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}, pages = {917--936}, publisher = {Elsevier}, volume = {27}, number = {3}, year = {2010}, doi = {10.1016/j.anihpc.2010.01.005}, mrnumber = {2629886}, zbl = {1217.34075}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.anihpc.2010.01.005/} }
TY - JOUR AU - Zhu, Changrong AU - Luo, Guangping AU - Lan, Kunquan TI - Multiple homoclinic solutions for singular differential equations JO - Annales de l'I.H.P. Analyse non linéaire PY - 2010 SP - 917 EP - 936 VL - 27 IS - 3 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2010.01.005/ DO - 10.1016/j.anihpc.2010.01.005 LA - en ID - AIHPC_2010__27_3_917_0 ER -
%0 Journal Article %A Zhu, Changrong %A Luo, Guangping %A Lan, Kunquan %T Multiple homoclinic solutions for singular differential equations %J Annales de l'I.H.P. Analyse non linéaire %D 2010 %P 917-936 %V 27 %N 3 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2010.01.005/ %R 10.1016/j.anihpc.2010.01.005 %G en %F 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, Volume 27 (2010) no. 3, pp. 917-936. doi : 10.1016/j.anihpc.2010.01.005. http://www.numdam.org/articles/10.1016/j.anihpc.2010.01.005/
[1] Exponential dichotomies, heteroclinic orbits, and Melnikov functions, J. Differential Equations 86 (1990), 342-366 | MR | Zbl
, ,[2] Heteroclinic orbits in systems with slowing varying coefficients, J. Differential Equations 105 (1993), 1-29 | MR | Zbl
, ,[3] Chaos in the Duffing equation, J. Differential Equations 101 (1993), 276-301 | MR | Zbl
, ,[4] 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 | Zbl
, ,[5] Bifurcation from a homoclinic orbit in parabolic differential equations, Proc. Roy. Soc. Edinburgh 103A (1986), 265-274 | MR | Zbl
,[6] Methods of Bifurcation Theory, Springer-Verlag, New York (1982) | MR
, ,[7] An example of bifurcation to homoclinic orbits, J. Differential Equations 37 (1980), 351-373 | MR | Zbl
, , ,[8] Transversal bounded solutions in systems with normal and slow variables, J. Differential Equations 165 (2000), 123-142 | MR | Zbl
, ,[9] Homoclinic solutions for autonomous systems in arbitrary dimension, SIAM J. Math. Anal. 23 no. 3 (1992), 702-721 | MR | Zbl
,[10] Homoclinic solutions for autonomous ordinary differential equations with nonautonomous perturbations, J. Differential Equations 122 (1995), 1-26 | MR | Zbl
,[11] Homoclinic solutions and chaos in ordinary differential equations with singular perturbations, Trans. Amer. Math. Soc. 350 (1998), 3797-3814 | MR | Zbl
,[12] Introduction to dynamic bifurcation, Bifurcation Theory and Application, Lecture Notes in Mathematics vol. 1057, Springer-Verlag, Berlin (1984) | MR | Zbl
,[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] Exponential dichotomies and transversal homoclinic points, J. Differential Equations 55 (1984), 225-256 | MR | Zbl
,[15] Homoclinic points near degenerate homoclinics, Nonlinearity 8 (1995), 1133-1141 | MR | Zbl
, ,[16] Bifurcation of degenerate homoclinics, Results in Mathematics 21 (1992), 211-223 | MR | Zbl
,[17] Center manifolds and contractions on a scale of Banach spaces, J. Funct. Anal. 72 (1987), 209-224 | MR | Zbl
, ,[18] Linearly independent homoclinic bifurcations parameterized by a small function, J. Differential Equations 240 (2007), 38-57 | MR | Zbl
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