Logistic map

It would appear that the tradeoffs between these two requirements are optimized at the phase transition. Langton also cites a very similar relationship found by Crutchfield [crutch90] between a measure of machine complexity and the (per-symbol) entropy for the logistic map. The fact that the complexity/entropy relationship is so similar between two different classes of dynamical systems in turn suggests that what we are observing may be of fundamental importance complexity generically increases with randomness up until a phase transition is reached, beyond which further increases in randomness decrease complexity. We will have many occasions to return to this basic idea. 

In many ways, May s sentiment echoes the basic philosophy behind the study of CA, the elementary versions of which, as we have seen, are among the simplest conceivable dynamical systems. There are indeed many parallels and similarities between the behaviors of discrete-time dissipative dynamical systems and generic irreversible CA, not the least of which is the ability of both to give rise to enormously complicated behavior in an attractive fashion. In the subsections below, we introduce a variety of concepts and terminology in the context of two prototypical discrete-time mapping systems the one-dimensional Logistic map, and the two-dimensional Henon map. 

Behavior for a > aoo- What happens for a > Qoo The simple answer is that the logistic map exhibits a transition to chaos, with a variety of different attractors for Qoo < a < 4 exhibiting exponential divergence of nearby points. To leave it at that, however, would surely bo a great disservice to the extraordinarily beautiful manner in which this trairsition takes place. 

Let us begin by considering the stability of homogeneous solutions and/or initial-conditions i.e. by considering the stability of a simple-diffusive CML when cri(O) = a for all sites i , where a is a fixed point of the local logistic map F(cr) = acr(l—cr). Following Waller and Kapral [kapral84], we first recast equations 8.23 and 8.24... 

A natural question is how does the local period-doubling behavior cf the logistic map translate to its CML-version incarnation Without loss of generality, let us consider the Laplacian-coupled version of the logistic-driven CML ... 

This route should already be familiar to us from our discussion of the logistic map in chapter 4, Prom that chapter, we recall that the Feigenbaum route calls for a sequence of period-doubling bifurcations pitchfork bifurcations versus the Hopf bifurcations of the Landau-Hopf route) such that if subharmonic bifurcations are observed at Reynolds numbers TZi and 7 2, another can be expected at TZ determined by... 

This difference equation is called a logistic map, and represents a simple deterministic system, where given a yi one can calculate the consequent point y% and so on. We are interested in solutions yi > 0 with 6 > 0. This model describes the dynamics of a single species population [32]. For this map, the fixed points y on the first iteration are solutions of... 

The period doubling route to chaos is best illustrated with the help of the logistic map... 

This is a difference equation widely used as a model in ecology and population dynamics (May (1974, 1987), Gleick (1987), Devaney (1992), Ott (1993)). Let Xn be the (normalized) number of individuals of some biological species present in year n. Then, the prescription (1.2.1) predicts the number of individuals in the following year n -I-1. The logistic map... 

Fig. 1.9. Sensitivity of the logistic map to (a) initial conditions and (b) perturbation of the mapping itself.

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