The Logistic Map

Rt is ironic that such an intensely computational mathematical science as chaos theory owes much of its modern origin to calculations that were performed not on a large mainframe computer but rather a simple programmable pocket calculator, a Hewlett-Packard HP-65 Feigenbaum would explain years later that had the series of intermediate calculations not been carried out sufficiently slowly, it is likely that most of the key observations would have been missed [feigSOa]. 

As long as the single control parameter, a, is less than or equal to four, the orbit of any point xq (0,1) remains bounded on the unit interval. Notice also, that like the Bernoulli map, this map is noninvertible there are two antecedents, Xn and x, for each point Xn+i- We now want to study the behavior of orbits as a function of the parameter a. That the behavior of this map strongly depends on the value of a is easily appreciated from figure 4.3, which shows sample evolutions for Q = 0.5, 2.5, 3.3, 4. 

Fixed Point Solutions We begin by asking whether there are any values of a for which the system has fixed points. Solving the fixed-point equation 

Given a fixed point, x, the subsequent evolution of a nearby point, x  

We find that, regardless of the initial point Xq, the deviations from the fixed point x g decrease exponentially fast for all a 1 i.e., all points Xq e (0,1) are attracted to the fixed point x = 0 lim oo = 0. 

4 Control Parameters and the Emergence of Artificial Life 9.4.1 The Logistic Map 

For most efficient evolutionary design, this implies that there wLU exist an optimum condition, at which the compromise of the two conflicting requirements is found. Langton defined a complexity parameter A that determines such a conditionI l 

We learned in Chapter 8 of excitable systems and active media and in the previous sections of this chapter of complex autocatalytic reaction networks. Computational systems have been designed l that provide insights into the general condition by which complex behavior emerges. A more general understanding of the dynamic features that determine complex behavior is obtained from analysis of the the so-called logistic map[ land Wolfram s two-dimensional cellular automata studies 

It will appear especially from Wolfram s work that a priori prediction of macroscopic behavior even for many particle systems that follow simple interaction rules is often not possible. The behavior can be sensitive to initial conditions and disturbances. These are, of course, conditions that are optimum for a learning system, where microscopic rules have to be adapted to macroscopic requirements. The optimum condition for emergence of life-like multiplication appears close to conditions where the system behavior becomes unpredictable. An important feature of a living cell is its finite lifetime. Sustained existence 

In order to introduce the concept of a control parameter, we will analyze first the logistic map to illustrate more precisely the different phases of self organization that one can distinguish. We will follow closely the work by NicolisP l. The logistic map is a generalized equation that describes the development of a population x due to growth and decline by competition between the species  

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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. 

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. 

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... 

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.

The chaotic behaviour of box C shows that questions of measurement theory and the concept of predictabifity are not just at the foundations of quantum mechanics, but enter in an equally profound way already on the classical level. This was recently emphasized by Sommerer and Ott in an article by Naeye (1994). They argue that in addition to the problem of predictability the problem of reproducability of measurements in classically chaotic systems has to be discussed. The results of Fig. 1.9 indicate that the logistic map displays similar complexity. In fact, regions which act sensitively to initial conditions, intertwined with regions where prediction is possible, are generic in classical particle dynamics. 

Nonlinear mapping functions, such as the function fr of the logistic mapping discussed in Section 1.2, are the most important and the most useful type of mapping functions for the theory of chaos. Although usually quite innocuous in appearance fr x), e.g., is a simple quadratic function of x), they can produce astonishingly complex orbits when used in iteration prescriptions such as (2.2.1). We encountered examples of this complexity in Section 1.2 (see Figs. 1.7 - 1.9). 

For example, xq = 1 — 1/r is a fixed point of the logistic map fr. The fixed point is a special type oiperiodic orbit. Suppose that for a particular mapping function / we have... 

In order to gain some experience with the new concepts introduced above, we will now discuss the stability properties of the fixed points and periodic orbits of the logistic mapping. The following is also a more in-depth presentation of the period doubling scenario briefly discussed in Section 1.2. 

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