Bifurcations and Transitions to Chaos in the Three-Dimensional Lotka–Volterra Map

Abstract

The three-dimensional cyclically invariant Lotka–Volterra map is studied, pointing out, by analytic and numeric techniques, the bifurcations of fixed points that have no counterpart in the corresponding system of ordinary differential equations. In particular, the Hopf-bifurcation of the three-population equilibrium point, and the related dynamics in a bounded invariant domain, are analysed. The analytic description of stable periodic orbits in a weak-resonant case is given, and the transitions to chaotic attractors via sequences of Hopf-bifurcations and period-doublings are discussed.