Homoclinic Solutions and Chaos in Ordinary Differential Equations with Singular Perturbations

Gruendler, Joseph

doi:10.1090/S0002-9947-98-02211-9

Homoclinic Solutions and Chaos in Ordinary Differential Equations with Singular Perturbations
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Trans. Amer. Math. Soc. 350 (1998), 3797-3814 Request permission

Abstract:

Ordinary differential equations are considered which contain a singular perturbation. It is assumed that when the perturbation parameter is zero, the equation has a hyperbolic equilibrium and homoclinic solution. No restriction is placed on the dimension of the phase space or on the dimension of intersection of the stable and unstable manifolds. A bifurcation function is established which determines nonzero values of the perturbation parameter for which the homoclinic solution persists. It is further shown that when the vector field is periodic and a transversality condition is satisfied, the homoclinic solution to the perturbed equation produces a transverse homoclinic orbit in the period map. The techniques used are those of exponential dichotomies, Lyapunov-Schmidt reduction and scales of Banach spaces. A much simplified version of this latter theory is developed suitable for the present case. This work generalizes some recent results of Battelli and Palmer.

References

Flaviano Battelli and Kenneth J. Palmer, Chaos in the Duffing equation, J. Differential Equations 101 (1993), no. 2, 276–301. MR 1204329, DOI 10.1006/jdeq.1993.1013
Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
W. A. Coppel, Dichotomies in stability theory, Lecture Notes in Mathematics, Vol. 629, Springer-Verlag, Berlin-New York, 1978. MR 0481196
J. Dieudonné, Foundations of modern analysis, Pure and Applied Mathematics, Vol. 10-I, Academic Press, New York-London, 1969. Enlarged and corrected printing. MR 0349288
Joseph Gruendler, Homoclinic solutions for autonomous dynamical systems in arbitrary dimension, SIAM J. Math. Anal. 23 (1992), no. 3, 702–721. MR 1158829, DOI 10.1137/0523036
Joseph Gruendler, Homoclinic solutions for autonomous ordinary differential equations with nonautonomous perturbations, J. Differential Equations 122 (1995), no. 1, 1–26. MR 1356127, DOI 10.1006/jdeq.1995.1136
Joseph Gruendler, The existence of transverse homoclinic solutions for higher order equations, J. Differential Equations 130 (1996), no. 2, 307–320. MR 1410890, DOI 10.1006/jdeq.1996.0145
Jack K. Hale, Ordinary differential equations, 2nd ed., Robert E. Krieger Publishing Co., Inc., Huntington, N.Y., 1980. MR 587488
J. Knobloch, Bifurcation of Degenerate Homoclinics in Reversible and Conservative Systems, Preprint No. M 15/94, Technical University of Ilmenau, PSF 327, D 98684 Ilmenau, Germany, 1994.
Kenneth J. Palmer, Exponential dichotomies and transversal homoclinic points, J. Differential Equations 55 (1984), no. 2, 225–256. MR 764125, DOI 10.1016/0022-0396(84)90082-2
U. Schalk and J. Knobloch, Homoclinic points near degenerate homoclinics, Nonlinearity 8 (1995), no. 6, 1133–1141. MR 1363403, DOI 10.1088/0951-7715/8/6/014
Patricio L. Felmer, Periodic solutions of spatially periodic Hamiltonian systems, J. Differential Equations 98 (1992), no. 1, 143–168. MR 1168976, DOI 10.1016/0022-0396(92)90109-Z
A. Vanderbauwhede and S. A. van Gils, Center manifolds and contractions on a scale of Banach spaces, J. Funct. Anal. 72 (1987), no. 2, 209–224. MR 886811, DOI 10.1016/0022-1236(87)90086-3