Two-dimensional heteroclinic attractor in the generalized Lotka–Volterra system. (5th April 2016) - Afraimovich, Valentin S; Moses, Gregory; Young, Todd
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, …, 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 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.