Applied Mathematics and Nonlinear Sciences
Journal license

Journal

Applied Mathematics and Nonlinear Sciences


Volume
& Issue

Volume 4, Issue 1


Published
on

June 28, 2019


Pages

279-288


DOI

Article

Predicting the separation of time scales in a heteroclinic network

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Authors

Maximilian Voit Affiliation:
Department of Physics and Earth Sciences, Jacobs University Bremen, 28759 Bremen Germany
and Hildegard Meyer-Ortmanns Affiliation:
Department of Physics and Earth Sciences, Jacobs University Bremen, 28759 Bremen Germany


Abstract

We consider a heteroclinic network in the framework of winnerless competition, realized by generalized Lotka-Volterra equations. By an appropriate choice of predation rates we impose a structural hierarchy so that the network consists of a heteroclinic cycle of three heteroclinic cycles which connect saddles on the basic level. As we have demonstrated in previous work, the structural hierarchy can induce a hierarchy in time scales such that slow oscillations modulate fast oscillations of species concentrations. Here we derive a Poincaré map to determine analytically the number of revolutions of the trajectory within one heteroclinic cycle on the basic level, before it switches to the heteroclinic connection on the second level. This provides an understanding of which parameters control the separation of time scales and determine the decisions of the trajectory at branching points of this network.


Keywords

winnerless competition, generalized Lotka-Volterra equations, heteroclinic dynamics, structural and temporal hierarchies, 37A60, 37D40


Citation

Voit, M. & Meyer-Ortmanns, H. (2019). Predicting the separation of time scales in a heteroclinic network. Applied Mathematics and Nonlinear Sciences, 4(1), 279–288. https://doi.org/10.2478/AMNS.2019.1.00024

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