Lin's method and homoclinic bifurcations for functional differential equations of mixed type

Indiana University Mathematics Journal

Volumn 58, Issue 6, 2009, Pages 2433-2487

  Hupkes, H J ^a,b   Lunel, S M Verduyn ^a  

^a LEIDEN UNIVERSITY   (Netherlands)

^b BROWN UNIVERSITY   (United States)

Author keywords

Advanced and retarded arguments; Exponential dichotomies; Finite dimensional reduction; Homoclinic bifurcation; Lin's method; Mixed type functional differential equation; Orbit flip bifurcation

Indexed keywords

EID: 77949824358     PISSN: 00222518     EISSN: None     Source Type: Journal    
DOI: 10.1512/iumj.2009.58.3661     Document Type: Article

References (22)

1. S.-N. CHOW and B. DENG, Bifurcation of a unique stable periodic orbit from a homoclinic orbit in infinite-dimensional systems, Trans. Amer. Math. Soc. 312 (1989), 539-587, http://dx.doi.org/10.2307/2001001. MR 988882 (90e:58115)
2. O. DIEKMANN, S.A. VAN GILS, S.M. VERDUYN LUNEL, and H.-O. WALTHER, Delay Equations, Functional Complex, and Nonlinear Analysis, Applied Mathematical Sciences, vol.110, Springer-Verlag, New York, 1995, ISBN 0-387-94416-8. MR 1345150 (97a:34001)
3. M. GEORGI, Bifurcations from homoclinic orbits to non-hyperbolic equilibria in reversible lattice differential equations, Nonlinearity 21 (2008), 735-763, http://dx.doi.org/10.1088/0951-7715/21/4/005. MR 2399823 (2009d:37136)
4. J.K. HALE and S.M. VERDUYN LUNEL, Introduction to Functional-differential Equations, Applied Mathematical Sciences, vol.99, Springer-Verlag, New York, 1993, ISBN 0-387-94076-6. MR 1243878 (94m:34169)
5. J. HÄRTERICH, B. SANDSTEDE, and A. SCHEEL, Exponential dichotomies for linear nonautonomous functional differential equations of mixed type, Indiana Univ. Math. J. 51 (2002), 1081-1109, http://dx.doi.org/10.1512/iumj. 2002.51.2188. MR 1947869 (2004e:34103)
6. A.J. HOMBURG, H. KOKUBU, and M. KRUPA, The cusp horseshoe and its bifurcations in the unfolding of an inclination-flip homoclinic orbit, Ergodic Theory Dynam. Systems 14 (1994), 667-693, http://dx.doi.org/10.1017/ S0143385700008117. MR 1304138 (96a:58134)
7. H.J. HUPKES, Invariant Manifolds and Applications for Functional Differential Equations of Mixed Type, Leiden University, 2008 (Ph.D. Thesis).
8. H.J. HUPKES, E. AUGERAUD-VÉRON, and S.M. VERDUYN LUNEL, Center projections for smooth difference equations of mixed type, J. Differential Equations 244 (2008), 803-835, http://dx.doi.org/10.1016/j.jde.2007.10.033. MR 2391345 (2008m:39029)
9. H.J. HUPKES and S.M. VERDUYN LUNEL, Center manifolds for periodic functional differential equations of mixed type, J. Differential Equations 245 (2008), 1526-1565, http://dx.doi.org/10.1016/j.jde.2008.02.026. MR 2436452
10. -, Center manifold theory for functional differential equations of mixed type, J. Dynam. Differential Equations 19 (2007), 497-560, http://dx.doi.org/10. 1007/s10884-006-9055-9. MR 2333418 (2008c:34156)
11. M. KlSAKA, H. KOKUBU, and H. OKA, Bifurcations to N-homoclinic orbits and N-periodic orbits in vector fields, J. Dynam. Differential Equations 5 (1993), 305-357, http://dx.doi.org/10.1007/BF01053164. MR 1223451 (94m:58158)
12. X.-B. LIN, Using Melnikov's method to solve Šilnikov's problems, Proc. Roy. Soc. Edinburgh Sect. A116 (1990), 295-325. MR 1084736 (92b:58195)
13. J. MALLET-PARET, The Fredholm alternative fir functional-differential equations of mixed type, J. Dynam. Differential Equations 11 (1999), 1-47, http://dx.doi.org/10.1023/A:1021889401235. MR 1680463 (2000j:34107)
14. J. MALLET-PARET and S.M. VERDUYN-LUNEL, Exponential Dichotomies and Wiener-Hopf Factorizations for Mixed-Type Functional Differential Equations, J. Diff. Eq. (to appear).
15. B. SANDSTEDE, Verzweigungstheorie Homokliner Verdopplungen, PhD Thesis, 1993.
16. L.P. SIL'NIKOV, Some instances of generation of periodic motions in n-space, Dokl. Akad. Nauk SSSR 143 (1962), 289-292. MR 0136812 (25 #273) (Russian)
17. -, A case of the existence of a denumerable set of periodic motions, Dokl. Akad. Nauk SSSR 160 (1965), 558-561. MR 0173047 (30 #3262) (Russian)
18. -, Generation of a periodic motion from the trajectory going from the state of equilibrium of the saddle-saddle type into the same state, Dokl. Akad. Nauk SSSR 170 (1966), 49-52. MR 0211005 (35 #1890) (Russian)
19. -, The generation of a periodic motion from a trajectory which is doubly asymptotic to a saddle type equilibrium state, Mat. Sb. (N.S.) 77 (119) (1968), 461-472. MR 0255922 (41 #582) (Russian)
20. -, A certain new type of bifurcation of multidimensional dynamic systems, Dokl. Akad. Nauk SSSR 189 (1969), 59-62. MR 0262629 (41 #7234) (Russian)
21. -, On the question of the structure of an extended neighborhood of a structurally stable state of equilibrium of saddle-focus type, Mat. Sb. (N.S.) 81 (123) (1970), 92-103. MR 0259275 (41 #3917) (Russian)
22. E. YANAGIDA, Branching of double pulse solutions from single pulse solutions in nerve axon equations, J. Differential Equations 66 (1987), 243-262, http://dx.doi.org/10.1016/0022-0396(87)90034-90039 MR 871997 (88c:35078)