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, $O_k$, $k=1,\dots ,p$, have two dimensional unstable manifolds that contain orbits connecting each $O_k$ to the next two equilibrium points $O_{k+1}$ and $O_{k+2}$ in the chain ($O_{p+1}=O_1$). We show that the union of these equilibria and their unstable manifolds form a $2$-dimensional surface with 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.

Cited by in corpus (6)

• Stability of cycling behaviour near a heteroclinic network model of Rock-Paper-Scissors-Lizard-Spock
• Heteroclinic Dynamics of Localized Frequency Synchrony: Heteroclinic Cycles for Small Populations
• Mind-to-mind heteroclinic coordination: model of sequential episodic memory initiation
• Asymptotic stability of robust heteroclinic networks
• Numerical continuation of spiral waves in heteroclinic networks of cyclic dominance
• Analysis of dynamics near heteroclinic networks in ${\mathbb{R}}^4$ with a projected map