The Logistic Map

Recall the exotic dynamics of the logistic map (see for example, Rasband, 1990; Peitgen, et al., 1992; Beck & Schlögl, 1993):

Xk+1 = a Xk · (1 – Xk)

that is, the chain of ultimately stable (and unstable) values X8(a) found iterating the map, where Xks denotes the normalized size of a population at generation k and a is a free parameter having values between 0 and 4:

When a = 1, the logistic parabola is below the one to one line (added to aid in the calculations), and then X8 = 0 (Figure 1);

When 1 < a = 3, the parabola is above the line X = Y and X8 = (a – 1)/a , the non-zero intersection between the curve and the straight line (Figure 2);

When 3 < a = 3.449…, X8 = {X8(1), X8(2)} and the population settles into an oscillation repeating every two generations (Figure 3);

When 3.449… < a = 3.544…, X8 = {X8(3), X8(4), X8(5), X8(6)}. The population ultimately repeats every four generations, and the dynamics have experienced a bifurcation (Figure 4);

When a is increased up to a value a8= 3.5699…, successive bifurcations in powers of two happen quickly, that is, the dynamics repeat exactly every 2n generations, for any value of n;

When a8 < a = 4, behavior is found either periodic or non periodic. For instance, for a = 3.6 an infinite strange attractor with a whole in the middle is found (Figure 5);

When a = 3.83, X8 = {X8(1), X8(2), X8(3)} and the dynamics oscillate every 3 generations (Figure 6);

When a = 4, the most common behavior is non periodic and a dense strange attractor over the interval [0, 1] is found (Figure 7).

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Fig. 1: Convergence to the origin

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Fig. 2: Convergence to a fixed point

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Fig. 3: Convergence to a 2-cycle.

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Fig. 4: Convergence to a 4-cycle

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Fig. 5: Convergence to a “dusty” non-repetitive attractor

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Fig. 6: Convergence to a 3-cycle

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Fig. 7: Convergence to a maximal non-repetitive set

At the end, the cascade of stable period-doubling bifurcations (before a8) and the emergence of chaos (strange attractors) intertwined with periodic behavior (including any period greater than two) is summarized via the celebrated Feigenbaum’s diagram (Figure 8).

This is so named after Mitchell Feigenbaum who showed that the bifurcation openings and their durations happen universally for a class of unimodal maps according to two universal constants F1 and F2, as follows (Feigenbaum, 1978) (refer to Figure 9):

dn/dn+1 ? F1 = -2.5029…, ?n/?n+1 ? F2 = 4.6692…

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Fig. 8: Bifurcations tree for the logistic map

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Fig. 9: Successive bifurcations

x̄ corresponds to supercycles, i.e., ½ for the logistic map

For example, other “fig trees” guided by F1 and F2 and for the two simple mappings f(X) = a X · (1 — X 3) and f(X) = a X · (1 – X)3 are shown below[1]. Notice how such contain: a straight “root,” a bent “branch,” bifurcation branches, and then, in an orderly intertwined fashion, following Sharkovskii’s order (see for example, Rasband, 1990; Peitgen, et al., 1992; Beck & Schlögl, 1993), periodic branches, and the ever dusty “foliage of chaos,” where the unforgiving condition of sensitivity to initial conditions rules.

Chaos theory and our quest for peace

As the dynamics of the logistic map describe several physical processes (see for instance, Cvitanovic, 1989; Bai-Lin. 1984), including fluid turbulence as induced by heating, that is, convection, it is pertinent to consider such a simple and universal mechanism to study how “chaos” and its related condition of “violence” may arise in the world.

Given that the key parameter a, associated with the amount of heat (Libchaber & Maurer, 1978), dictates the ultimate organization of the fluid, we may see that it is wise to keep it small (in the world, and within each one of us) in order to avoid undesirable “nonlinearities.” For although the allegorical fig trees exhibit clear order in their pathway towards disorder, we may appreciate in the uneasy jumping on strange attractors (and also on periodic ones), the anxious and foolish frustration we often experience (so many times deterministically!) when we, by choosing to live in a hurry, travel from place to place to place in “high heat” without finding our “root.”

In this spirit, the best solution for each one of us is to slow down altogether the pace of life, coming down the tree, so that by not crossing the main thresholdX = Y, that is, by choosing a = 1, we may surely live without turbulence and chaos in the robust state symbolized by X8 = 0[2,3]. For there is a marked difference between a seemingly laminar condition as it happens through tangent bifurcations (see for example, Rasband, 1990; Peitgen, et al., 1992; Beck & Schlögl, 1993) and being truly at peace, for the former invariably contains dramatic bursts of chaos and ample intermittency (see for example, Rasband, 1990; Peitgen, et al., 1992; Beck & Schlögl, 1993).

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Fig. 10: Bifurcation diagrams for two simple nonlinear maps

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Fig. 11: Orbits of the logistic map (a = 4) ending up at zero

As zero, that is, converging to the origin, is identified as the desired state, it is sensible to realize that such an organization, a trivial solution for X8(a), even if unstable, may be reached even when the worst chaos engulfs us (a = 4). For the precise dynamics of the pre-images of zero do not wander for ever in high heat, but rather find the way to the root through a delicate hopscotch by the middle[4] (Figure 11). For it is tragic indeed to “oscillate for ever” (Figure 12). And more tragic yet to be close to “the point” and miss it altogether forever[5] (Figure 13). For the butterfly effect, with all probability and contrary to the illusion that it provides us with options, leaves us irremediably trapped in dust.

At the end, the emergence of the modern science of complexity helps us visualize our ancient choices. It is indeed best for us to live in serenity and in a simple manner, not amplifying and hence heeding the voice. For only the conscious order of Love does not suffer the destiny of arrogant stubbornness that justly receives the same “bad luck” of a parabolic tree that did not have any fruit, the same one that with its tender branch(es) and budding leaves, also announces horrendous times, but also very good ones, times of joy and of friendship.