Two-dimensional heteroclinic attractor in the generalized Lotka–Volterra system

Abstract

We study a simple dynamical model exhibiting sequential dynamics. We show that in this model there exist sets of parameter values for which a cyclic chain of saddle equilibria, Ok, , have two-dimensional unstable manifolds that contain orbits connecting each Ok to the next two equilibrium points Ok+1 and Ok+2 in the chain (). We show that the union of these equilibria and their unstable manifolds form a two-dimensional surface with a boundary that is homeomorphic to a cylinder if p is even and a Möbius strip if p is odd. If, further, each equilibrium in the chain satisfies a condition called ‘dissipativity’, then this surface is asymptotically stable.