Bifurcation of a unique stable periodic orbit from a homoclinic orbit in infinite-dimensional systems
HTML articles powered by AMS MathViewer
- by Shui-Nee Chow and Bo Deng PDF
- Trans. Amer. Math. Soc. 312 (1989), 539-587 Request permission
Abstract:
Under some generic conditions, we show how a unique stable periodic orbit can bifurcate from a homoclinic orbit for semilinear parabolic equations and retarded functional differential equations. This is a generalization of a result of Šil’nikov for ordinary differential equations.References
- A. A. Andronov, E. A. Leontovich, I. I. Gordon, and A. G. Maĭer, Theory of bifurcations of dynamic systems on a plane, Halsted Press [John Wiley & Sons], New York-Toronto; Israel Program for Scientific Translations, Jerusalem-London, 1973. Translated from the Russian. MR 0344606 C. M. J. Blazqueg, Bifurcation from homoclinic orbits in parabolic equations, preprint.
- Shui Nee Chow and Jack K. Hale, Methods of bifurcation theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 251, Springer-Verlag, New York-Berlin, 1982. MR 660633
- John W. Evans, Neil Fenichel, and John A. Feroe, Double impulse solutions in nerve axon equations, SIAM J. Appl. Math. 42 (1982), no. 2, 219–234. MR 650218, DOI 10.1137/0142016 J. A. Feroe, Temporal stability of solitary impulse solutions of a nerve equation, Biophys. J. 21 (1978), 102-110.
- Tetsuo Furumochi, Existence of periodic solutions of one-dimensional differential-delay equations, Tohoku Math. J. (2) 30 (1978), no. 1, 13–35. MR 470418, DOI 10.2748/tmj/1178230094
- Jack K. Hale, Functional differential equations, Applied Mathematical Sciences, Vol. 3, Springer-Verlag New York, New York-Heidelberg, 1971. MR 0466837
- Daniel Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. MR 610244 Ju. I. Neimark and L. P. Šil’nikov, A case of generation of periodic motions, Soviet Math. Dokl. 6 (1965), 1261-1264. L. P. Šil’nikov, On the generation of periodic motion from trajectories doubly asymptotic to an equilibrium state of saddle type, Mat. Sb. 77 (1968), 427-438.
- Hans-Otto Walther, Bifurcation from a heteroclinic solution in differential delay equations, Trans. Amer. Math. Soc. 290 (1985), no. 1, 213–233. MR 787962, DOI 10.1090/S0002-9947-1985-0787962-4 —, Bifurcation from saddle connection in functional differential equations: an approach with inclination lemmas, preprint.
Additional Information
- © Copyright 1989 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 312 (1989), 539-587
- MSC: Primary 58F14; Secondary 34G20, 34K15, 35B32, 35R20
- DOI: https://doi.org/10.1090/S0002-9947-1989-0988882-6
- MathSciNet review: 988882