Two dimensional heteroclinic attractor in the generalized Lotka-Volterra system

TL;DR

This paper analyzes a dynamical system based on the generalized Lotka-Volterra model, revealing a 2D heteroclinic attractor structure formed by saddle equilibria and their unstable manifolds, which can be stable and resemble a cylinder or Möbius strip.

Contribution

It demonstrates the existence and stability of a 2D heteroclinic attractor in a generalized Lotka-Volterra system with novel topological properties.

Findings

Existence of a 2D surface formed by saddle equilibria and their unstable manifolds.

The surface is homeomorphic to a cylinder or Möbius strip depending on p.

Under dissipativity, the surface is asymptotically stable.

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, $O_k$, $k=1, \ldots, p$, 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 M\"{o}bius strip if $p$ is odd. If, further, each equilibrium in the chain satisfies a condition called ``dissipativity," then this surface is asymptotically stable.