How to find a codimension-one heterocliniccycle between two periodic orbits

Wenjun Zhang; Bernd Krauskopf; Vivien Kirk

doi:10.3934/dcds.2012.32.2825

Article Contents

2012, Volume 32, Issue 8: 2825-2851. Doi: 10.3934/dcds.2012.32.2825

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How to find a codimension-one heteroclinic cycle between two periodic orbits

1.
Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland 1142

Received: July 31, 2011

Revised: October 31, 2011

Published: July 2012

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Abstract

Global bifurcations involving saddle periodic orbits have recently been recognized as being involved in various new types of organizing centers for complicated dynamics. The main emphasis has been on heteroclinic connections between saddle equilibria and saddle periodic orbits, called EtoP orbits for short, which can be found in vector fields in $\mathbb{R}^3$. Thanks to the development of dedicated numerical techniques, EtoP orbits have been found in a number of three-dimensional model vector fields arising in applications.
We are concerned here with the case of heteroclinic connections between two saddle periodic orbits, called PtoP orbits for short. A homoclinic orbit from a periodic orbit to itself is an example of a PtoP connection, but is generically structurally stable in a phase space of any dimension. The issue that we address here is that, until now, no example of a concrete vector field with a non-structurally stable PtoP connection was known. We present an example of a PtoP heteroclinic cycle of codimension one between two different saddle periodic orbits in a four-dimensional vector field model of intracellular calcium dynamics. We first show that this model is a good candidate system for the existence of such a PtoP cycle and then demonstrate how a PtoP cycle can be detected and continued in system parameters using a numerical setup that is based on Lin's method.

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Mathematics Subject Classification: 34C37, 34C23, 37G15.

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