Full characterization of a strange attractor: Chaotic dynamics in low-dimensional replicator systems

Abstract

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 expressions are derived for the Lyapunov exponents of the flow. In the centre of the chaotic regime the strange attractor was characterized by the Lyapunov dimension ( D L = 2.06 ± 0.02) and the Rényi fractal dimensions. The set corresponding to the attractor represents a multifractal. Its singularity spectrum is computed. One route in parameter space leading into the chaotic regime and crossing it was studied in detail. It leads to a Feigenbaum-type cascade of bifurcations. Before the chaos is fully developed the dynamic system passes an internal crisis through a period-two tangent bifurcation. Then a second sequence of period-doubling bifurcations leading to another chaotic regime is observed, which eventually ends in a crisis and then after a rather complicated periodic regime the attractor finally disappears. Trajectories in the intermediate periodic regime show transient chaotic bursts of intermittency type. A series of one-dimensional maps is derived from a properly chosen Poincaré 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 perturbation theory. The most striking result is the gradual disappearance of complex dynamics with increasing mutation rates. The parameter space spanned by the mutation rate and selected parameters of the replication network is split in regions of complex periodic behaviour near the chaotic regime. Fractal boundaries of these regions are likely to occur but the data available do not allow definitive conclusions yet.