<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>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.</p></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.

Predicting the separation of time scales in a heteroclinic network

Applied Mathematics and Nonlinear Sciences

This work is licensed under the Creative Commons Attribution 4.0 International License.

{"article-title":"Predicting the separation of time scales in a heteroclinic network"}

We consider a heteroclinic network in the framework of winnerless competition, realized by generalized Lotka-Volterra equations. By an appropriate choice of...

Predicting the separation of time scales in a heteroclinic network

Article Category: Papers dedicated to the memory of Valetnin Afraymovich (1945-2018)

Published Online: Jun 28, 2019

Page range: 279 - 288

Received: Nov 26, 2019

Accepted: May 03, 2019

DOI: https://doi.org/10.2478/AMNS.2019.1.00024

Keywords
winnerless competition, generalized Lotka-Volterra equations, heteroclinic dynamics, structural and temporal hierarchies

© 2019 Maximilian Voit, Hildegard Meyer-Ortmanns, published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Predicting the separation of time scales in a heteroclinic network

Article Category: Papers dedicated to the memory of Valetnin Afraymovich (1945-2018)

Published Online: Jun 28, 2019

Page range: 279 - 288

Received: Nov 26, 2019

Accepted: May 03, 2019

DOI: https://doi.org/10.2478/AMNS.2019.1.00024

Keywordswinnerless competition, generalized Lotka-Volterra equations, heteroclinic dynamics, structural and temporal hierarchies

© 2019 Maximilian Voit, Hildegard Meyer-Ortmanns, published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Keywords
winnerless competition, generalized Lotka-Volterra equations, heteroclinic dynamics, structural and temporal hierarchies