Homoclinic branch switching: A numerical implementation of Lin's method

BE Oldeman; AR Champneys; B Krauskopf

doi:10.1142/S0218127403008326

Homoclinic branch switching: A numerical implementation of Lin's method

BE Oldeman, AR Champneys, B Krauskopf

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Abstract

We present a numerical method for branch switching between homoclinic orbits to equilibria of ODEs computed via numerical continuation. Starting from a 1-homoclinic orbit our method allows us to find and follow an N-homoclinic orbit, for any N>1 (if it exists nearby). This scheme is based on Lin's method and it is robust and reliable.

The method is implemented in AUTO/HOMCONT. A system of ordinary differential equations introduced by Sandstede featuring inclination and orbit flip bifurcations and homoclinic-doubling cascades, is used as a test bed for the algorithm. It is also successfully applied to reliably find multihump traveling wave solutions in the FitzHugh–Nagumo nerve-axon equations and in a fourth-order Hamiltonian system arising as a model for water waves.

Translated title of the contribution	Homoclinic branch switching: A numerical implementation of Lin's method
Original language	English
Pages (from-to)	2977-2999
Number of pages	23
Journal	International Journal of Bifurcation and Chaos
Volume	13
Issue number	10
DOIs	https://doi.org/10.1142/S0218127403008326
Publication status	Published - 2003

Keywords

Homoclinic bifurcations
numerical continuation
branch switching

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10.1142/S0218127403008326

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Cite this

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title = "Homoclinic branch switching: A numerical implementation of Lin's method",

abstract = "We present a numerical method for branch switching between homoclinic orbits to equilibria of ODEs computed via numerical continuation. Starting from a 1-homoclinic orbit our method allows us to find and follow an N-homoclinic orbit, for any N>1 (if it exists nearby). This scheme is based on Lin's method and it is robust and reliable.The method is implemented in AUTO/HOMCONT. A system of ordinary differential equations introduced by Sandstede featuring inclination and orbit flip bifurcations and homoclinic-doubling cascades, is used as a test bed for the algorithm. It is also successfully applied to reliably find multihump traveling wave solutions in the FitzHugh–Nagumo nerve-axon equations and in a fourth-order Hamiltonian system arising as a model for water waves.",

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author = "BE Oldeman and AR Champneys and B Krauskopf",

year = "2003",

doi = "10.1142/S0218127403008326",

language = "English",

volume = "13",

pages = "2977--2999",

journal = "International Journal of Bifurcation and Chaos",

issn = "0218-1274",

publisher = "World Scientific Publishing Co.",

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TY - JOUR

T1 - Homoclinic branch switching

T2 - A numerical implementation of Lin's method

AU - Oldeman, BE

AU - Champneys, AR

AU - Krauskopf, B

PY - 2003

Y1 - 2003

N2 - We present a numerical method for branch switching between homoclinic orbits to equilibria of ODEs computed via numerical continuation. Starting from a 1-homoclinic orbit our method allows us to find and follow an N-homoclinic orbit, for any N>1 (if it exists nearby). This scheme is based on Lin's method and it is robust and reliable.The method is implemented in AUTO/HOMCONT. A system of ordinary differential equations introduced by Sandstede featuring inclination and orbit flip bifurcations and homoclinic-doubling cascades, is used as a test bed for the algorithm. It is also successfully applied to reliably find multihump traveling wave solutions in the FitzHugh–Nagumo nerve-axon equations and in a fourth-order Hamiltonian system arising as a model for water waves.

AB - We present a numerical method for branch switching between homoclinic orbits to equilibria of ODEs computed via numerical continuation. Starting from a 1-homoclinic orbit our method allows us to find and follow an N-homoclinic orbit, for any N>1 (if it exists nearby). This scheme is based on Lin's method and it is robust and reliable.The method is implemented in AUTO/HOMCONT. A system of ordinary differential equations introduced by Sandstede featuring inclination and orbit flip bifurcations and homoclinic-doubling cascades, is used as a test bed for the algorithm. It is also successfully applied to reliably find multihump traveling wave solutions in the FitzHugh–Nagumo nerve-axon equations and in a fourth-order Hamiltonian system arising as a model for water waves.

KW - Homoclinic bifurcations

KW - numerical continuation

KW - branch switching

U2 - 10.1142/S0218127403008326

DO - 10.1142/S0218127403008326

M3 - Article (Academic Journal)

VL - 13

SP - 2977

EP - 2999

JO - International Journal of Bifurcation and Chaos

JF - International Journal of Bifurcation and Chaos

SN - 0218-1274

IS - 10

ER -

Homoclinic branch switching: A numerical implementation of Lin's method

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Keywords

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