Center manifolds for periodic functional differential equations of mixed type

Journal of Differential Equations

Volumn 245, Issue 6, 2008, Pages 1526-1565

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

^a LEIDEN UNIVERSITY   (Netherlands)

Author keywords

Advanced and retarded arguments; Center manifold; Finite dimensional reduction; Floquet theory; Hopf bifurcation; Mixed type functional differential equation

Indexed keywords

EID: 47949119554     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2008.02.026     Document Type: Article

References (34)

1. Abell K.A., Elmer C.E., Humphries A.R., and Van Vleck E.S. Computation of mixed type functional differential boundary value problems. SIAM J. Appl. Dyn. Syst. 4 (2005) 755-781
2. Bateman M.D., and Van Vleck E.S. Traveling wave solutions to a coupled system of spatially discrete Nagumo equations. SIAM J. Appl. Math. 66 (2006) 945-976
3. Bates P.W., and Chmaj A. A discrete convolution model for phase transitions. Arch. Ration. Mech. Anal. 150 (1999) 281-305
4. Bell J. Some threshold results for models of myelinated nerves. Math. Biosci. 54 (1981) 181-190
5. Cahn J.W. Theory of crystal growth and interface motion in crystalline materials. Acta Met. 8 (1960) 554-562
6. Cahn J.W., Mallet-Paret J., and Van Vleck E.S. Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice. SIAM J. Appl. Math. 59 (1999) 455-493
7. Chua L.O., and Roska T. The CNN paradigm. IEEE Trans. Circuits Syst. 40 (1993) 147-156
8. d'Albis H., and Augeraud-Véron E. Balanced cycles in an OLG model with a continuum of finitely-lived individuals. J. Econom. Theory 30 (2007) 181-186
9. Diekmann O., van Gils S.A., Verduyn Lunel S.M., and Walther H.O. Delay Equations (1995), Springer-Verlag, New York
10. Dieudonné J. Foundations of Modern Analysis (1960), Academic Press, New York
11. Elmer C.E., and Van Vleck E.S. A variant of Newton's method for the computation of traveling waves of bistable differential-difference equations. J. Dynam. Differential Equations 14 (2002) 493-517
12. Erneux T., and Nicolis G. Propagating waves in discrete bistable reaction-diffusion systems. Phys. D 67 (1993) 237-244
13. Hale J.K., and Verduyn Lunel S.M. Introduction to Functional Differential Equations (1993), Springer-Verlag, New York
14. Härterich J., Sandstede B., and Scheel A. Exponential dichotomies for linear non-autonomous functional differential equations of mixed type. Indiana Univ. Math. J. 51 (2002) 1081-1109
15. Hupkes H.J., Augeraud-Veron E., and Verduyn Lunel S.M. Center projections for smooth difference equations of mixed type. J. Differential Equations 244 (2008) 803-835
16. H.J. Hupkes, Invariant manifolds and applications for functional differential equations of mixed type, PhD thesis, University of Leiden, 2008
17. Hupkes H.J., and Verduyn Lunel S.M. Analysis of Newton's method to compute travelling waves in discrete media. J. Dynam. Differential Equations 17 (2005) 523-572
18. Hupkes H.J., and Verduyn Lunel S.M. Center manifold theory for functional differential equations of mixed type. J. Dynam. Differential Equations 19 (2007) 497-560
19. Kaashoek M.A., and Verduyn Lunel S.M. Characteristic matrices and spectral properties of evolutionary systems. Trans. Amer. Math. Soc. 334 (1992) 479-517
20. Keener J., and Sneed J. Mathematical Physiology (1998), Springer-Verlag, New York
21. Laplante J.P., and Erneux T. Propagation failure in arrays of coupled bistable chemical reactors. J. Phys. Chem. 96 (1992) 4931-4934
22. Luzyanina T., and Engelborghs K. Computing Floquet multipliers for functional differential equations. Internat. J. Bifur. Chaos 12 (2002) 2977-2989
23. Mallet-Paret J. The Fredholm alternative for functional differential equations of mixed type. J. Dynam. Differential Equations 11 (1999) 1-48
24. Mallet-Paret J. The global structure of traveling waves in spatially discrete dynamical systems. J. Dynam. Differential Equations 11 (1999) 49-128
25. J. Mallet-Paret, Crystallographic pinning: Direction dependent pinning in lattice differential equations, preprint, 2001
26. Mielke A. Floquet theory for, and bifurcations from spatially periodic patterns. Tatra Mt. Math. Publ. 4 (1994) 153-158
27. Rustichini A. Hopf bifurcation for functional-differential equations of mixed type. J. Dynam. Differential Equations 1 (1989) 145-177
28. Skubachevskii A.L., and Walther H.O. On the Floquet multipliers of periodic solutions to non-linear functional differential equations. J. Dynam. Differential Equations 18 (2006) 257-355
29. Strunz T., and Elmer F.J. Driven Frenkel-Kontorova model. I. Uniform sliding states and dynamical domains of different particle densities. Phys. Rev. E 58 (1998) 1601-1611
30. Strunz T., and Elmer F.J. Driven Frenkel-Kontorova model. II. Chaotic sliding and nonequilibrium melting and freezing. Phys. Rev. E 58 (1998) 1612-1620
31. Szalai R., Stépán G., and Hogan S.J. Continuation of bifurcations in periodic delay-differential equations using characteristic matrices. SIAM J. Sci. Comput. 28 (2006) 1301-1317
32. Vanderbauwhede A., and van Gils S.A. Center manifolds and contractions on a scale of Banach spaces. J. Functional Anal. 72 (1987) 209-224
33. Walther H.O. Bifurcation from periodic solutions in functional differential equations. Math. Z. 182 (1983) 269-289
34. Widder D.V. The Laplace Transform (1946), Princeton Univ. Press, Princeton, NJ