An Extension of the Stability Index for Traveling-Wave Solutions and Its Application to Bifurcations
Abstract
We treat the stability index for traveling-wave solutions of one-dimensional reaction-diffusion equations due to Alexander, Gardner, and Jones [J. Reine Angew. Math., 410 (1990), pp. 167--212]. An extension of the stability index which makes the index robust to perturbation is given and, using the extension, an additive formula for a gluing bifurcation of traveling waves is proven. Wealso consider certain heteroclinic bifurcations as an application, some specific examples of which are discussed.
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References
1.
J. Alexander, R. Gardner, and C. Jones, A topological invariant arising in the stability analysis of travelling waves, J. Reine Angew. Math., 410 (1990), pp. 167–212.
2.
J. Alexander, C. Jones, Existence and stability of asymptotically oscillatory double pulses, J. Reine Angew. Math., 446 (1994), 49–79
3.
J. Alexander, C. Jones, Existence and stability of asymptotically oscillatory triple pulses, Z. Angew. Math. Phys., 44 (1993), 189–200
4.
Peter Bates, Christopher Jones, Invariant manifolds for semilinear partial differential equations, Dynam. Report. Ser. Dynam. Systems Appl., Vol. 2, Wiley, Chichester, 1989, 1–38
5.
Bo Deng, The bifurcations of countable connections from a twisted heteroclinic loop, SIAM J. Math. Anal., 22 (1991), 653–679
6.
John Evans, Nerve axon equations. III. Stability of the nerve impulse, Indiana Univ. Math. J., 22 (1972/73), 577–593
7.
J. Evans, Nerve axon equations IV: The stable and unstable impulses, Indiana Univ. Math. J., 24 (1975), pp. 1169–1190.
8.
J.‐M. Gambaudo, P. Glendinning, C. Tresser, The gluing bifurcation. I. Symbolic dynamics of the closed curves, Nonlinearity, 1 (1988), 203–214
9.
Daniel Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, Vol. 840, Springer‐Verlag, 1981iv+348
10.
Hideo Ikeda, Singular pulse wave bifurcations from front and back waves in bistable reaction‐diffusion systems, Methods Appl. Anal., 3 (1996), 285–317
11.
H. Ikeda, Existence and stability of pulse waves bifurcated from front and back waves in reaction‐diffusion systems, preprint.
12.
Christopher Jones, Stability of the travelling wave solution of the FitzHugh‐Nagumo system, Trans. Amer. Math. Soc., 286 (1984), 431–469
13.
Yukio Kan‐on, Parameter dependence of propagation speed of travelling waves for competition‐diffusion equations, SIAM J. Math. Anal., 26 (1995), 340–363
14.
Yukio Kan‐on, Existence of standing waves for competition‐diffusion equations, Japan J. Indust. Appl. Math., 13 (1996), 117–133
15.
Y. Kan‐on, Instability of stationary solutions for Lotka–Volterra competition model with diffusion, preprint.
16.
Yukio Kan‐on, Qing Fang, Stability of monotone travelling waves for competition‐diffusion equations, Japan J. Indust. Appl. Math., 13 (1996), 343–349
17.
Hiroshi Kokubu, Homoclinic and heteroclinic bifurcations of vector fields, Japan J. Appl. Math., 5 (1988), 455–501
18.
Hiroshi Kokubu, Yasumasa Nishiura, Hiroe Oka, Heteroclinic and homoclinic bifurcations in bistable reaction diffusion systems, J. Differential Equations, 86 (1990), 260–341
19.
Shunsaku Nii, ‐homoclinic bifurcations for homoclinic orbits changing their twisting, J. Dynam. Differential Equations, 8 (1996), 549–572
20.
Eiji Yanagida, Stability of fast travelling pulse solutions of the FitzHugh‐Nagumo equations, J. Math. Biol., 22 (1985), 81–104
21.
Eiji Yanagida, Kenjiro Maginu, Stability of double‐pulse solutions in nerve axon equations, SIAM J. Appl. Math., 49 (1989), 1158–1173
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Published In
SIAM Journal on Mathematical Analysis
Pages: 402 - 433
ISSN (online): 1095-7154
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Copyright © 1997 Society for Industrial and Applied Mathematics.
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Published online: 1 August 2006
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