Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Homoclinic Solutions and Chaos
in Ordinary Differential Equations
with Singular Perturbations

Author: Joseph Gruendler
Journal: Trans. Amer. Math. Soc. 350 (1998), 3797-3814
MSC (1991): Primary 34E15, 34C37, 58F13
MathSciNet review: 1475684
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. F. Battelli and K. Palmer, Chaos in the Duffing Equation, J. Differential Equations 101 (1993), 276-301. MR 93k:34099
  • 2. Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955. MR 16:1022b
  • 3. W. A. Coppel, Dichotomies in Stability Theory, Lecture Notes in Mathematics, v. 629, Springer-Verlag, 1978. MR 58:1332
  • 4. J. Dieudonné, Foundations of Modern Analysis, 2nd (enlarged and corrected) printing, Series in Pure and Applied Mathematics, v. 10-I, Academic Press, New York and London, 1969. MR 50:1782
  • 5. J. R. Gruendler, Homoclinic Solutions for Autonomous Dynamical Systems in Arbitrary Dimension, SIAM J. Math. Anal. 23(3) (1992), 702-721. MR 93e:34068
  • 6. J. R. Gruendler, Homoclinic Solutions for Autonomous Ordinary Differential Equations with Nonautonomous Perturbations, J. Differential Equations 122(1) (1995), 1-26. MR 96j:58123
  • 7. J. R. Gruendler, The Existence of Transverse Homoclinic Solutions for Higher Order Equations, J. Differential Equations 130 (1996), 307-320. MR 97g:34059
  • 8. J. K. Hale, Ordinary Differential Equations, 2nd ed., Robert E. Krieger, New York, 1980. MR 82e:34001
  • 9. 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.
  • 10. K. Palmer, Exponential Dichotomies and Transversal Homoclinic Points, J. Differential Equations 55 (1984), 225-256. MR 86d:58088
  • 11. U. Schalk and J. Knobloch, Homoclinic Points Near Degenerate Homoclinics, Nonlinearity 8 (1995), 1133-1141. MR 97a:58142
  • 12. A. Vanderbauwhede, Bifurcation of Degenerate Homoclinics, Results in Mathematics 21 (1992), 211-223. MR 93d:58131
  • 13. A. Vanderbauwhede and S. A. Van Gils, Center Manifolds and Contractions on a Scale of Banach Spaces, J. Funct. Anal. 72 (1987), 209-224. MR 88d:58085

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 34E15, 34C37, 58F13

Retrieve articles in all journals with MSC (1991): 34E15, 34C37, 58F13

Additional Information

Joseph Gruendler
Affiliation: Department of Mathematics, North Carolina A&T State University, Greensboro, North Carolina 27411

Keywords: Ordinary differential equations, homoclinic solutions, bifurcations, singular perturbations
Received by editor(s): December 28, 1995
Received by editor(s) in revised form: October 23, 1996
Additional Notes: This work supported in part by the J. William Fulbright Commission.
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society