743.667 Td /F6 9.8 Tf [(P)] TJ ET
BT 296.149 741.602 Td /F6 8.7 Tf [(t)] TJ ET
BT 299.036 743.667 Td /F1 9.8 Tf [(= )] TJ ET
BT 307.440 743.667 Td /F6 9.8 Tf [(P)] TJ ET
BT 313.943 743.667 Td /F6 8.7 Tf [(t)] TJ ET
BT 316.829 743.667 Td /F4 8.7 Tf [(+1)] TJ ET
BT 329.119 743.667 Td /F4 9.8 Tf [(.)] TJ ET
BT 26.250 724.262 Td /F1 9.8 Tf [(I remember from math courses that such parabolic equations were studied in calculus but usually in the form of differential )] TJ ET
BT 26.250 712.357 Td /F1 9.8 Tf [(equations that were not particularly interesting nor revealing but instead consisted of just finding solutions, graphically displaying )] TJ ET
BT 26.250 700.452 Td /F1 9.8 Tf [(the equations, the mechanisms of differentiation, and so forth \(unless we were fortunate enough to have taken classes in )] TJ ET
BT 26.250 688.548 Td /F1 9.8 Tf [(dynamical systems but they were few at that time, for undergraduates at least\). What May discovered and labeled complicated )] TJ ET
BT 26.250 676.643 Td /F1 9.8 Tf [(dynamics in the title of his classic paper was in radical contrast to what most of us encountered in calculus. His findings were )] TJ ET
BT 26.250 664.738 Td /F1 9.8 Tf [(due not to the usual exploration of changes of the values of the population but instead arose from playing around with the )] TJ ET
BT 26.250 652.833 Td /F1 9.8 Tf [(values of the parameter )] TJ ET
BT 131.384 652.833 Td /F4 9.8 Tf [(a)] TJ ET
BT 136.805 652.833 Td /F1 9.8 Tf [(. He refers to this manipulation of the parameter )] TJ ET
BT 345.982 652.833 Td /F6 9.8 Tf [(a)] TJ ET
BT 351.403 652.833 Td /F1 9.8 Tf [( tuning the nonlinearity of the logistic map since )] TJ ET
BT 26.250 640.929 Td /F1 9.8 Tf [(the steepness of the tangent line indicates how nonlinear, i.e., how complicated \(in his terminology\) things are going to get )] TJ ET
BT 26.250 629.024 Td /F1 9.8 Tf [(when bifurcation occurs.)] TJ ET
BT 26.250 609.619 Td /F1 9.8 Tf [(By tuning the nonlinearity of the logistic map, May found, against expectations, that as the values of the parameter )] TJ ET
BT 530.793 609.619 Td /F6 9.8 Tf [(a)] TJ ET
BT 26.250 597.714 Td /F1 9.8 Tf [(increased, the values of )] TJ ET
BT 134.641 597.714 Td /F6 9.8 Tf [(P)] TJ ET
BT 141.144 597.714 Td /F6 8.7 Tf [(t)] TJ ET
BT 144.030 595.650 Td /F4 8.7 Tf [(+1)] TJ ET
BT 153.910 597.714 Td /F1 9.8 Tf [(did not consistently change in the usual manner but instead could get trapped at some particular )] TJ ET
BT 26.250 585.810 Td /F1 9.8 Tf [(value\(s\) or circumscribed set of values and stay there no matter how many times the iteration involved took place, or they could )] TJ ET
BT 26.250 573.905 Td /F1 9.8 Tf [(in contrast display sudden jumps in values These trapped and jumped values, as May found out, hinged on the values of the )] TJ ET
BT 26.250 562.000 Td /F1 9.8 Tf [(parameters and whether they kept the system inside an attractor or instead prompted a bifurcation or jump out of the attractor. )] TJ ET
BT 26.250 550.095 Td /F1 9.8 Tf [(are constrained by attractors because they metaphorically attract the values of the variables in the long-run. It is only when )] TJ ET
BT 26.250 538.191 Td /F1 9.8 Tf [(the parameter values reach a critical threshold that they can prompt bifurcation or the emergence or disappearance of new )] TJ ET
BT 26.250 526.286 Td /F1 9.8 Tf [(attractors.)] TJ ET
BT 26.250 489.683 Td /F4 12.0 Tf [(Complexity from simplicity?)] TJ ET
BT 26.250 469.729 Td /F1 9.8 Tf [(It is generally thought that one of the most significant findings stemming from chaos theory is the idea that highly complex )] TJ ET
BT 26.250 457.824 Td /F1 9.8 Tf [(outcomes can emerge from very simple beginnings, a concept finding expression in the title of Mays classic paper and in the )] TJ ET
BT 26.250 445.920 Td /F1 9.8 Tf [(notion of deterministic chaos \(in the sense that whatever is determined by a causal linkagefunctional iteration consisting of )] TJ ET
BT 26.250 434.015 Td /F1 9.8 Tf [(such a causal linkageshould only generate results proportionate to whatever rudimentary elements are at work in the )] TJ ET
BT 26.250 422.110 Td /F1 9.8 Tf [(causation\). Yet, this is not what happens. Rather, deterministic chaos brings forth something as complex as a random seeming )] TJ ET
BT 26.250 410.205 Td /F1 9.8 Tf [(pattern, i.e., chaos. In Mays terminology, functional iteration engenders this complicatedness out of simple although )] TJ ET
BT 26.250 398.301 Td /F1 9.8 Tf [(complicated for May was not so much chaos per se as the operation of functional iteration, indeed, the entire bifurcation )] TJ ET
BT 26.250 386.396 Td /F1 9.8 Tf [(scenario generated by changes in the bifurcation parameter. Should we then, not attempt to understand what May )] TJ ET
BT 26.250 374.491 Td /F1 9.8 Tf [(accomplished as falling in line behind the slogan of complexity from simplicity.)] TJ ET
BT 26.250 355.086 Td /F1 9.8 Tf [(It is not only in chaos theory and its associated fields that this implication has found a foothold. We can hear it in other )] TJ ET
BT 26.250 343.182 Td /F1 9.8 Tf [(complexity science branches as well. For instance, it is very frequently heard in emergentist circles that emergence, in many of )] TJ ET
BT 26.250 331.277 Td /F1 9.8 Tf [(its different contexts, can only occur when a certain threshold of complexity \(this threshold measured by different metrics )] TJ ET
BT 26.250 319.372 Td /F1 9.8 Tf [(according to what kind of emergence is being queried\). Perhaps the most extreme, recent statement of such a viewpoint is the )] TJ ET
BT 26.250 307.467 Td /F1 9.8 Tf [(claim that consciousness is an emergent phenomena that can only come about when evolution has produced some sufficiently )] TJ ET
BT 26.250 295.563 Td /F1 9.8 Tf [(critical level of complexity, e.g., some supposed metric of neural connectivity or neural networking \(see a very recent version of )] TJ ET
BT 26.250 283.658 Td /F1 9.8 Tf [(this thesis formulated in terms of integrated information in the work of Giulio Tononi and Cristof Koch)] TJ ET
0.267 0.267 0.267 rg
BT 465.712 285.165 Td /F4 8.7 Tf [(4)] TJ ET
0.271 0.267 0.267 rg
BT 470.530 283.658 Td /F1 9.8 Tf [(\).)] TJ ET
BT 26.250 264.253 Td /F1 9.8 Tf [(The two classic papers being republished in )] TJ ET
BT 218.081 264.253 Td /F5 9.8 Tf [(E:CO)] TJ ET
BT 241.920 264.253 Td /F1 9.8 Tf [( should be of aid for us only only in grasping how functional iteration in )] TJ ET
BT 26.250 252.348 Td /F1 9.8 Tf [(nonlinear maps takes placethey should also help give the lie to any such conclusions of complexity coming from simplicity. )] TJ ET
BT 26.250 240.444 Td /F1 9.8 Tf [(That is, these two classic papers can be read as showing how the logistic map and similar complexity images in actuality )] TJ ET
BT 26.250 228.539 Td /F1 9.8 Tf [(contain a huge reservoir of complex dynamics that is made manifest under the right conditions. In other words, what May and )] TJ ET
BT 26.250 216.634 Td /F1 9.8 Tf [(Feigenbaum have done is to further the project of discerning how complex patterns emerge from antecedent patterns or as )] TJ ET
BT 26.250 204.729 Td /F1 9.8 Tf [(Turing described it in the arising of novel morphogenic patterns: developing from one pattern into another, rather than from )] TJ ET
BT 26.250 192.825 Td /F1 9.8 Tf [(homogeneity into a pattern.)] TJ ET
0.267 0.267 0.267 rg
BT 147.647 194.332 Td /F4 8.7 Tf [(5)] TJ ET
0.271 0.267 0.267 rg
BT 26.250 173.420 Td /F1 9.8 Tf [(This may seem like a semantical issue but I am contending it is much deeper since the popularization of the sloganeering of )] TJ ET
BT 26.250 161.515 Td /F1 9.8 Tf [(complexity coming out of simplicity can get in the way of appreciating just how much complexity, complicatedness, )] TJ ET
BT 26.250 149.610 Td /F1 9.8 Tf [(complexification are going on already or have been going on. For instance, consider functional iteration: it is fundamentally a )] TJ ET
BT 26.250 137.706 Td /F1 9.8 Tf [(complexifying operation in the sense likened to kneading dough to make bread, an operation that aims at optimal mixing of )] TJ ET
BT 26.250 125.801 Td /F1 9.8 Tf [(ingredients, especially diffusing yeast throughout the flour where the yeast can do its work. This can be seen vividly in the )] TJ ET
BT 26.250 113.896 Td /F1 9.8 Tf [(closely related horseshoe map of Steve Smale which illustrates the folding nonlinearity of functional iteration by way of a )] TJ ET
BT 26.250 101.991 Td /F1 9.8 Tf [(horseshoe which is elongated, then folded over, the result being elongated and then folded over again and so on.)] TJ ET
Q
q
15.000 36.992 577.500 740.008 re W n
0.271 0.267 0.267 rg
BT 26.250 767.476 Td /F1 9.8 Tf [(keeps the population from exploding beyond any sensible interpretation. The plot is called a cobweb diagram since the way the )] TJ ET
BT 26.250 755.571 Td /F1 9.8 Tf [(solutions change discretely looks like a cobweb \(see the parabolic curves in Mays paper, Figure 1 on p. 2\). The birth/death )] TJ ET
BT 26.250 743.667 Td /F1 9.8 Tf [(parameter )] TJ ET
BT 73.391 743.667 Td /F6 9.8 Tf [(a)] TJ ET
BT 78.812 743.667 Td /F1 9.8 Tf [( shows up as the slope of the tangent line, when )] TJ ET
BT 289.646 743.667 Td /F6 9.8 Tf [(P)] TJ ET
BT 296.149 741.602 Td /F6 8.7 Tf [(t)] TJ ET
BT 299.036 743.667 Td /F1 9.8 Tf [(= )] TJ ET
BT 307.440 743.667 Td /F6 9.8 Tf [(P)] TJ ET
BT 313.943 743.667 Td /F6 8.7 Tf [(t)] TJ ET
BT 316.829 743.667 Td /F4 8.7 Tf [(+1)] TJ ET
BT 329.119 743.667 Td /F4 9.8 Tf [(.)] TJ ET
BT 26.250 724.262 Td /F1 9.8 Tf [(I remember from math courses that such parabolic equations were studied in calculus but usually in the form of differential )] TJ ET
BT 26.250 712.357 Td /F1 9.8 Tf [(equations that were not particularly interesting nor revealing but instead consisted of just finding solutions, graphically displaying )] TJ ET
BT 26.250 700.452 Td /F1 9.8 Tf [(the equations, the mechanisms of differentiation, and so forth \(unless we were fortunate enough to have taken classes in )] TJ ET
BT 26.250 688.548 Td /F1 9.8 Tf [(dynamical systems but they were few at that time, for undergraduates at least\). What May discovered and labeled complicated )] TJ ET
BT 26.250 676.643 Td /F1 9.8 Tf [(dynamics in the title of his classic paper was in radical contrast to what most of us encountered in calculus. His findings were )] TJ ET
BT 26.250 664.738 Td /F1 9.8 Tf [(due not to the usual exploration of changes of the values of the population but instead arose from playing around with the )] TJ ET
BT 26.250 652.833 Td /F1 9.8 Tf [(values of the parameter )] TJ ET
BT 131.384 652.833 Td /F4 9.8 Tf [(a)] TJ ET
BT 136.805 652.833 Td /F1 9.8 Tf [(. He refers to this manipulation of the parameter )] TJ ET
BT 345.982 652.833 Td /F6 9.8 Tf [(a)] TJ ET
BT 351.403 652.833 Td /F1 9.8 Tf [( tuning the nonlinearity of the logistic map since )] TJ ET
BT 26.250 640.929 Td /F1 9.8 Tf [(the steepness of the tangent line indicates how nonlinear, i.e., how complicated \(in his terminology\) things are going to get )] TJ ET
BT 26.250 629.024 Td /F1 9.8 Tf [(when bifurcation occurs.)] TJ ET
BT 26.250 609.619 Td /F1 9.8 Tf [(By tuning the nonlinearity of the logistic map, May found, against expectations, that as the values of the parameter )] TJ ET
BT 530.793 609.619 Td /F6 9.8 Tf [(a)] TJ ET
BT 26.250 597.714 Td /F1 9.8 Tf [(increased, the values of )] TJ ET
BT 134.641 597.714 Td /F6 9.8 Tf [(P)] TJ ET
BT 141.144 597.714 Td /F6 8.7 Tf [(t)] TJ ET
BT 144.030 595.650 Td /F4 8.7 Tf [(+1)] TJ ET
BT 153.910 597.714 Td /F1 9.8 Tf [(did not consistently change in the usual manner but instead could get trapped at some particular )] TJ ET
BT 26.250 585.810 Td /F1 9.8 Tf [(value\(s\) or circumscribed set of values and stay there no matter how many times the iteration involved took place, or they could )] TJ ET
BT 26.250 573.905 Td /F1 9.8 Tf [(in contrast display sudden jumps in values These trapped and jumped values, as May found out, hinged on the values of the )] TJ ET
BT 26.250 562.000 Td /F1 9.8 Tf [(parameters and whether they kept the system inside an attractor or instead prompted a bifurcation or jump out of the attractor. )] TJ ET
BT 26.250 550.095 Td /F1 9.8 Tf [(are constrained by attractors because they metaphorically attract the values of the variables in the long-run. It is only when )] TJ ET
BT 26.250 538.191 Td /F1 9.8 Tf [(the parameter values reach a critical threshold that they can prompt bifurcation or the emergence or disappearance of new )] TJ ET
BT 26.250 526.286 Td /F1 9.8 Tf [(attractors.)] TJ ET
BT 26.250 489.683 Td /F4 12.0 Tf [(Complexity from simplicity?)] TJ ET
BT 26.250 469.729 Td /F1 9.8 Tf [(It is generally thought that one of the most significant findings stemming from chaos theory is the idea that highly complex )] TJ ET
BT 26.250 457.824 Td /F1 9.8 Tf [(outcomes can emerge from very simple beginnings, a concept finding expression in the title of Mays classic paper and in the )] TJ ET
BT 26.250 445.920 Td /F1 9.8 Tf [(notion of deterministic chaos \(in the sense that whatever is determined by a causal linkagefunctional iteration consisting of )] TJ ET
BT 26.250 434.015 Td /F1 9.8 Tf [(such a causal linkageshould only generate results proportionate to whatever rudimentary elements are at work in the )] TJ ET
BT 26.250 422.110 Td /F1 9.8 Tf [(causation\). Yet, this is not what happens. Rather, deterministic chaos brings forth something as complex as a random seeming )] TJ ET
BT 26.250 410.205 Td /F1 9.8 Tf [(pattern, i.e., chaos. In Mays terminology, functional iteration engenders this complicatedness out of simple although )] TJ ET
BT 26.250 398.301 Td /F1 9.8 Tf [(complicated for May was not so much chaos per se as the operation of functional iteration, indeed, the entire bifurcation )] TJ ET
BT 26.250 386.396 Td /F1 9.8 Tf [(scenario generated by changes in the bifurcation parameter. Should we then, not attempt to understand what May )] TJ ET
BT 26.250 374.491 Td /F1 9.8 Tf [(accomplished as falling in line behind the slogan of complexity from simplicity.)] TJ ET
BT 26.250 355.086 Td /F1 9.8 Tf [(It is not only in chaos theory and its associated fields that this implication has found a foothold. We can hear it in other )] TJ ET
BT 26.250 343.182 Td /F1 9.8 Tf [(complexity science branches as well. For instance, it is very frequently heard in emergentist circles that emergence, in many of )] TJ ET
BT 26.250 331.277 Td /F1 9.8 Tf [(its different contexts, can only occur when a certain threshold of complexity \(this threshold measured by different metrics )] TJ ET
BT 26.250 319.372 Td /F1 9.8 Tf [(according to what kind of emergence is being queried\). Perhaps the most extreme, recent statement of such a viewpoint is the )] TJ ET
BT 26.250 307.467 Td /F1 9.8 Tf [(claim that consciousness is an emergent phenomena that can only come about when evolution has produced some sufficiently )] TJ ET
BT 26.250 295.563 Td /F1 9.8 Tf [(critical level of complexity, e.g., some supposed metric of neural connectivity or neural networking \(see a very recent version of )] TJ ET
BT 26.250 283.658 Td /F1 9.8 Tf [(this thesis formulated in terms of integrated information in the work of Giulio Tononi and Cristof Koch)] TJ ET
0.267 0.267 0.267 rg
BT 465.712 285.165 Td /F4 8.7 Tf [(4)] TJ ET
0.271 0.267 0.267 rg
BT 470.530 283.658 Td /F1 9.8 Tf [(\).)] TJ ET
BT 26.250 264.253 Td /F1 9.8 Tf [(The two classic papers being republished in )] TJ ET
BT 218.081 264.253 Td /F5 9.8 Tf [(E:CO)] TJ ET
BT 241.920 264.253 Td /F1 9.8 Tf [( should be of aid for us only only in grasping how functional iteration in )] TJ ET
BT 26.250 252.348 Td /F1 9.8 Tf [(nonlinear maps takes placethey should also help give the lie to any such conclusions of complexity coming from simplicity. )] TJ ET
BT 26.250 240.444 Td /F1 9.8 Tf [(That is, these two classic papers can be read as showing how the logistic map and similar complexity images in actuality )] TJ ET
BT 26.250 228.539 Td /F1 9.8 Tf [(contain a huge reservoir of complex dynamics that is made manifest under the right conditions. In other words, what May and )] TJ ET
BT 26.250 216.634 Td /F1 9.8 Tf [(Feigenbaum have done is to further the project of discerning how complex patterns emerge from antecedent patterns or as )] TJ ET
BT 26.250 204.729 Td /F1 9.8 Tf [(Turing described it in the arising of novel morphogenic patterns: developing from one pattern into another, rather than from )] TJ ET
BT 26.250 192.825 Td /F1 9.8 Tf [(homogeneity into a pattern.)] TJ ET
0.267 0.267 0.267 rg
BT 147.647 194.332 Td /F4 8.7 Tf [(5)] TJ ET
0.271 0.267 0.267 rg
BT 26.250 173.420 Td /F1 9.8 Tf [(This may seem like a semantical issue but I am contending it is much deeper since the popularization of the sloganeering of )] TJ ET
BT 26.250 161.515 Td /F1 9.8 Tf [(complexity coming out of simplicity can get in the way of appreciating just how much complexity, complicatedness, )] TJ ET
BT 26.250 149.610 Td /F1 9.8 Tf [(complexification are going on already or have been going on. For instance, consider functional iteration: it is fundamentally a )] TJ ET
BT 26.250 137.706 Td /F1 9.8 Tf [(complexifying operation in the sense likened to kneading dough to make bread, an operation that aims at optimal mixing of )] TJ ET
BT 26.250 125.801 Td /F1 9.8 Tf [(ingredients, especially diffusing yeast throughout the flour where the yeast can do its work. This can be seen vividly in the )] TJ ET
BT 26.250 113.896 Td /F1 9.8 Tf [(closely related horseshoe map of Steve Smale which illustrates the folding nonlinearity of functional iteration by way of a )] TJ ET
BT 26.250 101.991 Td /F1 9.8 Tf [(horseshoe which is elongated, then folded over, the result being elongated and then folded over again and so on.)] TJ ET
Q
q
0.000 0.000 0.000 rg
BT 291.710 19.825 Td /F1 11.0 Tf [(3)] TJ ET
BT 25.000 19.825 Td /F1 11.0 Tf [(Emergence: Complexity and Organization)] TJ ET
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15.000 27.229 577.500 749.771 re W n
0.271 0.267 0.267 rg
BT 26.250 767.476 Td /F1 9.8 Tf [(The following is an algebraic expansion of the logistic map which I generated to better visualize the complexity inherent )] TJ ET
BT 26.250 755.571 Td /F1 9.8 Tf [(\(perhaps hidden is a better term\) in the seeming simplicity of the map as the first generation is taken to the the )] TJ ET
BT 510.172 755.571 Td /F6 9.8 Tf [(P)] TJ ET
BT 516.675 755.571 Td /F6 8.7 Tf [(t)] TJ ET
BT 519.561 755.571 Td /F4 8.7 Tf [(+4)] TJ ET
BT 529.441 755.571 Td /F1 9.8 Tf [( generation )] TJ ET
BT 26.250 743.667 Td /F1 9.8 Tf [(by way of the operation of functional iteration \(or pumping back-into\) performed at each generation, i.e., here expressing )] TJ ET
BT 549.211 743.667 Td /F6 9.8 Tf [(P)] TJ ET
BT 555.714 743.667 Td /F6 8.7 Tf [(t)] TJ ET
BT 558.600 741.602 Td /F4 8.7 Tf [(+4)] TJ ET
BT 26.250 731.762 Td /F1 9.8 Tf [(in terms only of)] TJ ET
BT 92.355 731.762 Td /F6 9.8 Tf [( P)] TJ ET
BT 101.569 729.698 Td /F6 8.7 Tf [(t)] TJ ET
BT 104.455 731.762 Td /F1 9.8 Tf [(:)] TJ ET
BT 26.250 708.081 Td /F1 9.8 Tf [(Eqn. 2 )] TJ ET
q
109.500 0 0 17.250 57.148 704.631 cm /I8 Do
Q
BT 26.250 683.331 Td /F1 9.8 Tf [(Eqn. 3 )] TJ ET
q
224.250 0 0 17.250 57.148 679.881 cm /I9 Do
Q
BT 26.250 658.581 Td /F1 9.8 Tf [(Eqn. 4 )] TJ ET
q
452.250 0 0 17.250 57.148 655.131 cm /I10 Do
Q
BT 26.250 611.031 Td /F1 9.8 Tf [(Eqn. 5 )] TJ ET
q
404.250 0 0 45.750 57.148 601.881 cm /I11 Do
Q
BT 26.250 584.857 Td /F1 9.8 Tf [(Eqn. 5 is only the fourth generation yet it is already a mess of complicatedness or complexity! Certainly, from the vantage point )] TJ ET
BT 26.250 572.952 Td /F1 9.8 Tf [(had by starting at )] TJ ET
BT 103.753 572.952 Td /F6 9.8 Tf [(P)] TJ ET
BT 110.256 570.888 Td /F6 8.7 Tf [(t)] TJ ET
BT 113.142 572.952 Td /F1 9.8 Tf [( \(say, the zeroth year of when a population is measured\), this initial condition obviously appears as the )] TJ ET
BT 26.250 561.048 Td /F1 9.8 Tf [(simplest facet of this expansion of the logistic map following its functional iteration. With that apparent simplicity as a starting )] TJ ET
BT 26.250 549.143 Td /F1 9.8 Tf [(point, then in comparison with Eqn. 5, it appears that a gargantuan complication has indeed ensued. But such a conclusion fails )] TJ ET
BT 26.250 537.238 Td /F1 9.8 Tf [(to take into consideration the fact that )] TJ ET
BT 191.015 537.238 Td /F6 9.8 Tf [(P)] TJ ET
BT 197.519 535.174 Td /F6 8.7 Tf [(t)] TJ ET
BT 200.405 537.238 Td /F1 9.8 Tf [( was arbitrarily chosen as the zeroth year of the population being measured. What is )] TJ ET
BT 26.250 525.333 Td /F1 9.8 Tf [(left out is the complication involved in arriving at the value of )] TJ ET
BT 288.535 525.333 Td /F6 9.8 Tf [(P)] TJ ET
BT 295.038 523.269 Td /F6 8.7 Tf [(t)] TJ ET
BT 297.924 525.333 Td /F1 9.8 Tf [( before any operation of functional iteration shown in the )] TJ ET
BT 26.250 513.429 Td /F1 9.8 Tf [(sequence of Eqn. 2 to Eqn. 5 above had taken place applied \(taking )] TJ ET
BT 321.646 513.429 Td /F6 9.8 Tf [(P)] TJ ET
BT 328.149 511.364 Td /F6 8.7 Tf [(t)] TJ ET
BT 331.035 513.429 Td /F1 9.8 Tf [( to )] TJ ET
BT 347.298 513.429 Td /F6 9.8 Tf [(P)] TJ ET
BT 353.801 513.429 Td /F6 8.7 Tf [(t)] TJ ET
BT 356.687 511.364 Td /F4 8.7 Tf [(+4)] TJ ET
BT 366.567 513.429 Td /F1 9.8 Tf [(\). This complication is left out for an economical )] TJ ET
BT 26.250 501.524 Td /F1 9.8 Tf [(reason since, otherwise, one could not escape the enormous complication resulting from a need to keep updating the value of )] TJ ET
BT 26.250 489.619 Td /F1 9.8 Tf [(whatever )] TJ ET
BT 68.516 489.619 Td /F6 9.8 Tf [(P)] TJ ET
BT 75.019 487.555 Td /F6 8.7 Tf [(t)] TJ ET
BT 77.906 489.619 Td /F1 9.8 Tf [( happens to be at any arbitrary time in terms of )] TJ ET
BT 284.918 489.619 Td /F6 9.8 Tf [(P)] TJ ET
BT 291.421 487.555 Td /F6 8.7 Tf [(t+n)] TJ ET
BT 304.663 489.619 Td /F1 9.8 Tf [(.)] TJ ET
BT 26.250 470.214 Td /F1 9.8 Tf [(For this and other reasons, it should really come as no surprise \(and definitely not after Mays brilliant exposition\) that the )] TJ ET
BT 26.250 458.310 Td /F1 9.8 Tf [(logistic map and other similar one-humped equations are thoroughly imbued with complicated dynamics orcomplexity from )] TJ ET
BT 26.250 446.405 Td /F1 9.8 Tf [(the get-go. The finding of chaos along the way of a period-doubling route or other routes, therefore is just more confirmation )] TJ ET
BT 26.250 434.500 Td /F1 9.8 Tf [(that the logistic map or equation contains an inherent complexity playing itself out as the complex systems modeled by it evolve, )] TJ ET
BT 26.250 422.595 Td /F1 9.8 Tf [(change, adapt, grow, die, and so forth.)] TJ ET
BT 26.250 403.191 Td /F1 9.8 Tf [(Moreover, it is not only its functional iteration that renders the logistic map a complexifying operation, it is perhaps even more so )] TJ ET
BT 26.250 391.286 Td /F1 9.8 Tf [(because of the bifurcation scenario that May so amply describes. The possibility of bifurcation into new attractors was not )] TJ ET
BT 26.250 379.381 Td /F1 9.8 Tf [(expected at Mays time but now, because of his and others work in dynamical systems research, we expect this kind of )] TJ ET
BT 26.250 367.476 Td /F1 9.8 Tf [(bifurcation complexification. What a surprise it was to many of us brought up on calculus and the prominent role of linearity in )] TJ ET
BT 26.250 355.572 Td /F1 9.8 Tf [(studying it. Who knew that the values we were seeking were confined to these attractors? We didnt even know that values )] TJ ET
BT 26.250 343.667 Td /F1 9.8 Tf [(resulting from computing these equations were confined at all! The complexity was there all along but all we saw was the )] TJ ET
BT 26.250 331.762 Td /F1 9.8 Tf [(simplicity of what we took for granted. For the fact is that we are always already steeped in a nonlinear complex world replete )] TJ ET
BT 26.250 319.857 Td /F1 9.8 Tf [(with complex systems and all of the phenomena complex systems exhibit. And the causes of the complex mathematical )] TJ ET
BT 26.250 307.953 Td /F1 9.8 Tf [(dynamics of the equations are primarily found in the structure of the mathematics and not just in interaction with external )] TJ ET
BT 26.250 296.048 Td /F1 9.8 Tf [(causative agents.)] TJ ET
BT 26.250 276.643 Td /F1 9.8 Tf [(Mays insightful analysis of the bifurcation scenario of the logistic map circles around the issue of instability, that new attractors )] TJ ET
BT 26.250 264.738 Td /F1 9.8 Tf [(arise when attracting equilibrium points become unstable and lead to divergence, not convergence. May shows how this )] TJ ET
BT 26.250 252.834 Td /F1 9.8 Tf [(instability leading to bifurcation is a function of the parameter in steepening the nonlinearity and the functions internal )] TJ ET
BT 26.250 240.929 Td /F1 9.8 Tf [(mathematical structure. So along with nonlinearity, complicatedness, bifurcation and so on, we also live in a world abounding in )] TJ ET
BT 26.250 229.024 Td /F1 9.8 Tf [(instability islands of stability in oceans of instability, islands of instability in oceans of stability. A physicist of no less a )] TJ ET
BT 26.250 217.119 Td /F1 9.8 Tf [(luminary status than Enrico Fermi \(who was also well known as a genius mathematician for his ability to surmise what the )] TJ ET
BT 26.250 205.215 Td /F1 9.8 Tf [(solutions of mathematical problems would look like before even methods for solving them had been developed!\) was of the )] TJ ET
BT 26.250 193.310 Td /F1 9.8 Tf [(strong conviction that future theories would involve these kinds of nonlinearity and complicatedness and therefore pushed hard )] TJ ET
BT 26.250 181.405 Td /F1 9.8 Tf [(for the development of computers which could hopefully make some headway using numerical methods.)] TJ ET
BT 26.250 144.803 Td /F4 12.0 Tf [(Conclusion: Complexity gets more complex)] TJ ET
BT 26.250 124.848 Td /F1 9.8 Tf [(When chaos theory first splashed on the scene in the 1980s \(in the popular press, that is\), among those given to speculation )] TJ ET
BT 26.250 112.944 Td /F1 9.8 Tf [(about weighty metaphysical themes, it was considered to have revolutionary implications particularly in so far as it now )] TJ ET
BT 26.250 101.039 Td /F1 9.8 Tf [(appeared that nature had a capacity for generating surprisingly complex, even random dynamics out of much simpler initial )] TJ ET
BT 26.250 89.134 Td /F1 9.8 Tf [(states. Chaos suggested a plethora of reconceptualizations of the relation of order to disorder, predictability to unpredictability, )] TJ ET
BT 26.250 77.229 Td /F1 9.8 Tf [(determinism to stochasticity, and stability to instability. It has become a commonplace to hear, not just within complexity science )] TJ ET
BT 26.250 65.325 Td /F1 9.8 Tf [(circles but elsewhere in science and philosophy, about some drive or proclivity or tendency, no matter how slight, in nature )] TJ ET
BT 26.250 53.420 Td /F1 9.8 Tf [(pushing for complexity to arise from the simple to the complex \(not unlike Spencers early evolutionary claim of the )] TJ ET
BT 26.250 41.515 Td /F1 9.8 Tf [(heterogeneous springing forth from the homogenous.\) We can see such an inference, for example, in emergence-tinged )] TJ ET
Q
q
15.000 27.229 577.500 749.771 re W n
0.271 0.267 0.267 rg
BT 26.250 767.476 Td /F1 9.8 Tf [(The following is an algebraic expansion of the logistic map which I generated to better visualize the complexity inherent )] TJ ET
BT 26.250 755.571 Td /F1 9.8 Tf [(\(perhaps hidden is a better term\) in the seeming simplicity of the map as the first generation is taken to the the )] TJ ET
BT 510.172 755.571 Td /F6 9.8 Tf [(P)] TJ ET
BT 516.675 755.571 Td /F6 8.7 Tf [(t)] TJ ET
BT 519.561 755.571 Td /F4 8.7 Tf [(+4)] TJ ET
BT 529.441 755.571 Td /F1 9.8 Tf [( generation )] TJ ET
BT 26.250 743.667 Td /F1 9.8 Tf [(by way of the operation of functional iteration \(or pumping back-into\) performed at each generation, i.e., here expressing )] TJ ET
BT 549.211 743.667 Td /F6 9.8 Tf [(P)] TJ ET
BT 555.714 743.667 Td /F6 8.7 Tf [(t)] TJ ET
BT 558.600 741.602 Td /F4 8.7 Tf [(+4)] TJ ET
BT 26.250 731.762 Td /F1 9.8 Tf [(in terms only of)] TJ ET
BT 92.355 731.762 Td /F6 9.8 Tf [( P)] TJ ET
BT 101.569 729.698 Td /F6 8.7 Tf [(t)] TJ ET
BT 104.455 731.762 Td /F1 9.8 Tf [(:)] TJ ET
BT 26.250 708.081 Td /F1 9.8 Tf [(Eqn. 2 )] TJ ET
q
109.500 0 0 17.250 57.148 704.631 cm /I8 Do
Q
BT 26.250 683.331 Td /F1 9.8 Tf [(Eqn. 3 )] TJ ET
q
224.250 0 0 17.250 57.148 679.881 cm /I9 Do
Q
BT 26.250 658.581 Td /F1 9.8 Tf [(Eqn. 4 )] TJ ET
q
452.250 0 0 17.250 57.148 655.131 cm /I10 Do
Q
BT 26.250 611.031 Td /F1 9.8 Tf [(Eqn. 5 )] TJ ET
q
404.250 0 0 45.750 57.148 601.881 cm /I11 Do
Q
BT 26.250 584.857 Td /F1 9.8 Tf [(Eqn. 5 is only the fourth generation yet it is already a mess of complicatedness or complexity! Certainly, from the vantage point )] TJ ET
BT 26.250 572.952 Td /F1 9.8 Tf [(had by starting at )] TJ ET
BT 103.753 572.952 Td /F6 9.8 Tf [(P)] TJ ET
BT 110.256 570.888 Td /F6 8.7 Tf [(t)] TJ ET
BT 113.142 572.952 Td /F1 9.8 Tf [( \(say, the zeroth year of when a population is measured\), this initial condition obviously appears as the )] TJ ET
BT 26.250 561.048 Td /F1 9.8 Tf [(simplest facet of this expansion of the logistic map following its functional iteration. With that apparent simplicity as a starting )] TJ ET
BT 26.250 549.143 Td /F1 9.8 Tf [(point, then in comparison with Eqn. 5, it appears that a gargantuan complication has indeed ensued. But such a conclusion fails )] TJ ET
BT 26.250 537.238 Td /F1 9.8 Tf [(to take into consideration the fact that )] TJ ET
BT 191.015 537.238 Td /F6 9.8 Tf [(P)] TJ ET
BT 197.519 535.174 Td /F6 8.7 Tf [(t)] TJ ET
BT 200.405 537.238 Td /F1 9.8 Tf [( was arbitrarily chosen as the zeroth year of the population being measured. What is )] TJ ET
BT 26.250 525.333 Td /F1 9.8 Tf [(left out is the complication involved in arriving at the value of )] TJ ET
BT 288.535 525.333 Td /F6 9.8 Tf [(P)] TJ ET
BT 295.038 523.269 Td /F6 8.7 Tf [(t)] TJ ET
BT 297.924 525.333 Td /F1 9.8 Tf [( before any operation of functional iteration shown in the )] TJ ET
BT 26.250 513.429 Td /F1 9.8 Tf [(sequence of Eqn. 2 to Eqn. 5 above had taken place applied \(taking )] TJ ET
BT 321.646 513.429 Td /F6 9.8 Tf [(P)] TJ ET
BT 328.149 511.364 Td /F6 8.7 Tf [(t)] TJ ET
BT 331.035 513.429 Td /F1 9.8 Tf [( to )] TJ ET
BT 347.298 513.429 Td /F6 9.8 Tf [(P)] TJ ET
BT 353.801 513.429 Td /F6 8.7 Tf [(t)] TJ ET
BT 356.687 511.364 Td /F4 8.7 Tf [(+4)] TJ ET
BT 366.567 513.429 Td /F1 9.8 Tf [(\). This complication is left out for an economical )] TJ ET
BT 26.250 501.524 Td /F1 9.8 Tf [(reason since, otherwise, one could not escape the enormous complication resulting from a need to keep updating the value of )] TJ ET
BT 26.250 489.619 Td /F1 9.8 Tf [(whatever )] TJ ET
BT 68.516 489.619 Td /F6 9.8 Tf [(P)] TJ ET
BT 75.019 487.555 Td /F6 8.7 Tf [(t)] TJ ET
BT 77.906 489.619 Td /F1 9.8 Tf [( happens to be at any arbitrary time in terms of )] TJ ET
BT 284.918 489.619 Td /F6 9.8 Tf [(P)] TJ ET
BT 291.421 487.555 Td /F6 8.7 Tf [(t+n)] TJ ET
BT 304.663 489.619 Td /F1 9.8 Tf [(.)] TJ ET
BT 26.250 470.214 Td /F1 9.8 Tf [(For this and other reasons, it should really come as no surprise \(and definitely not after Mays brilliant exposition\) that the )] TJ ET
BT 26.250 458.310 Td /F1 9.8 Tf [(logistic map and other similar one-humped equations are thoroughly imbued with complicated dynamics orcomplexity from )] TJ ET
BT 26.250 446.405 Td /F1 9.8 Tf [(the get-go. The finding of chaos along the way of a period-doubling route or other routes, therefore is just more confirmation )] TJ ET
BT 26.250 434.500 Td /F1 9.8 Tf [(that the logistic map or equation contains an inherent complexity playing itself out as the complex systems modeled by it evolve, )] TJ ET
BT 26.250 422.595 Td /F1 9.8 Tf [(change, adapt, grow, die, and so forth.)] TJ ET
BT 26.250 403.191 Td /F1 9.8 Tf [(Moreover, it is not only its functional iteration that renders the logistic map a complexifying operation, it is perhaps even more so )] TJ ET
BT 26.250 391.286 Td /F1 9.8 Tf [(because of the bifurcation scenario that May so amply describes. The possibility of bifurcation into new attractors was not )] TJ ET
BT 26.250 379.381 Td /F1 9.8 Tf [(expected at Mays time but now, because of his and others work in dynamical systems research, we expect this kind of )] TJ ET
BT 26.250 367.476 Td /F1 9.8 Tf [(bifurcation complexification. What a surprise it was to many of us brought up on calculus and the prominent role of linearity in )] TJ ET
BT 26.250 355.572 Td /F1 9.8 Tf [(studying it. Who knew that the values we were seeking were confined to these attractors? We didnt even know that values )] TJ ET
BT 26.250 343.667 Td /F1 9.8 Tf [(resulting from computing these equations were confined at all! The complexity was there all along but all we saw was the )] TJ ET
BT 26.250 331.762 Td /F1 9.8 Tf [(simplicity of what we took for granted. For the fact is that we are always already steeped in a nonlinear complex world replete )] TJ ET
BT 26.250 319.857 Td /F1 9.8 Tf [(with complex systems and all of the phenomena complex systems exhibit. And the causes of the complex mathematical )] TJ ET
BT 26.250 307.953 Td /F1 9.8 Tf [(dynamics of the equations are primarily found in the structure of the mathematics and not just in interaction with external )] TJ ET
BT 26.250 296.048 Td /F1 9.8 Tf [(causative agents.)] TJ ET
BT 26.250 276.643 Td /F1 9.8 Tf [(Mays insightful analysis of the bifurcation scenario of the logistic map circles around the issue of instability, that new attractors )] TJ ET
BT 26.250 264.738 Td /F1 9.8 Tf [(arise when attracting equilibrium points become unstable and lead to divergence, not convergence. May shows how this )] TJ ET
BT 26.250 252.834 Td /F1 9.8 Tf [(instability leading to bifurcation is a function of the parameter in steepening the nonlinearity and the functions internal )] TJ ET
BT 26.250 240.929 Td /F1 9.8 Tf [(mathematical structure. So along with nonlinearity, complicatedness, bifurcation and so on, we also live in a world abounding in )] TJ ET
BT 26.250 229.024 Td /F1 9.8 Tf [(instability islands of stability in oceans of instability, islands of instability in oceans of stability. A physicist of no less a )] TJ ET
BT 26.250 217.119 Td /F1 9.8 Tf [(luminary status than Enrico Fermi \(who was also well known as a genius mathematician for his ability to surmise what the )] TJ ET
BT 26.250 205.215 Td /F1 9.8 Tf [(solutions of mathematical problems would look like before even methods for solving them had been developed!\) was of the )] TJ ET
BT 26.250 193.310 Td /F1 9.8 Tf [(strong conviction that future theories would involve these kinds of nonlinearity and complicatedness and therefore pushed hard )] TJ ET
BT 26.250 181.405 Td /F1 9.8 Tf [(for the development of computers which could hopefully make some headway using numerical methods.)] TJ ET
BT 26.250 144.803 Td /F4 12.0 Tf [(Conclusion: Complexity gets more complex)] TJ ET
BT 26.250 124.848 Td /F1 9.8 Tf [(When chaos theory first splashed on the scene in the 1980s \(in the popular press, that is\), among those given to speculation )] TJ ET
BT 26.250 112.944 Td /F1 9.8 Tf [(about weighty metaphysical themes, it was considered to have revolutionary implications particularly in so far as it now )] TJ ET
BT 26.250 101.039 Td /F1 9.8 Tf [(appeared that nature had a capacity for generating surprisingly complex, even random dynamics out of much simpler initial )] TJ ET
BT 26.250 89.134 Td /F1 9.8 Tf [(states. Chaos suggested a plethora of reconceptualizations of the relation of order to disorder, predictability to unpredictability, )] TJ ET
BT 26.250 77.229 Td /F1 9.8 Tf [(determinism to stochasticity, and stability to instability. It has become a commonplace to hear, not just within complexity science )] TJ ET
BT 26.250 65.325 Td /F1 9.8 Tf [(circles but elsewhere in science and philosophy, about some drive or proclivity or tendency, no matter how slight, in nature )] TJ ET
BT 26.250 53.420 Td /F1 9.8 Tf [(pushing for complexity to arise from the simple to the complex \(not unlike Spencers early evolutionary claim of the )] TJ ET
BT 26.250 41.515 Td /F1 9.8 Tf [(heterogeneous springing forth from the homogenous.\) We can see such an inference, for example, in emergence-tinged )] TJ ET
Q
q
15.000 27.229 577.500 749.771 re W n
0.271 0.267 0.267 rg
BT 26.250 767.476 Td /F1 9.8 Tf [(The following is an algebraic expansion of the logistic map which I generated to better visualize the complexity inherent )] TJ ET
BT 26.250 755.571 Td /F1 9.8 Tf [(\(perhaps hidden is a better term\) in the seeming simplicity of the map as the first generation is taken to the the )] TJ ET
BT 510.172 755.571 Td /F6 9.8 Tf [(P)] TJ ET
BT 516.675 755.571 Td /F6 8.7 Tf [(t)] TJ ET
BT 519.561 755.571 Td /F4 8.7 Tf [(+4)] TJ ET
BT 529.441 755.571 Td /F1 9.8 Tf [( generation )] TJ ET
BT 26.250 743.667 Td /F1 9.8 Tf [(by way of the operation of functional iteration \(or pumping back-into\) performed at each generation, i.e., here expressing )] TJ ET
BT 549.211 743.667 Td /F6 9.8 Tf [(P)] TJ ET
BT 555.714 743.667 Td /F6 8.7 Tf [(t)] TJ ET
BT 558.600 741.602 Td /F4 8.7 Tf [(+4)] TJ ET
BT 26.250 731.762 Td /F1 9.8 Tf [(in terms only of)] TJ ET
BT 92.355 731.762 Td /F6 9.8 Tf [( P)] TJ ET
BT 101.569 729.698 Td /F6 8.7 Tf [(t)] TJ ET
BT 104.455 731.762 Td /F1 9.8 Tf [(:)] TJ ET
BT 26.250 708.081 Td /F1 9.8 Tf [(Eqn. 2 )] TJ ET
q
109.500 0 0 17.250 57.148 704.631 cm /I8 Do
Q
BT 26.250 683.331 Td /F1 9.8 Tf [(Eqn. 3 )] TJ ET
q
224.250 0 0 17.250 57.148 679.881 cm /I9 Do
Q
BT 26.250 658.581 Td /F1 9.8 Tf [(Eqn. 4 )] TJ ET
q
452.250 0 0 17.250 57.148 655.131 cm /I10 Do
Q
BT 26.250 611.031 Td /F1 9.8 Tf [(Eqn. 5 )] TJ ET
q
404.250 0 0 45.750 57.148 601.881 cm /I11 Do
Q
BT 26.250 584.857 Td /F1 9.8 Tf [(Eqn. 5 is only the fourth generation yet it is already a mess of complicatedness or complexity! Certainly, from the vantage point )] TJ ET
BT 26.250 572.952 Td /F1 9.8 Tf [(had by starting at )] TJ ET
BT 103.753 572.952 Td /F6 9.8 Tf [(P)] TJ ET
BT 110.256 570.888 Td /F6 8.7 Tf [(t)] TJ ET
BT 113.142 572.952 Td /F1 9.8 Tf [( \(say, the zeroth year of when a population is measured\), this initial condition obviously appears as the )] TJ ET
BT 26.250 561.048 Td /F1 9.8 Tf [(simplest facet of this expansion of the logistic map following its functional iteration. With that apparent simplicity as a starting )] TJ ET
BT 26.250 549.143 Td /F1 9.8 Tf [(point, then in comparison with Eqn. 5, it appears that a gargantuan complication has indeed ensued. But such a conclusion fails )] TJ ET
BT 26.250 537.238 Td /F1 9.8 Tf [(to take into consideration the fact that )] TJ ET
BT 191.015 537.238 Td /F6 9.8 Tf [(P)] TJ ET
BT 197.519 535.174 Td /F6 8.7 Tf [(t)] TJ ET
BT 200.405 537.238 Td /F1 9.8 Tf [( was arbitrarily chosen as the zeroth year of the population being measured. What is )] TJ ET
BT 26.250 525.333 Td /F1 9.8 Tf [(left out is the complication involved in arriving at the value of )] TJ ET
BT 288.535 525.333 Td /F6 9.8 Tf [(P)] TJ ET
BT 295.038 523.269 Td /F6 8.7 Tf [(t)] TJ ET
BT 297.924 525.333 Td /F1 9.8 Tf [( before any operation of functional iteration shown in the )] TJ ET
BT 26.250 513.429 Td /F1 9.8 Tf [(sequence of Eqn. 2 to Eqn. 5 above had taken place applied \(taking )] TJ ET
BT 321.646 513.429 Td /F6 9.8 Tf [(P)] TJ ET
BT 328.149 511.364 Td /F6 8.7 Tf [(t)] TJ ET
BT 331.035 513.429 Td /F1 9.8 Tf [( to )] TJ ET
BT 347.298 513.429 Td /F6 9.8 Tf [(P)] TJ ET
BT 353.801 513.429 Td /F6 8.7 Tf [(t)] TJ ET
BT 356.687 511.364 Td /F4 8.7 Tf [(+4)] TJ ET
BT 366.567 513.429 Td /F1 9.8 Tf [(\). This complication is left out for an economical )] TJ ET
BT 26.250 501.524 Td /F1 9.8 Tf [(reason since, otherwise, one could not escape the enormous complication resulting from a need to keep updating the value of )] TJ ET
BT 26.250 489.619 Td /F1 9.8 Tf [(whatever )] TJ ET
BT 68.516 489.619 Td /F6 9.8 Tf [(P)] TJ ET
BT 75.019 487.555 Td /F6 8.7 Tf [(t)] TJ ET
BT 77.906 489.619 Td /F1 9.8 Tf [( happens to be at any arbitrary time in terms of )] TJ ET
BT 284.918 489.619 Td /F6 9.8 Tf [(P)] TJ ET
BT 291.421 487.555 Td /F6 8.7 Tf [(t+n)] TJ ET
BT 304.663 489.619 Td /F1 9.8 Tf [(.)] TJ ET
BT 26.250 470.214 Td /F1 9.8 Tf [(For this and other reasons, it should really come as no surprise \(and definitely not after Mays brilliant exposition\) that the )] TJ ET
BT 26.250 458.310 Td /F1 9.8 Tf [(logistic map and other similar one-humped equations are thoroughly imbued with complicated dynamics orcomplexity from )] TJ ET
BT 26.250 446.405 Td /F1 9.8 Tf [(the get-go. The finding of chaos along the way of a period-doubling route or other routes, therefore is just more confirmation )] TJ ET
BT 26.250 434.500 Td /F1 9.8 Tf [(that the logistic map or equation contains an inherent complexity playing itself out as the complex systems modeled by it evolve, )] TJ ET
BT 26.250 422.595 Td /F1 9.8 Tf [(change, adapt, grow, die, and so forth.)] TJ ET
BT 26.250 403.191 Td /F1 9.8 Tf [(Moreover, it is not only its functional iteration that renders the logistic map a complexifying operation, it is perhaps even more so )] TJ ET
BT 26.250 391.286 Td /F1 9.8 Tf [(because of the bifurcation scenario that May so amply describes. The possibility of bifurcation into new attractors was not )] TJ ET
BT 26.250 379.381 Td /F1 9.8 Tf [(expected at Mays time but now, because of his and others work in dynamical systems research, we expect this kind of )] TJ ET
BT 26.250 367.476 Td /F1 9.8 Tf [(bifurcation complexification. What a surprise it was to many of us brought up on calculus and the prominent role of linearity in )] TJ ET
BT 26.250 355.572 Td /F1 9.8 Tf [(studying it. Who knew that the values we were seeking were confined to these attractors? We didnt even know that values )] TJ ET
BT 26.250 343.667 Td /F1 9.8 Tf [(resulting from computing these equations were confined at all! The complexity was there all along but all we saw was the )] TJ ET
BT 26.250 331.762 Td /F1 9.8 Tf [(simplicity of what we took for granted. For the fact is that we are always already steeped in a nonlinear complex world replete )] TJ ET
BT 26.250 319.857 Td /F1 9.8 Tf [(with complex systems and all of the phenomena complex systems exhibit. And the causes of the complex mathematical )] TJ ET
BT 26.250 307.953 Td /F1 9.8 Tf [(dynamics of the equations are primarily found in the structure of the mathematics and not just in interaction with external )] TJ ET
BT 26.250 296.048 Td /F1 9.8 Tf [(causative agents.)] TJ ET
BT 26.250 276.643 Td /F1 9.8 Tf [(Mays insightful analysis of the bifurcation scenario of the logistic map circles around the issue of instability, that new attractors )] TJ ET
BT 26.250 264.738 Td /F1 9.8 Tf [(arise when attracting equilibrium points become unstable and lead to divergence, not convergence. May shows how this )] TJ ET
BT 26.250 252.834 Td /F1 9.8 Tf [(instability leading to bifurcation is a function of the parameter in steepening the nonlinearity and the functions internal )] TJ ET
BT 26.250 240.929 Td /F1 9.8 Tf [(mathematical structure. So along with nonlinearity, complicatedness, bifurcation and so on, we also live in a world abounding in )] TJ ET
BT 26.250 229.024 Td /F1 9.8 Tf [(instability islands of stability in oceans of instability, islands of instability in oceans of stability. A physicist of no less a )] TJ ET
BT 26.250 217.119 Td /F1 9.8 Tf [(luminary status than Enrico Fermi \(who was also well known as a genius mathematician for his ability to surmise what the )] TJ ET
BT 26.250 205.215 Td /F1 9.8 Tf [(solutions of mathematical problems would look like before even methods for solving them had been developed!\) was of the )] TJ ET
BT 26.250 193.310 Td /F1 9.8 Tf [(strong conviction that future theories would involve these kinds of nonlinearity and complicatedness and therefore pushed hard )] TJ ET
BT 26.250 181.405 Td /F1 9.8 Tf [(for the development of computers which could hopefully make some headway using numerical methods.)] TJ ET
BT 26.250 144.803 Td /F4 12.0 Tf [(Conclusion: Complexity gets more complex)] TJ ET
BT 26.250 124.848 Td /F1 9.8 Tf [(When chaos theory first splashed on the scene in the 1980s \(in the popular press, that is\), among those given to speculation )] TJ ET
BT 26.250 112.944 Td /F1 9.8 Tf [(about weighty metaphysical themes, it was considered to have revolutionary implications particularly in so far as it now )] TJ ET
BT 26.250 101.039 Td /F1 9.8 Tf [(appeared that nature had a capacity for generating surprisingly complex, even random dynamics out of much simpler initial )] TJ ET
BT 26.250 89.134 Td /F1 9.8 Tf [(states. Chaos suggested a plethora of reconceptualizations of the relation of order to disorder, predictability to unpredictability, )] TJ ET
BT 26.250 77.229 Td /F1 9.8 Tf [(determinism to stochasticity, and stability to instability. It has become a commonplace to hear, not just within complexity science )] TJ ET
BT 26.250 65.325 Td /F1 9.8 Tf [(circles but elsewhere in science and philosophy, about some drive or proclivity or tendency, no matter how slight, in nature )] TJ ET
BT 26.250 53.420 Td /F1 9.8 Tf [(pushing for complexity to arise from the simple to the complex \(not unlike Spencers early evolutionary claim of the )] TJ ET
BT 26.250 41.515 Td /F1 9.8 Tf [(heterogeneous springing forth from the homogenous.\) We can see such an inference, for example, in emergence-tinged )] TJ ET
Q
q
0.000 0.000 0.000 rg
BT 291.710 19.825 Td /F1 11.0 Tf [(4)] TJ ET
BT 25.000 19.825 Td /F1 11.0 Tf [(Emergence: Complexity and Organization)] TJ ET
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BT 26.250 767.476 Td /F1 9.8 Tf [(conceptions of how life began out of much simpler autocatalytic chemical reactions, or even how the universe itself and its vast )] TJ ET
BT 26.250 755.571 Td /F1 9.8 Tf [(complicatedness and complexity emerged from the much simpler dynamics of a supposed singularity and a subsequent process )] TJ ET
BT 26.250 743.667 Td /F1 9.8 Tf [(of inflation. Not a few esteemed physicists or cosmologists seem to take such a cosmic drive for complexity for granted, even )] TJ ET
BT 26.250 731.762 Td /F1 9.8 Tf [(among the more sober and thoughtful like Frank Wilzcek.)] TJ ET
BT 26.250 712.357 Td /F1 9.8 Tf [(Helping to empower this supposed cosmic engine of complexity have been further discoveries in chaos theory which have )] TJ ET
BT 26.250 700.452 Td /F1 9.8 Tf [(uncovered even greater depths of complicatedness. Here Ill briefly described two such trends. First, according to Palmer, )] TJ ET
BT 26.250 688.548 Td /F1 9.8 Tf [(Doring, and Seregin)] TJ ET
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BT 112.957 690.055 Td /F4 8.7 Tf [(6)] TJ ET
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BT 117.775 688.548 Td /F1 9.8 Tf [( in their careful and thoughtful examination of Edward Lorenzs early work on the butterfly effect \(i.e., )] TJ ET
BT 26.250 676.643 Td /F1 9.8 Tf [(sensitivity to initial conditions\) at the end of the sixties and during the early seventies, it appears Lorenz was not yet finished )] TJ ET
BT 26.250 664.738 Td /F1 9.8 Tf [(with his earlier work of 1963 since over the next decade, he ventured into a far more unpredictable arena, that of multi-scale )] TJ ET
BT 26.250 652.833 Td /F1 9.8 Tf [(systems \(e.g., in a presentation from he gave in 1972\).)] TJ ET
BT 26.250 633.429 Td /F1 9.8 Tf [(Remember that in the original butterfly effect, unpredictability in chaotic sysems showed itself in sensitive dependence on initial )] TJ ET
BT 26.250 621.524 Td /F1 9.8 Tf [(conditions that would lead to a blowing-up of even the most miniscule margin of imprecision in a measurementhence a )] TJ ET
BT 26.250 609.619 Td /F1 9.8 Tf [(vastly different outcome than had been predicted. Although various ways were offered to mitigate the explosion of imprecision )] TJ ET
BT 26.250 597.714 Td /F1 9.8 Tf [(\(e.g., see the brilliant method recommended by Shaw)] TJ ET
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BT 257.101 599.222 Td /F4 8.7 Tf [(7)] TJ ET
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BT 261.919 597.714 Td /F1 9.8 Tf [(\), damage had been done to the age-old linkage of two of the )] TJ ET
BT 26.250 585.810 Td /F1 9.8 Tf [(cornerstones of scientific explanation, namely, determinism in the sense of a causal linkage and predictability of the outcome of )] TJ ET
BT 26.250 573.905 Td /F1 9.8 Tf [(the causation acting on the initial conditions. Lorenz, however, asked if there were other sources of unpredictability that would )] TJ ET
BT 26.250 562.000 Td /F1 9.8 Tf [(not even yield to improvements in the measurement of initial conditions, that is, were not due to just sensitive dependence on )] TJ ET
BT 26.250 550.095 Td /F1 9.8 Tf [(initial conditions.)] TJ ET
BT 26.250 530.691 Td /F1 9.8 Tf [(What he came up with was the presence of a certain kind of multi-scale constituency in a system \(again using weather systems )] TJ ET
BT 26.250 518.786 Td /F1 9.8 Tf [(as his forte\), which rendered unpredictability to a much greater extent than the chaos he had first discovered. An example of )] TJ ET
BT 26.250 506.881 Td /F1 9.8 Tf [(what he had in mind for multi-scale systems)] TJ ET
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BT 215.351 508.388 Td /F4 8.7 Tf [(6)] TJ ET
0.271 0.267 0.267 rg
BT 220.170 506.881 Td /F1 9.8 Tf [( would be a hurricane which scales at say 600 miles in diameter but contains )] TJ ET
BT 26.250 494.976 Td /F1 9.8 Tf [(mesoscale structures whose scales might be a fifty miles or less \(embedded super cell sub-cyclonic activity\), individual cloud )] TJ ET
BT 26.250 483.072 Td /F1 9.8 Tf [(formations with scales of miles, turbulent sub-cloud eddies with scales appreciably smaller, and so on. Lorenz pointed out that )] TJ ET
BT 26.250 471.167 Td /F1 9.8 Tf [(small errors in measurements of the finer structures tend to grow more rapidly and when these errors are large enough, they )] TJ ET
BT 26.250 459.262 Td /F1 9.8 Tf [(can effect measurements of coarser structures. Thus measure imprecision itself jumps up scales, at each new scale rapidly )] TJ ET
BT 26.250 447.357 Td /F1 9.8 Tf [(expanding, on up the scaling hierarchy.)] TJ ET
BT 26.250 427.953 Td /F1 9.8 Tf [(Moreover, at some point, because what is being studied is fluid dynamics, there would be a need to bring in heavy mathematical )] TJ ET
BT 26.250 416.048 Td /F1 9.8 Tf [(equipment like the Navier Stokes initial-boundary value approach to partial differential equations, and like methods. But this )] TJ ET
BT 26.250 404.143 Td /F1 9.8 Tf [(could quickly bring the whole mathematical modeling enterprise up against intractabitily, and thereby wind-up against )] TJ ET
BT 26.250 392.238 Td /F1 9.8 Tf [(intractability. It is this multi-scale complexity which Palmer, Doring, and Seregin conclude is what Lorenz meant by the real )] TJ ET
BT 26.250 380.334 Td /F1 9.8 Tf [(butterfly effect. Whereas improvements in measuring initial conditions might be of help in dealing with the explosion of )] TJ ET
BT 26.250 368.429 Td /F1 9.8 Tf [(unpredictability due to sensitivity to initial conditions, in Lorenzian multi-scale systems predictability estimates would not be )] TJ ET
BT 26.250 356.524 Td /F1 9.8 Tf [(extended in any significant way by such methods. Hence, certain formally deterministic fluid systems which possess many )] TJ ET
BT 26.250 344.619 Td /F1 9.8 Tf [(scales of motion are observationally indistinguishable from indeterministic systems.)] TJ ET
BT 26.250 325.215 Td /F1 9.8 Tf [(At around the same time as Lorenz was doing his real butterfly effect work and that both May and Feigenbaum were exploring )] TJ ET
BT 26.250 313.310 Td /F1 9.8 Tf [(the functional iteration route to chaos via bifurcation Edward Ott)] TJ ET
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BT 300.469 314.817 Td /F4 8.7 Tf [(8)] TJ ET
0.271 0.267 0.267 rg
BT 305.287 313.310 Td /F1 9.8 Tf [(, a colleague of James Yorke \(who had coined the term )] TJ ET
BT 26.250 301.405 Td /F1 9.8 Tf [(chaos in 1975\), was drawn toward another source kind unpredictability related to chaos but that went one better even than )] TJ ET
BT 26.250 289.500 Td /F1 9.8 Tf [(Lorenzs real butterfly effect: a )] TJ ET
BT 166.055 289.500 Td /F5 9.8 Tf [(riddled)] TJ ET
BT 195.315 289.500 Td /F1 9.8 Tf [( basin of attraction. A basin of attraction refers to all possible values taken by the )] TJ ET
BT 26.250 277.596 Td /F1 9.8 Tf [(variable\(s\) in question \(such as P in the logistic map\) which through the dynamical operations effectuated at particular values of )] TJ ET
BT 26.250 265.691 Td /F1 9.8 Tf [(the control parameter \(e.g., the birth/death rate in the logistic function\), are attracted to the attractors. For instance, the basin of )] TJ ET
BT 26.250 253.786 Td /F1 9.8 Tf [(attraction for a population which doesnt change at its yearly update \(Mays equilibrium fixed point\) are all the values possible for )] TJ ET
BT 26.250 241.881 Td /F1 9.8 Tf [(the population which wind-up at the same value at the fixed point attractor, that is, remain the same at the yearly measurement.)] TJ ET
BT 26.250 222.477 Td /F1 9.8 Tf [(But, in a riddled basin of attraction, there are points arbitrarily close to but not the same as the other initial conditions in the )] TJ ET
BT 26.250 210.572 Td /F1 9.8 Tf [(basin of attraction. However, these close-by initial points generate phase orbits that go to a different attractor! In measuring a )] TJ ET
BT 26.250 198.667 Td /F1 9.8 Tf [(system in this case, because the initial point is arbitrarily close to another and thus susceptible to measurement imprecision, it )] TJ ET
BT 26.250 186.762 Td /F1 9.8 Tf [(can be chosen instead as the value of the variable that will go through the nonlinear operation \(even the blow-up from sensitive )] TJ ET
BT 26.250 174.858 Td /F1 9.8 Tf [(dependence on initial conditions\) but leading to an entirely different phase portrait. As Ott puts it, the existence of riddled basins )] TJ ET
BT 26.250 162.953 Td /F1 9.8 Tf [(calls into question the repeatability of experiments for how does one know the initial conditions are of this particular basin of )] TJ ET
BT 26.250 151.048 Td /F1 9.8 Tf [(attraction and not that one?)] TJ ET
BT 26.250 131.643 Td /F1 9.8 Tf [(Bifurcations, sensitivity to initial conditions, multi-scale amplificafions, riddled basins of attraction, instabilitythese terms and )] TJ ET
BT 26.250 119.739 Td /F1 9.8 Tf [(their associated images would have us believe we are condemned to a hurly burly, chaotic, unstable universe with nowhere to )] TJ ET
BT 26.250 107.834 Td /F1 9.8 Tf [(stand that is determinate, stable, safe from the fluctuations of nature. Yet, there have been parallel developments as well which )] TJ ET
BT 26.250 95.929 Td /F1 9.8 Tf [(have uncovered contrasting metaphoric descriptions of complex systems. In the next issue of E:CO we will look at one such )] TJ ET
BT 26.250 84.024 Td /F1 9.8 Tf [(example where constancy replaces flux, and universality replaces the vicissitudes of specific contexts, notably the famous )] TJ ET
BT 26.250 72.120 Td /F1 9.8 Tf [(Feigenbaum constants which have shown universal constants at work in dynamical systems that reveal nature also consists of )] TJ ET
BT 26.250 60.215 Td /F1 9.8 Tf [(reservoirs of constancy.)] TJ ET
BT 26.250 40.810 Td /F1 9.8 Tf [(Mays original article was published as R.M. May \(1976\). Simple mathematical models with very complicated dynamics, )] TJ ET
Q
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BT 26.250 767.476 Td /F1 9.8 Tf [(conceptions of how life began out of much simpler autocatalytic chemical reactions, or even how the universe itself and its vast )] TJ ET
BT 26.250 755.571 Td /F1 9.8 Tf [(complicatedness and complexity emerged from the much simpler dynamics of a supposed singularity and a subsequent process )] TJ ET
BT 26.250 743.667 Td /F1 9.8 Tf [(of inflation. Not a few esteemed physicists or cosmologists seem to take such a cosmic drive for complexity for granted, even )] TJ ET
BT 26.250 731.762 Td /F1 9.8 Tf [(among the more sober and thoughtful like Frank Wilzcek.)] TJ ET
BT 26.250 712.357 Td /F1 9.8 Tf [(Helping to empower this supposed cosmic engine of complexity have been further discoveries in chaos theory which have )] TJ ET
BT 26.250 700.452 Td /F1 9.8 Tf [(uncovered even greater depths of complicatedness. Here Ill briefly described two such trends. First, according to Palmer, )] TJ ET
BT 26.250 688.548 Td /F1 9.8 Tf [(Doring, and Seregin)] TJ ET
0.267 0.267 0.267 rg
BT 112.957 690.055 Td /F4 8.7 Tf [(6)] TJ ET
0.271 0.267 0.267 rg
BT 117.775 688.548 Td /F1 9.8 Tf [( in their careful and thoughtful examination of Edward Lorenzs early work on the butterfly effect \(i.e., )] TJ ET
BT 26.250 676.643 Td /F1 9.8 Tf [(sensitivity to initial conditions\) at the end of the sixties and during the early seventies, it appears Lorenz was not yet finished )] TJ ET
BT 26.250 664.738 Td /F1 9.8 Tf [(with his earlier work of 1963 since over the next decade, he ventured into a far more unpredictable arena, that of multi-scale )] TJ ET
BT 26.250 652.833 Td /F1 9.8 Tf [(systems \(e.g., in a presentation from he gave in 1972\).)] TJ ET
BT 26.250 633.429 Td /F1 9.8 Tf [(Remember that in the original butterfly effect, unpredictability in chaotic sysems showed itself in sensitive dependence on initial )] TJ ET
BT 26.250 621.524 Td /F1 9.8 Tf [(conditions that would lead to a blowing-up of even the most miniscule margin of imprecision in a measurementhence a )] TJ ET
BT 26.250 609.619 Td /F1 9.8 Tf [(vastly different outcome than had been predicted. Although various ways were offered to mitigate the explosion of imprecision )] TJ ET
BT 26.250 597.714 Td /F1 9.8 Tf [(\(e.g., see the brilliant method recommended by Shaw)] TJ ET
0.267 0.267 0.267 rg
BT 257.101 599.222 Td /F4 8.7 Tf [(7)] TJ ET
0.271 0.267 0.267 rg
BT 261.919 597.714 Td /F1 9.8 Tf [(\), damage had been done to the age-old linkage of two of the )] TJ ET
BT 26.250 585.810 Td /F1 9.8 Tf [(cornerstones of scientific explanation, namely, determinism in the sense of a causal linkage and predictability of the outcome of )] TJ ET
BT 26.250 573.905 Td /F1 9.8 Tf [(the causation acting on the initial conditions. Lorenz, however, asked if there were other sources of unpredictability that would )] TJ ET
BT 26.250 562.000 Td /F1 9.8 Tf [(not even yield to improvements in the measurement of initial conditions, that is, were not due to just sensitive dependence on )] TJ ET
BT 26.250 550.095 Td /F1 9.8 Tf [(initial conditions.)] TJ ET
BT 26.250 530.691 Td /F1 9.8 Tf [(What he came up with was the presence of a certain kind of multi-scale constituency in a system \(again using weather systems )] TJ ET
BT 26.250 518.786 Td /F1 9.8 Tf [(as his forte\), which rendered unpredictability to a much greater extent than the chaos he had first discovered. An example of )] TJ ET
BT 26.250 506.881 Td /F1 9.8 Tf [(what he had in mind for multi-scale systems)] TJ ET
0.267 0.267 0.267 rg
BT 215.351 508.388 Td /F4 8.7 Tf [(6)] TJ ET
0.271 0.267 0.267 rg
BT 220.170 506.881 Td /F1 9.8 Tf [( would be a hurricane which scales at say 600 miles in diameter but contains )] TJ ET
BT 26.250 494.976 Td /F1 9.8 Tf [(mesoscale structures whose scales might be a fifty miles or less \(embedded super cell sub-cyclonic activity\), individual cloud )] TJ ET
BT 26.250 483.072 Td /F1 9.8 Tf [(formations with scales of miles, turbulent sub-cloud eddies with scales appreciably smaller, and so on. Lorenz pointed out that )] TJ ET
BT 26.250 471.167 Td /F1 9.8 Tf [(small errors in measurements of the finer structures tend to grow more rapidly and when these errors are large enough, they )] TJ ET
BT 26.250 459.262 Td /F1 9.8 Tf [(can effect measurements of coarser structures. Thus measure imprecision itself jumps up scales, at each new scale rapidly )] TJ ET
BT 26.250 447.357 Td /F1 9.8 Tf [(expanding, on up the scaling hierarchy.)] TJ ET
BT 26.250 427.953 Td /F1 9.8 Tf [(Moreover, at some point, because what is being studied is fluid dynamics, there would be a need to bring in heavy mathematical )] TJ ET
BT 26.250 416.048 Td /F1 9.8 Tf [(equipment like the Navier Stokes initial-boundary value approach to partial differential equations, and like methods. But this )] TJ ET
BT 26.250 404.143 Td /F1 9.8 Tf [(could quickly bring the whole mathematical modeling enterprise up against intractabitily, and thereby wind-up against )] TJ ET
BT 26.250 392.238 Td /F1 9.8 Tf [(intractability. It is this multi-scale complexity which Palmer, Doring, and Seregin conclude is what Lorenz meant by the real )] TJ ET
BT 26.250 380.334 Td /F1 9.8 Tf [(butterfly effect. Whereas improvements in measuring initial conditions might be of help in dealing with the explosion of )] TJ ET
BT 26.250 368.429 Td /F1 9.8 Tf [(unpredictability due to sensitivity to initial conditions, in Lorenzian multi-scale systems predictability estimates would not be )] TJ ET
BT 26.250 356.524 Td /F1 9.8 Tf [(extended in any significant way by such methods. Hence, certain formally deterministic fluid systems which possess many )] TJ ET
BT 26.250 344.619 Td /F1 9.8 Tf [(scales of motion are observationally indistinguishable from indeterministic systems.)] TJ ET
BT 26.250 325.215 Td /F1 9.8 Tf [(At around the same time as Lorenz was doing his real butterfly effect work and that both May and Feigenbaum were exploring )] TJ ET
BT 26.250 313.310 Td /F1 9.8 Tf [(the functional iteration route to chaos via bifurcation Edward Ott)] TJ ET
0.267 0.267 0.267 rg
BT 300.469 314.817 Td /F4 8.7 Tf [(8)] TJ ET
0.271 0.267 0.267 rg
BT 305.287 313.310 Td /F1 9.8 Tf [(, a colleague of James Yorke \(who had coined the term )] TJ ET
BT 26.250 301.405 Td /F1 9.8 Tf [(chaos in 1975\), was drawn toward another source kind unpredictability related to chaos but that went one better even than )] TJ ET
BT 26.250 289.500 Td /F1 9.8 Tf [(Lorenzs real butterfly effect: a )] TJ ET
BT 166.055 289.500 Td /F5 9.8 Tf [(riddled)] TJ ET
BT 195.315 289.500 Td /F1 9.8 Tf [( basin of attraction. A basin of attraction refers to all possible values taken by the )] TJ ET
BT 26.250 277.596 Td /F1 9.8 Tf [(variable\(s\) in question \(such as P in the logistic map\) which through the dynamical operations effectuated at particular values of )] TJ ET
BT 26.250 265.691 Td /F1 9.8 Tf [(the control parameter \(e.g., the birth/death rate in the logistic function\), are attracted to the attractors. For instance, the basin of )] TJ ET
BT 26.250 253.786 Td /F1 9.8 Tf [(attraction for a population which doesnt change at its yearly update \(Mays equilibrium fixed point\) are all the values possible for )] TJ ET
BT 26.250 241.881 Td /F1 9.8 Tf [(the population which wind-up at the same value at the fixed point attractor, that is, remain the same at the yearly measurement.)] TJ ET
BT 26.250 222.477 Td /F1 9.8 Tf [(But, in a riddled basin of attraction, there are points arbitrarily close to but not the same as the other initial conditions in the )] TJ ET
BT 26.250 210.572 Td /F1 9.8 Tf [(basin of attraction. However, these close-by initial points generate phase orbits that go to a different attractor! In measuring a )] TJ ET
BT 26.250 198.667 Td /F1 9.8 Tf [(system in this case, because the initial point is arbitrarily close to another and thus susceptible to measurement imprecision, it )] TJ ET
BT 26.250 186.762 Td /F1 9.8 Tf [(can be chosen instead as the value of the variable that will go through the nonlinear operation \(even the blow-up from sensitive )] TJ ET
BT 26.250 174.858 Td /F1 9.8 Tf [(dependence on initial conditions\) but leading to an entirely different phase portrait. As Ott puts it, the existence of riddled basins )] TJ ET
BT 26.250 162.953 Td /F1 9.8 Tf [(calls into question the repeatability of experiments for how does one know the initial conditions are of this particular basin of )] TJ ET
BT 26.250 151.048 Td /F1 9.8 Tf [(attraction and not that one?)] TJ ET
BT 26.250 131.643 Td /F1 9.8 Tf [(Bifurcations, sensitivity to initial conditions, multi-scale amplificafions, riddled basins of attraction, instabilitythese terms and )] TJ ET
BT 26.250 119.739 Td /F1 9.8 Tf [(their associated images would have us believe we are condemned to a hurly burly, chaotic, unstable universe with nowhere to )] TJ ET
BT 26.250 107.834 Td /F1 9.8 Tf [(stand that is determinate, stable, safe from the fluctuations of nature. Yet, there have been parallel developments as well which )] TJ ET
BT 26.250 95.929 Td /F1 9.8 Tf [(have uncovered contrasting metaphoric descriptions of complex systems. In the next issue of E:CO we will look at one such )] TJ ET
BT 26.250 84.024 Td /F1 9.8 Tf [(example where constancy replaces flux, and universality replaces the vicissitudes of specific contexts, notably the famous )] TJ ET
BT 26.250 72.120 Td /F1 9.8 Tf [(Feigenbaum constants which have shown universal constants at work in dynamical systems that reveal nature also consists of )] TJ ET
BT 26.250 60.215 Td /F1 9.8 Tf [(reservoirs of constancy.)] TJ ET
BT 26.250 40.810 Td /F1 9.8 Tf [(Mays original article was published as R.M. May \(1976\). Simple mathematical models with very complicated dynamics, )] TJ ET
Q
q
15.000 26.524 577.500 750.476 re W n
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BT 26.250 767.476 Td /F1 9.8 Tf [(conceptions of how life began out of much simpler autocatalytic chemical reactions, or even how the universe itself and its vast )] TJ ET
BT 26.250 755.571 Td /F1 9.8 Tf [(complicatedness and complexity emerged from the much simpler dynamics of a supposed singularity and a subsequent process )] TJ ET
BT 26.250 743.667 Td /F1 9.8 Tf [(of inflation. Not a few esteemed physicists or cosmologists seem to take such a cosmic drive for complexity for granted, even )] TJ ET
BT 26.250 731.762 Td /F1 9.8 Tf [(among the more sober and thoughtful like Frank Wilzcek.)] TJ ET
BT 26.250 712.357 Td /F1 9.8 Tf [(Helping to empower this supposed cosmic engine of complexity have been further discoveries in chaos theory which have )] TJ ET
BT 26.250 700.452 Td /F1 9.8 Tf [(uncovered even greater depths of complicatedness. Here Ill briefly described two such trends. First, according to Palmer, )] TJ ET
BT 26.250 688.548 Td /F1 9.8 Tf [(Doring, and Seregin)] TJ ET
0.267 0.267 0.267 rg
BT 112.957 690.055 Td /F4 8.7 Tf [(6)] TJ ET
0.271 0.267 0.267 rg
BT 117.775 688.548 Td /F1 9.8 Tf [( in their careful and thoughtful examination of Edward Lorenzs early work on the butterfly effect \(i.e., )] TJ ET
BT 26.250 676.643 Td /F1 9.8 Tf [(sensitivity to initial conditions\) at the end of the sixties and during the early seventies, it appears Lorenz was not yet finished )] TJ ET
BT 26.250 664.738 Td /F1 9.8 Tf [(with his earlier work of 1963 since over the next decade, he ventured into a far more unpredictable arena, that of multi-scale )] TJ ET
BT 26.250 652.833 Td /F1 9.8 Tf [(systems \(e.g., in a presentation from he gave in 1972\).)] TJ ET
BT 26.250 633.429 Td /F1 9.8 Tf [(Remember that in the original butterfly effect, unpredictability in chaotic sysems showed itself in sensitive dependence on initial )] TJ ET
BT 26.250 621.524 Td /F1 9.8 Tf [(conditions that would lead to a blowing-up of even the most miniscule margin of imprecision in a measurementhence a )] TJ ET
BT 26.250 609.619 Td /F1 9.8 Tf [(vastly different outcome than had been predicted. Although various ways were offered to mitigate the explosion of imprecision )] TJ ET
BT 26.250 597.714 Td /F1 9.8 Tf [(\(e.g., see the brilliant method recommended by Shaw)] TJ ET
0.267 0.267 0.267 rg
BT 257.101 599.222 Td /F4 8.7 Tf [(7)] TJ ET
0.271 0.267 0.267 rg
BT 261.919 597.714 Td /F1 9.8 Tf [(\), damage had been done to the age-old linkage of two of the )] TJ ET
BT 26.250 585.810 Td /F1 9.8 Tf [(cornerstones of scientific explanation, namely, determinism in the sense of a causal linkage and predictability of the outcome of )] TJ ET
BT 26.250 573.905 Td /F1 9.8 Tf [(the causation acting on the initial conditions. Lorenz, however, asked if there were other sources of unpredictability that would )] TJ ET
BT 26.250 562.000 Td /F1 9.8 Tf [(not even yield to improvements in the measurement of initial conditions, that is, were not due to just sensitive dependence on )] TJ ET
BT 26.250 550.095 Td /F1 9.8 Tf [(initial conditions.)] TJ ET
BT 26.250 530.691 Td /F1 9.8 Tf [(What he came up with was the presence of a certain kind of multi-scale constituency in a system \(again using weather systems )] TJ ET
BT 26.250 518.786 Td /F1 9.8 Tf [(as his forte\), which rendered unpredictability to a much greater extent than the chaos he had first discovered. An example of )] TJ ET
BT 26.250 506.881 Td /F1 9.8 Tf [(what he had in mind for multi-scale systems)] TJ ET
0.267 0.267 0.267 rg
BT 215.351 508.388 Td /F4 8.7 Tf [(6)] TJ ET
0.271 0.267 0.267 rg
BT 220.170 506.881 Td /F1 9.8 Tf [( would be a hurricane which scales at say 600 miles in diameter but contains )] TJ ET
BT 26.250 494.976 Td /F1 9.8 Tf [(mesoscale structures whose scales might be a fifty miles or less \(embedded super cell sub-cyclonic activity\), individual cloud )] TJ ET
BT 26.250 483.072 Td /F1 9.8 Tf [(formations with scales of miles, turbulent sub-cloud eddies with scales appreciably smaller, and so on. Lorenz pointed out that )] TJ ET
BT 26.250 471.167 Td /F1 9.8 Tf [(small errors in measurements of the finer structures tend to grow more rapidly and when these errors are large enough, they )] TJ ET
BT 26.250 459.262 Td /F1 9.8 Tf [(can effect measurements of coarser structures. Thus measure imprecision itself jumps up scales, at each new scale rapidly )] TJ ET
BT 26.250 447.357 Td /F1 9.8 Tf [(expanding, on up the scaling hierarchy.)] TJ ET
BT 26.250 427.953 Td /F1 9.8 Tf [(Moreover, at some point, because what is being studied is fluid dynamics, there would be a need to bring in heavy mathematical )] TJ ET
BT 26.250 416.048 Td /F1 9.8 Tf [(equipment like the Navier Stokes initial-boundary value approach to partial differential equations, and like methods. But this )] TJ ET
BT 26.250 404.143 Td /F1 9.8 Tf [(could quickly bring the whole mathematical modeling enterprise up against intractabitily, and thereby wind-up against )] TJ ET
BT 26.250 392.238 Td /F1 9.8 Tf [(intractability. It is this multi-scale complexity which Palmer, Doring, and Seregin conclude is what Lorenz meant by the real )] TJ ET
BT 26.250 380.334 Td /F1 9.8 Tf [(butterfly effect. Whereas improvements in measuring initial conditions might be of help in dealing with the explosion of )] TJ ET
BT 26.250 368.429 Td /F1 9.8 Tf [(unpredictability due to sensitivity to initial conditions, in Lorenzian multi-scale systems predictability estimates would not be )] TJ ET
BT 26.250 356.524 Td /F1 9.8 Tf [(extended in any significant way by such methods. Hence, certain formally deterministic fluid systems which possess many )] TJ ET
BT 26.250 344.619 Td /F1 9.8 Tf [(scales of motion are observationally indistinguishable from indeterministic systems.)] TJ ET
BT 26.250 325.215 Td /F1 9.8 Tf [(At around the same time as Lorenz was doing his real butterfly effect work and that both May and Feigenbaum were exploring )] TJ ET
BT 26.250 313.310 Td /F1 9.8 Tf [(the functional iteration route to chaos via bifurcation Edward Ott)] TJ ET
0.267 0.267 0.267 rg
BT 300.469 314.817 Td /F4 8.7 Tf [(8)] TJ ET
0.271 0.267 0.267 rg
BT 305.287 313.310 Td /F1 9.8 Tf [(, a colleague of James Yorke \(who had coined the term )] TJ ET
BT 26.250 301.405 Td /F1 9.8 Tf [(chaos in 1975\), was drawn toward another source kind unpredictability related to chaos but that went one better even than )] TJ ET
BT 26.250 289.500 Td /F1 9.8 Tf [(Lorenzs real butterfly effect: a )] TJ ET
BT 166.055 289.500 Td /F5 9.8 Tf [(riddled)] TJ ET
BT 195.315 289.500 Td /F1 9.8 Tf [( basin of attraction. A basin of attraction refers to all possible values taken by the )] TJ ET
BT 26.250 277.596 Td /F1 9.8 Tf [(variable\(s\) in question \(such as P in the logistic map\) which through the dynamical operations effectuated at particular values of )] TJ ET
BT 26.250 265.691 Td /F1 9.8 Tf [(the control parameter \(e.g., the birth/death rate in the logistic function\), are attracted to the attractors. For instance, the basin of )] TJ ET
BT 26.250 253.786 Td /F1 9.8 Tf [(attraction for a population which doesnt change at its yearly update \(Mays equilibrium fixed point\) are all the values possible for )] TJ ET
BT 26.250 241.881 Td /F1 9.8 Tf [(the population which wind-up at the same value at the fixed point attractor, that is, remain the same at the yearly measurement.)] TJ ET
BT 26.250 222.477 Td /F1 9.8 Tf [(But, in a riddled basin of attraction, there are points arbitrarily close to but not the same as the other initial conditions in the )] TJ ET
BT 26.250 210.572 Td /F1 9.8 Tf [(basin of attraction. However, these close-by initial points generate phase orbits that go to a different attractor! In measuring a )] TJ ET
BT 26.250 198.667 Td /F1 9.8 Tf [(system in this case, because the initial point is arbitrarily close to another and thus susceptible to measurement imprecision, it )] TJ ET
BT 26.250 186.762 Td /F1 9.8 Tf [(can be chosen instead as the value of the variable that will go through the nonlinear operation \(even the blow-up from sensitive )] TJ ET
BT 26.250 174.858 Td /F1 9.8 Tf [(dependence on initial conditions\) but leading to an entirely different phase portrait. As Ott puts it, the existence of riddled basins )] TJ ET
BT 26.250 162.953 Td /F1 9.8 Tf [(calls into question the repeatability of experiments for how does one know the initial conditions are of this particular basin of )] TJ ET
BT 26.250 151.048 Td /F1 9.8 Tf [(attraction and not that one?)] TJ ET
BT 26.250 131.643 Td /F1 9.8 Tf [(Bifurcations, sensitivity to initial conditions, multi-scale amplificafions, riddled basins of attraction, instabilitythese terms and )] TJ ET
BT 26.250 119.739 Td /F1 9.8 Tf [(their associated images would have us believe we are condemned to a hurly burly, chaotic, unstable universe with nowhere to )] TJ ET
BT 26.250 107.834 Td /F1 9.8 Tf [(stand that is determinate, stable, safe from the fluctuations of nature. Yet, there have been parallel developments as well which )] TJ ET
BT 26.250 95.929 Td /F1 9.8 Tf [(have uncovered contrasting metaphoric descriptions of complex systems. In the next issue of E:CO we will look at one such )] TJ ET
BT 26.250 84.024 Td /F1 9.8 Tf [(example where constancy replaces flux, and universality replaces the vicissitudes of specific contexts, notably the famous )] TJ ET
BT 26.250 72.120 Td /F1 9.8 Tf [(Feigenbaum constants which have shown universal constants at work in dynamical systems that reveal nature also consists of )] TJ ET
BT 26.250 60.215 Td /F1 9.8 Tf [(reservoirs of constancy.)] TJ ET
BT 26.250 40.810 Td /F1 9.8 Tf [(Mays original article was published as R.M. May \(1976\). Simple mathematical models with very complicated dynamics, )] TJ ET
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BT 25.000 19.825 Td /F1 11.0 Tf [(Emergence: Complexity and Organization)] TJ ET
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BT 26.250 767.476 Td /F1 9.8 Tf [(Nature, 261: 459-67. It can be downloaded from )] TJ ET
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BT 235.992 767.476 Td /F1 9.8 Tf [(here)] TJ ET
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BT 26.250 730.874 Td /F4 12.0 Tf [(References)] TJ ET
BT 26.250 703.419 Td /F1 9.8 Tf [(1.)] TJ ET
BT 38.132 703.419 Td /F1 9.8 Tf [(Goldstein, J. \(2011\). "Emergence in complex systems," in P. Allen, S. Maguire, and B. McKelvey \(eds.\), The Sage Handbook )] TJ ET
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