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Bifurcation of homoclinics
Author:
Jacobo Pejsachowicz
Journal:
Proc. Amer. Math. Soc. 136 (2008), 111-118
MSC (2000):
Primary 34C23, 58E07; Secondary 37G20, 47A53
Posted:
September 27, 2007
MathSciNet review:
2350395
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Additional Information
Abstract: We show that homoclinic trajectories of nonautonomous vector fields parametrized by a circle bifurcate from the stationary solution when the asymptotic stable bundles of the linearization at plus and minus infinity are ``twisted'' in different ways.
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- 1.
- Abbondandolo A., Majer P., Ordinary differential operators and Fredholm pairs, Math. Z. 243 (2003), 525-562. MR 1970015 (2004d:58014)
- 2.
- Atiyah M. F., K-Theory, Benjamin, New York, 1967. MR 0224083 (36:7130)
- 3.
- Berry M. V. ,Geometric amplitude factors in adiabatic quantum transitions, Proc. Roy. Soc. London Ser. A 430 (1990), 405-411. MR 1068305 (91k:81045)
- 4.
- Coppel W. A., Dichotomies in stability theory, Lect. Notes in Math. 629, Springer Verlag, Berlin 1978. MR 0481196 (58:1332)
- 5.
- Crandall M. G., Rabinowitz P. H., Bifurcation from simple eigenvalues, J. Functional Analysis 8(1971), 321-340. MR 0288640 (44:5836)
- 6.
- Fitzpatrick P. M., Pejsachowicz, J., Nonorientability of the index bundle and several-parameter bifurcation. J. of Functional Anal. 98 (1991), 42-58. MR 1111193 (92g:58019)
- 7.
- Fitzpatrick P. M., Pejsachowicz J., Parity and generalized multiplicity, Trans. Amer. Math. Soc. 326 (1991), 281-305. MR 1030507 (91j:58038)
- 8.
- Morris J. R., Nonlinear ordinary and partial differential equations on unbounded domains, PhD thesis, University of Pittsburgh, 2005.
- 9.
- Mrozek M., Reineck J. F., Srzednicki R., The Conley index over the circle, J. Dynam. Differential Equations 12 (2000), 385-409. MR 1790660 (2002m:37022)
- 10.
- Palmer K. J., Exponential dichotomies and transversal homoclinic points, J. Diff. Eq. 104 (1988), 149-156. MR 958058 (89k:34052)
- 11.
- Pejsachowicz J., Rabier P. J., Degree theory for
 -Fredholm mappings of index 0, Journal d'Analyse Mathématique 76 (1998), 289-319. MR 1676979 (2000a:58026)
- 12.
- Sacker R. J., The splitting index for linear differential systems, J. Diff. Eq. 33 (1979), no. 3, 368-405. MR 543706 (81f:34020)
- 13.
- Secchi S., Stuart, C. A., Global bifurcation of homoclinic solutions of Hamiltonian systems, Discrete Contin. Dyn. Syst. 9 (2003), 1493-1518. MR 2017678 (2005b:37138)
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Additional Information
Jacobo Pejsachowicz
Affiliation:
Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
Email:
jacobo.pejsachowicz@polito.it
DOI:
http://dx.doi.org/10.1090/S0002-9939-07-09088-0
PII:
S 0002-9939(07)09088-0
Keywords:
Differential equations,
homoclinics,
bifurcation,
index bundle
Received by editor(s):
August 3, 2006
Posted:
September 27, 2007
Communicated by:
Carmen C. Chicone
Article copyright:
© Copyright 2007 American Mathematical Society
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