Multiple homoclinic solutions for singular differential equations
Zhu, Changrong ; Luo, Guangping ; Lan, Kunquan
Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010), p. 917-936 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

The homoclinic bifurcations of ordinary differential equation under singular perturbations are considered. We use exponential dichotomy, Fredholm alternative and scales of Banach spaces to obtain various bifurcation manifolds with finite codimension in an appropriate infinite-dimensional space. When the perturbative term is taken from these bifurcation manifolds, the perturbed system has various coexistence of homoclinic solutions which are linearly independent.

@article{AIHPC_2010__27_3_917_0,
     author = {Zhu, Changrong and Luo, Guangping and Lan, Kunquan},
     title = {Multiple homoclinic solutions for singular differential equations},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {27},
     year = {2010},
     pages = {917-936},
     doi = {10.1016/j.anihpc.2010.01.005},
     mrnumber = {2629886},
     zbl = {1217.34075},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2010__27_3_917_0}
}

Zhu, Changrong; Luo, Guangping; Lan, Kunquan. Multiple homoclinic solutions for singular differential equations. Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) pp. 917-936. doi : 10.1016/j.anihpc.2010.01.005. http://gdmltest.u-ga.fr/item/AIHPC_2010__27_3_917_0/

[1] F. Battelli, C. Lazzari, Exponential dichotomies, heteroclinic orbits, and Melnikov functions, J. Differential Equations 86 (1990), 342-366 | MR 1064015 | Zbl 0713.34055

[2] F. Battelli, C. Lazzari, Heteroclinic orbits in systems with slowing varying coefficients, J. Differential Equations 105 (1993), 1-29 | MR 1237975 | Zbl 0784.34037

[3] F. Battelli, K.J. Palmer, Chaos in the Duffing equation, J. Differential Equations 101 (1993), 276-301 | MR 1204329 | Zbl 0772.34040

[4] F. Battelli, K.J. Palmer, Singular perturbations, transversality, and Silnikov saddle-focus homoclinic orbits, Special Issue Dedicated to Victor A. Pliss on the Occasion of his 70th Birthday J. Dynamics Differential Equations 15 (2003), 357-425 | MR 2046724 | Zbl 1040.34065

[5] C.M. Blazquez, Bifurcation from a homoclinic orbit in parabolic differential equations, Proc. Roy. Soc. Edinburgh 103A (1986), 265-274 | MR 866839 | Zbl 0636.35010

[6] S.N. Chow, J.K. Hale, Methods of Bifurcation Theory, Springer-Verlag, New York (1982) | MR 660633

[7] S.N. Chow, J.K. Hale, J. Mallet-Parret, An example of bifurcation to homoclinic orbits, J. Differential Equations 37 (1980), 351-373 | MR 589997 | Zbl 0439.34035

[8] M. Fečkan, J. Gruendler, Transversal bounded solutions in systems with normal and slow variables, J. Differential Equations 165 (2000), 123-142 | MR 1771791 | Zbl 0965.34035

[9] J. Gruendler, Homoclinic solutions for autonomous systems in arbitrary dimension, SIAM J. Math. Anal. 23 no. 3 (1992), 702-721 | MR 1158829 | Zbl 0767.34028

[10] J. Gruendler, Homoclinic solutions for autonomous ordinary differential equations with nonautonomous perturbations, J. Differential Equations 122 (1995), 1-26 | MR 1356127 | Zbl 0840.34045

[11] J. Gruendler, Homoclinic solutions and chaos in ordinary differential equations with singular perturbations, Trans. Amer. Math. Soc. 350 (1998), 3797-3814 | MR 1475684 | Zbl 0902.34038

[12] J.K. Hale, Introduction to dynamic bifurcation, Bifurcation Theory and Application, Lecture Notes in Mathematics vol. 1057, Springer-Verlag, Berlin (1984) | MR 753299 | Zbl 0544.58016

[13] J. Knobloch, Bifurcation of degenerate homoclinics in reversible and conservative systems, Preprint No M 15/94, Technical University of IImenau, PSF 327, D 98684 IImenau, Germany, 1984

[14] K.J. Palmer, Exponential dichotomies and transversal homoclinic points, J. Differential Equations 55 (1984), 225-256 | MR 764125 | Zbl 0508.58035

[15] U. Schalk, J. Knobloch, Homoclinic points near degenerate homoclinics, Nonlinearity 8 (1995), 1133-1141 | MR 1363403 | Zbl 0841.58050

[16] A. Vanderbauwhede, Bifurcation of degenerate homoclinics, Results in Mathematics 21 (1992), 211-223 | MR 1146644 | Zbl 0762.34022

[17] A. Vanderbauwhede, S.A. Van Gils, Center manifolds and contractions on a scale of Banach spaces, J. Funct. Anal. 72 (1987), 209-224 | MR 886811 | Zbl 0621.47050

[18] C. Zhu, W. Zhang, Linearly independent homoclinic bifurcations parameterized by a small function, J. Differential Equations 240 (2007), 38-57 | MR 2349164 | Zbl 1138.34024