Breathing pulses in singularly perturbed reaction-diffusion systems

The weakly nonlinear stability of pulses in general singularly perturbed reaction-diffusion systems near a Hopf bifurcation is determined using a centre manifold expansion. A general framework to obtain leading order expressions for the (Hopf) centre manifold expansion for scale separated, localised...

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Published inNonlinearity Vol. 28; no. 7; pp. 2211 - 2246
Main Author Veerman, Frits
Format Journal Article
LanguageEnglish
Published IOP Publishing 01.07.2015
Subjects
Breathing
breathing pulse
centre manifold expansion
Direct numerical simulation
Hopf bifurcation
Manifolds
Mathematical models
Nonlinearity
oscillon
reaction-diffusion
singular perturbation
Stability
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ISSN0951-7715
1361-6544
DOI10.1088/0951-7715/28/7/2211

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Abstract The weakly nonlinear stability of pulses in general singularly perturbed reaction-diffusion systems near a Hopf bifurcation is determined using a centre manifold expansion. A general framework to obtain leading order expressions for the (Hopf) centre manifold expansion for scale separated, localised structures is presented. Using the scale separated structure of the underlying pulse, directly calculable expressions for the Hopf normal form coefficients are obtained in terms of solutions to classical Sturm-Liouville problems. The developed theory is used to establish the existence of breathing pulses in a slowly nonlinear Gierer-Meinhardt system, and is confirmed by direct numerical simulation.
AbstractList The weakly nonlinear stability of pulses in general singularly perturbed reaction-diffusion systems near a Hopf bifurcation is determined using a centre manifold expansion. A general framework to obtain leading order expressions for the (Hopf) centre manifold expansion for scale separated, localised structures is presented. Using the scale separated structure of the underlying pulse, directly calculable expressions for the Hopf normal form coefficients are obtained in terms of solutions to classical Sturm-Liouville problems. The developed theory is used to establish the existence of breathing pulses in a slowly nonlinear Gierer-Meinhardt system, and is confirmed by direct numerical simulation.
Author Veerman, Frits
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StartPage 2211
SubjectTerms Breathing
breathing pulse
centre manifold expansion
Direct numerical simulation
Hopf bifurcation
Manifolds
Mathematical models
Nonlinearity
oscillon
reaction-diffusion
singular perturbation
Stability
Title Breathing pulses in singularly perturbed reaction-diffusion systems
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