91-00-004
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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