Hostname: page-component-5c6d5d7d68-tdptf Total loading time: 0 Render date: 2024-08-19T01:23:13.971Z Has data issue: false hasContentIssue false
  • English
  • Français

Horseshoes for autonomous Hamiltonian systems using the Melnikov integral

Published online by Cambridge University Press:  10 December 2009

Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper applies the Melnikov method to autonomous perturbations of completely integrable Hamiltonian systems. The forcing of the perturbed system is caused by internal oscillations which are not necessarily decoupled. A unified treatment is presented which relates some results of Holmes and Marsden with a result of Lerman and Umanskii. It is also shown that two forms of the Melnikov function by integrals are in fact equal.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 8 , Volume 8: Charles Conley Memorial Issue , December 1988 , pp. 395 - 409
Copyright
Copyright © Cambridge University Press 1988

References

REFERENCES

Arnold, V. I.. Instability of dynamical systems with many degrees of freedom. Sov. Math. Dokl. 5 (1963), 581–585.Google Scholar
de Carvalho, S. & Roussarie, R.. Some remarks about homoclinic points of second order differential equations. Geometric Dynamics, ed. Palis, J.. Lecture Notes in Mathematics 1007. Springer-Verlag, New York (1983), 88–95.Google Scholar
Churchill, R. & Rod, D., Pathology in dynamical systems III: analytic Hamiltonians. J. Differential Equations 37 (1980), 23–38.10.1016/0022-0396(80)90085-6Google Scholar
Devaney, R.. Homoclinic orbits in Hamiltonian systems. J. Differential Equations 21 (1976), 431–438.10.1016/0022-0396(76)90130-3Google Scholar
Gavrilov, N. K. & Silnikov, L. P.. On the three dimensional dynamical systems close to a system with a structurally unstable homoclinic curve I. Math. USSR Sbornik 17 (1972), 467–485;10.1070/SM1972v017n04ABEH001597Google Scholar
II. Math. USSR Sbornik 19 (1973), 139–156.10.1070/SM1973v019n01ABEH001741Google Scholar
Gruendler, J.. The existence of homoclinic orbits and the method of Melnikov for systems in R n. SIAM J. Math. Anal. 16 (1985), 907–931.10.1137/0516069Google Scholar
Guckenheimer, J. & Holmes, P.. Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields. Springer-Verlag, New York/Berlin/Heidelberg/Tokyo (1983).10.1007/978-1-4612-1140-2Google Scholar
Holmes, P. and Marsden, J.. Horseshoes in perturbations of Hamiltonian systems with two degrees of freedom. Commun. Math. Phys. 82 (1982), 523–544.10.1007/BF01961239Google Scholar
Holmes, P. & Marsden, J.. Melnikov's method and Arnold diffusion for perturbation of integrable Hamiltonian systems. J. Math. Phys. 23 (1982), 669–675.Google Scholar
Holmes, P. & Marsden, J.. Horseshoes and Arnold diffusion for Hamiltonian systems on Lie groups. Indiana Univ. Math. J. 32 (1983), 273–309.Google Scholar
Lerman, L. & Umanskii, la.. On the existence of separatrix loops in four-dimensional systems similar to the integrable Hamiltonian systems. Appl. Math. Mech. 47 (1984), 335–340.10.1016/0021-8928(83)90059-XGoogle Scholar
Marsden, J.. Chaos in dynamical systems by the Poincaré-Melnikov-Arnold method. Chaos in Nonlinear Dynamical Systems, ed. Chandra, J.. SIAM, Philadelphia (1984), 19–31.Google Scholar
Melnikov, V. K.. On the stability of the center for time periodic perturbations. Trans. Moscow Math. Soc. 12 (1963) 1–57.Google Scholar
Moser, J.. Convergent series expansion for quasi-periodic motion. Math. Annalen 169 (1967), 136–176.10.1007/BF01399536Google Scholar
Newhouse, S.. Diffeomorphisms with infinitely many sinks. Topology 13 (1974), 9–18.10.1016/0040-9383(74)90034-2Google Scholar
Newhouse, S. & Palis, J.. Cycles and bifurcation theory. Asterisque 31 (1976), 43–140.Google Scholar
Patterson, S.. Ω-stable limit set explosions. Trans. Amer. Math. Soc. 294 (1986), 775–798.Google Scholar
Robinson, C.. Bifurcation to infinitely many sinks. Commun. Math. Phys. 90 (1983), 433–459.Google Scholar
Salem, F., Marsden, J. & Varaiya, P.. Arnold diffusion in the swing equations of a power system. To appear.Google Scholar
Zehnder, E.. Moser's implicit function theorem in the framework of analytic smoothing. Math. Annalen 219 (1976), 105–121.10.1007/BF01351894Google Scholar