Abstract

This paper presents extensions and improvements of recently developed algorithms for the numerical analysis of orbits homoclinic to equilibria in ODEs and describes the implementation of these algorithms within the standard continuation package AUT086. This leads to a kind of toolbox, called HOMCONT, for analysing homoclinic bifurcations either as an aid to producing new theoretical results, or to understand dynamics arising from applications. This toolbox allows the continuation of codimension-one homoclinic orbits to hyperbolic or non-hyperbolic equilibria as well as detection and continuation of higher-order homoclinic singularities in more parameters. All known codimension-two cases that involve a unique homoclinic orbit are supported. Two specific example systems from ecology and chemical kinetics are analysed in some detail, allowing the reader to understand how to use the the toolbox for themselves. In the process, new results are also derived on these two particular models
Original languageEnglish
DOIs
Publication statusPublished - 1995

Bibliographical note

Additional information: Preprint of a paper later published by World Scientific (1996), International Journal of Bifurcation and Chaos, 6(5), pp. 867-887, ISSN 0218-1274

Sponsorship: The authors acknowledge the support of a Nuffield Foundation "Newly Appointed Science Lecturer" grant and a visiting fellowship grant from the EPSRC, UK

Keywords

  • homoclinic orbit
  • numerical analysis
  • bifurcation
  • continuation

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