The transition to chaos

A particularly important question is the exact nature of the transition to chaos at r 0 (r 1/2, respectively). For the box model discussed in this section, the transition to chaos is sudden (Bleher et al. (1990)). This means that at least some trajectories in box C are chaotic for any r with 0 r 1/2. Box C is regular only in two cases (i) for r = 0 (in which case C is identical to R), and (ii) for r = 1/2 (apart from the chaotic disconnected regions discussed above). No doubt this route to chaos is important, but rather abrupt. Other systems show more slowly developing, and thus more interesting, routes to chaos as a control parameter is varied. A particularly important route to chaos, the period doubling route to chaos is discussed in the following section. 

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 

Yet a third type of behaviour is displayed in Fig. 1.7(c). The iterates of xo never settle into any pattern but keep jumping irregularly in the interval [0,1]. Thus, depending on the value of the control parameter r, the iterates of (1.2.1) can display three qualitatively different kinds of asymptotic behaviour (i) convergent, (ii) cyclic, and (iii) chaotic. 

In many applications, and especially in population dynamics, one is not so much interested in the transient behaviour of (1.2.1), but rather in its asymptotic behaviour for n oo. Of special interest is the question whether the population described by (1.2.1) will settle down to some constant value Xoo(r) for n- oo, and here especially whether Xoo is finite or zero. In order for the as3onptotic value of Xn to be more significant, the result Xoo should be independent of the starting value xq of the population. This is possible if Xoo is an attractor of initial values 0 xq 1. For r r = 3.5699... this is indeed the case and Xoo(r) is defined independently of Xq. But, as we saw above, the function Xoo(r) is not always unique. In order to capture even cyclic asymptotic behaviour, the following procedure 3delds an excellent representation of Xqo (r) in a single 

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Reichl, L. E. The Transition to Chaos. (Springer-Verlag, New York, 2004). 

L. R. Reichl, The Transition to Chaos Quantum Manifestations, Springer, Berlin, 1992. 

M.L. Mehra, Random Matrices (Academic Press, New York 1967 and 1991) M. Carmeli, Statistical Theory and Random Matrices (Marcel Dekker, New York 1983) L.E. Reichl, The Transition to Chaos (Springer, New York 1992). 

Figure 19. Scattering resonances of 2F collinear models of the dissociation of H3 obtained by Manz et al. [143] on the Karplus-Porter surface and by Sadeghi and Skodje on a DMBE surface [132], The energy is defined with respect to the saddle point. The dashed lines mark the bifurcations of the transition to chaos 1, 5, and 3 are the numbers of shortest periodic orbits, or PODS, in each region.

Fig. 17. The transition to chaos (from a to d) observed in the work function during CO oxidation on Pt(l 10) while decreasing CO pressure. Chaos in the upper time series (d) was characterized by the Liapunov exponent, Kolmogorov entropy, and the embedding dimension (From Ref. 68.)...

This model was able to reproduce many kinds of non-linear behaviour including kinetic oscillations and the transition to chaos. Unfortunately the cumbersome character of the lattice gas... 

If the behaviour of the mutants Frl7 and HH201 did represent aperiodic oscillations, it would provide the first example of autonomous chaos at the cellular level, as well as an example of dynamic disease (Mackey Glass, 1977) in a unicellular organism. While the transition to chaos in Dictyostelium would result from some genetic mutation, the... 

N. De Leon and B. J. Berne,/. Chem. Phys., 75,3495 (1981). Intramolecular Rate Processes Isomerization Dynamics and the Transition to Chaos. 

In realistic models, of which those arising in chemistry are good otam-ples, the simple dynamics displayed by the sine circle map becomes more complex. It is possible that the sine circle map describes the dynamics of the system very close to and slightly beyond the transition to chaos, but that once one has gone well into the chaotic region in parameter space, this description no longer applies. 

A studyof a fairly simple model of an enzyme reaaion that exhibits chaotic behavior, the peroxidase-oxidase reaction, provides a good illustration of the role of circle map dynamics and mixed-mode oscillations in the transition to chaos. In the peroxidase-oxidase reaction, the peroxidase enzyme from horseradish (which, as its name implies, normally utilizes hydrogen peroxide as the electron acceptor) catalyzes an aerobic oxidation... 

Then we consider a path where the transition to chaos proceeds via the well - known cascade of period-doubling... 

Note the difference between the transition to chaos under the big lobe condition and without it in the second case the intervals Ai of chaotic dynamics may, in principle, interchange with the intervals where the system has only finitely many saddle and stable periodic orbits [151]. According to the reduction principle (Theorem 12.4), this occurs if, within some interval of u, the essential map... 

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