Journal Article

A hierarchical heteroclinic network

Voit, Maximilian, Meyer-Ortmanns, Hildegard

2018

Overview

We consider a heteroclinic network in the framework of winnerless competition of species. It consists of two levels of heteroclinic cycles. On the lower level, the heteroclinic cycle connects three saddles, each representing the survival of a single species; on the higher level, the cycle connects three such heteroclinic cycles, in which nine species are involved. We show how to tune the predation rates in order to generate the long time scales on the higher level from the shorter time scales on the lower level. Moreover, when we tune a single bifurcation parameter, first the motion along the lower and next along the higher-level heteroclinic cycles are replaced by a heteroclinic cycle between 3-species coexistence-fixed points and by a 9-species coexistence-fixed point, respectively. We also observe a similar impact of additive noise. Beyond its usual role of preventing the slowing-down of heteroclinic trajectories at small noise level, its increasing strength can replace the lower-level heteroclinic cycle by 3-species coexistence fixed-points, connected by an effective limit cycle, and for even stronger noise the trajectories converge to the 9-species coexistence-fixed point. The model has applications to systems in which slow oscillations modulate fast oscillations with sudden transitions between the temporary winners.

• Publisher: Springer Berlin Heidelberg,Springer Nature B.V
• Subject: Advances in Nonlinear Dynamics of Complex Networks: Adaptivity / Atomic / Bifurcations / Classical and Continuum Physics / Competition / Condensed Matter Physics / Delays / Materials Science / Measurement Science and Instrumentation / Molecular / Noise levels / Optical and Plasma Physics / Ordinary differential equations / Oscillations / Physics / Physics and Astronomy / Predation / Regular Article / Saddles / Stochasticity / Time