Bifurcation from degenerate homoclinics in periodically forced systems (original) (raw)

Abstract

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AI

This work investigates the bifurcation phenomena arising from degenerate homoclinic solutions in periodically forced systems described by ordinary differential equations. The analysis focuses on the conditions under which homoclinic solutions exist and how changes in parameters influence these solutions. A series of theorems are presented, detailing the necessary conditions and implications of these bifurcations, particularly emphasizing situations with varying numbers of parameters. The implications for future research involve understanding more complex structures in the presence of different forces and their effects on stability.

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References (15)

  1. F. BATTELLI & C. LAZZARI, Exponential dichotomies, heteroclinic orbits, and Melnikov functions, J. Differential Equations 86 (1990), 342-366.
  2. F. BATTELLI & K.J. PALMER, Tangencies between stable and unstable manifolds, Proc. Royal Soc. Edinburgh 121 A (1992), 73-90.
  3. C. CHICONE, Lyapunov-Schmidt reduction and Melnikov integrals for bifurcation of periodic solutions in coupled oscillators, J. Differential Equations 112 (1994), 407-447.
  4. S.N. CHOW & J.K. HALE, "Methods of Bifurcation Theory", Springer-Verlag, New York, 1982.
  5. M. FE ČKAN, Higher dimensional Melnikov mappings, Mathematica Slovaca (to appear).
  6. M. GOLUBITSKY & V. GUILLEMIN, "Stable Mappings and their Singularities", Springer- Verlag, New York, 1973.
  7. J. GRUENDLER, The existence of transverse homoclinic solutions for higher order equations, J. Differential Equations 130 (1996), 307-320.
  8. J. GRUENDLER, Homoclinic solutions for autonomous ordinary differential equations with nonautonomous perturbations, J. Differential Equations 122 (1995), 1-26.
  9. J. KNOBLOCH, Bifurcation of degenerate homoclinics in reversible and conservative systems, J. Dyn. Diff. Eqns. 9 (1997), 427-444.
  10. J. KNOBLOCH, B. MARX & M. EL MORSALANI, Characterization of homoclinic points bifurcating from degenerate homoclinic orbits, preprint (1997).
  11. J. KNOBLOCH & U. SCHALK, Homoclinic points near degenerate homoclinics, Nonlinearity 8 (1995), 1133-1141.
  12. B. MORIN, Formes canoniques des singularities d'une application différentiable, Comptes Rendus Acad. Sci., Paris 260 (1965), 5662-5665, 6503-6506.
  13. K.J. PALMER, Exponential dichotomies and transversal homoclinic points, J. Differential Equations 55 (1984), 225-256.
  14. K.J. PALMER, Existence of transversal homoclinic points in a degenerate case, Rocky Mount. J. Math. 20 (1990), 1099-1118.
  15. A. VANDERBAUWHEDE, Bifurcation of degenerate homoclinics, Results in Mathematics 21 (1992), 211-223.