Heteroclinic and homoclinic bifurcations in bistable reaction diffusion systems

Kokubu, Hiroshi; Nishiura, Yasumasa; Oka, Hiroe

doi:10.1016/0022-0396(90)90033-l

Cited by 31 publications

(33 citation statements)

References 14 publications

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“…(2) Proceed to more degenerate cases. Results in stage (1) go back to early works by Evans, Fenichel and Feroe [5], Yanagida [21], Yanagida and Maginu [22] Kokubu, Nishiura and Oka [13] and Deng [4]. Later results include Alexander and Jones [2] [3], Gardner [6], Kan-on [10] [11], Nii [14] [15] [16] and Sandstede [18] [19].…”

Section: Introductionmentioning

confidence: 77%

“…W U (Q)). In [13], a function Si(//) (resp. ^2 (1^)) is defined to analyze heteroclinic orbits from P to Q (resp.…”

Section: \0jmentioning

confidence: 99%

“…unstable) as a solution of the original system (1.1); the same applies for £2. In [13], the existence of homoclinic orbits to P and Q bifurcating from the transversal intersection of Mi 7^ and M2 7s)c are obtained by applying homoclinic bifurcation theorem by Kokubu [12]. That is, when Mi 7j|t and M2 7st!…”

Section: \0jmentioning

confidence: 99%

“…Later results include Alexander and Jones [2] [3], Gardner [6], Kan-on [10] [11], Nii [14] [15] [16] and Sandstede [18] [19]. [13] and [14] treat bifurcation of pulses from coexisting front and back waves under certain transversality conditions. Works belonging to stage (2) have already done by Sandstede, Alexander and Jones [20], Yew [23] and Nii and Sandstede [17].…”

Section: Introductionmentioning

confidence: 99%

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Pitchfork and Hopf bifurcations of traveling pulses generated by coexisting front and back waves

Nii¹

2000

Methods and Applications of Analysis

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Abstract. Pitchfork and Hopf bifurcations of traveling pulse solutions bifurcating from coexisting traveling front and back waves of reaction-diffusion systems are studied. It is assumed that the parameter set on which the traveling front exists and the set on which the back wave exists intersect non-transversally. As a result, more complicated bifurcations than the transversal case are proven to occur, including pitchfork and Hopf bifurcations of traveling pulses. Introduction.Recently, bifurcation phenomena of traveling waves and their stability in parabolic systems has been attracting attention. Such systems include reaction-diffusion systems, where the waves are thought to represent qualitative change of propagation of transition layers of chemical substances, and nerve axon equations, where they are thought to represent conduction of electric pulses. These waves are expected to be stable if they are observed experimentally. This problem has already commanded a large body of literature.One way to study this subject is to regard the bifurcations and the linear stability of traveling waves as bifurcations of homoclinic or heteroclinic solutions of ordinary differential equations.A natural strategy in this line of thought is as follows: (1 The subject of this paper is bifurcation from coexisting front and back waves which do not satisfy transversality conditions. This type of degeneracy appears in [13] and belongs to stage (2) above.In [13], the following type of one-dimensional reaction diffusion system is treated:[eTUt=e 2 u xx + f (u,v' i 0,j) \ vt =v xx +g (u,v',6,'y) where x € M. and t > 0. e and r are real positive parameters, with e a small parameter. The nullcline of / intersects with that of g at P - (up,vp), R - (UR,VR) and Q - (UQ^VQ) where vp < VR < VQ. P and Q are stable constant solutions of (1.1), which guarantees the bistability, while R is unstable. *

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Section: Introductionmentioning

confidence: 77%

“…W U (Q)). In [13], a function Si(//) (resp. ^2 (1^)) is defined to analyze heteroclinic orbits from P to Q (resp.…”

Section: \0jmentioning

confidence: 99%

Section: \0jmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Pitchfork and Hopf bifurcations of traveling pulses generated by coexisting front and back waves

Nii¹

2000

Methods and Applications of Analysis

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“…Their approach, which is based upon the classical theory of singular perturbations and matched asymptotic expansions, has led to some beautiful and deep rigorous results on the stability and bifurcation of waves in the singular limit [16]. In Sattinger's paper, stability with respect to perturbations with exponential spatial decay is proved in certain situations in which the linearization has continous spectrum passing through the origin.…”

mentioning

confidence: 99%

Book Review: Traveling wave solutions of parabolic systems

Gardner¹

1995

Bull. Amer. Math. Soc.

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446BOOK REVIEWS the predecessor volume [6] at hand. Many further results are given as exercises, with solutions or references.Besides, I found no serious lacuna in the index and very few misprints in the book, none of them bothersome. (At the author's request, I mention that the remark on page 274 should be deleted.) As in [5 and 6], for which Helgason received the 1988 Steele Prize for expository writing, the style is very fluent and pleasant, conducting the reader at a regular pace. I think the present book will be a most valuable (and reasonably priced) reference for anyone interested in Radon transforms and analysis on semisimple Lie groups. 4. V.

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Fast and Slow Waves in the FitzHugh–Nagumo Equation

Krupa

Sandstede

Szmolyan

1997

Journal of Differential Equations

114

116

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It is known that the FitzHugh Nagumo equation possesses fast and slow travelling waves. Fast waves are perturbations of singular orbits consisting of two pieces of slow manifolds and connections between them, whereas slow waves are perturbations of homoclinic orbits of the unperturbed system. We unfold a degenerate point where the two types of singular orbits coalesce forming a heteroclinic orbit of the unperturbed system. Let c denote the wave speed and = the singular perturbation parameter. We show that there exists a C 2 smooth curve of homoclinic orbits of the form (c, =(c)) connecting the fast wave branch to the slow wave branch. Additionally we show that this curve has a unique non-degenerate maximum. Our analysis is based on a Shilnikov coordinates result, extending the Exchange Lemma of Jones and Kopell. We also prove the existence of inclinationflip points for the travelling wave equation thus providing the evidence of the existence of n-homoclinic orbits (n-pulses for the FitzHugh Nagumo equation) for arbitrary n. Academic Press

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Heteroclinic and homoclinic bifurcations in bistable reaction diffusion systems

Cited by 31 publications

References 14 publications

Pitchfork and Hopf bifurcations of traveling pulses generated by coexisting front and back waves

Pitchfork and Hopf bifurcations of traveling pulses generated by coexisting front and back waves

Book Review: Traveling wave solutions of parabolic systems

Fast and Slow Waves in the FitzHugh–Nagumo Equation

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