Full Characterization of a Strange Attractor. Chaotic Dynamics in Low Dimensional Replicator Systems.

Wolfgang Schnabl, Peter F Stadler, Christian V Forst, Peter Schuster

Two chaotic attractors observed in Lotka-Volterra equations of dimension n=3 are shown to represent two different cross-sections of one and the same chaotic regime. The strange attractor is studied in the equivalent four dimensional catalytic replicator network. Analytical expression are derived for the Ljapunov exponents of the flow. In the centre of the chaotic regime the strange attractor is characterized by numerically computated R\'enyi fractal dimensions, Dq=2.04, 1.89, 1.65+-0.05 (q=0,1,2) as well as the Ljapunov dimension DL=2.06+-0.02. Accordingly it represents a multifractal. The fractal set is characterized by the singularity spectrum.
Two routes in parameter space leading into the chaotic regime were studied in detail, one corresponding to the Feigenbaum cascade of bifurcations. The second route is substantial different from this well known pathway and has some features in common with the intermittency route. A series of one-dimensional maps is derived from a properly chosen Poincar&eacut; cross-section which illustrates structural changes in the attractor.
Mutations are included in the catalytic replicator network and the changes in the dynamics observed are compared with the predictions of an approach based on pertubation theory. The most striking result is the gradual disappearance of complex dynamics with increasing mutation rates.

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