Two-dimensional heteroclinic attractor in the generalized Lotka–Volterra system. (5th April 2016)

• Record Type: Journal Article
• Authors: Afraimovich, Valentin S; Moses, Gregory; Young, Todd
• Is Part Of: Nonlinearity. Volume 29:Number 5(2016:May)
• Page Start: 1645
• Page End: 1644
• Publication Date: 2016-04-05
• Subjects: heteroclinic orbits -- Lotka–Volterra equations -- unstable manifolds; 34A34; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/0951-7715/29/5/1645
• Languages: English
• ISSNs: 0951-7715

Abstract

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.