Transition to chaos

K. Kassner, C. Misbah, H. Miiller-Krumbhaar. Transition to chaos in directional solidification. Phys Rev Lett 67 1551, 1991. 

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. 

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). 

Bifurcation Scenario Associated with Transition to Chaos... 

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.

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. 

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... 

---