A numerical toolbox for homoclinic bifurcation analysis

AR Champneys; YA Kuznetsov; B Sandstede

doi:10.1142/S0218127496000485

A numerical toolbox for homoclinic bifurcation analysis

AR Champneys, YA Kuznetsov, B Sandstede

Research output: Working paper › Working paper and Preprints

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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 language	English
DOIs	https://doi.org/10.1142/S0218127496000485
Publication status	Published - 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

Access to Document

10.1142/S0218127496000485

95r1Submitted manuscript, 424 KB

http://hdl.handle.net/1983/286

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title = "A numerical toolbox for homoclinic bifurcation analysis",

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",

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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 ",

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AU - Kuznetsov, YA

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N2 - 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

AB - 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

KW - homoclinic orbit

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