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The European Physical Journal Special Topics

, Volume 227, Issue 10–11, pp 1101–1115 | Cite as

A hierarchical heteroclinic network

Controlling the time evolution along its paths
  • Maximilian VoitEmail author
  • Hildegard Meyer-OrtmannsEmail author
Regular Article
Part of the following topical collections:
  1. Advances in Nonlinear Dynamics of Complex Networks: Adaptivity, Stochasticity, Delays

Abstract

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.

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Copyright information

© EDP Sciences, Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Physics and Earth Sciences, Jacobs University BremenBremenGermany

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