Abstract
An “intrinsic” Melnikov vector valued function is given, which can be used to detect homoclinic orbits in Hamiltonian perturbations of completely integrable systems. We use the description given by Prof. Nicole Desolneux-Moulis [1] of the dynamics along a singular leaf of the unperturbed system. As an example, it is shown that perturbations of the spherical pendulum on a rotating frame (or in a magnetic field) produce Silnikov’s spiralling chaos.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
N. Desolneux-Moulis, Dynamique des systemes Hamiltoniens complement integrables sur les varietés compactes,this Proceedings.
S. Wiggins, “Global bifurcations and chaos: analytical methods,” Springer-Verlag, New York, Berlin, Heidelberg, 1988.
L.M. Lerman and Ya.L. Umanskii, Structure of the Poisson action of R 2 on a four-dimensional symplectic manifold II, Sel. Math. Soy. 7 (1988), 39–48.
J. Koiller, Splitting of homoclinic Lagrangian submanifolds in perturbations of integrable Hamiltonian systems, Laboratdrio Nacional de Computaçäo Cientifica 36/84 (1984).
C. Robinson, Horseshoes for autonomous Hamiltonian systems using the Melnikov integral, Ergod. Th. and Dynam. Sys. 8 (1988), 395–409.
L.M. Lerman and Ya.L. Umanskii, On the existence of separatrix loops in four dimensional systems similar to the integrable Hamiltonian systems, PMM USSR 47 (1984), 335–340.
R. Devaney, Blue sky catastrophes in. reversible and Hamiltonian systems, Indiana Univ. Math. J. 26 (1970), 247–263.
A. T. Fomenko, The topology of surfaces of constant energy in integrable Hamiltonian systems and obstructions to integrability, Math. USSR Izvestiya 29 (1987), 629–658.
A.T. Fomenko and Kh. Tsishang, On typical topological properties of integrable Hamiltonian systems, Math. USSR Izvestiya 32 (1989), 385–412.
M.P. Kharlamov, Bifurcation of common levels of first integrals of the Kovalevskaya problem, PMM USSR 47 (1983), 737–743.
V. Rom-Kedar and S. Wiggins, Transport in two-dimensional maps, Arch. Rational Mech. Anal. 109 (1990), 239–298.
V. Rom-Kedar and S. Wiggins, Transport in two-dimensional maps, Arch. Rational Mech. Anal. 109 (1990), 239–298.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1991 Springer-Verlag New York, Inc.
About this paper
Cite this paper
Koiller, J. (1991). Melnikov Formulas For Nearly Integrable Hamiltonian Systems. In: Dazord, P., Weinstein, A. (eds) Symplectic Geometry, Groupoids, and Integrable Systems. Mathematical Sciences Research Institute Publications, vol 20. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9719-9_12
Download citation
DOI: https://doi.org/10.1007/978-1-4613-9719-9_12
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4613-9721-2
Online ISBN: 978-1-4613-9719-9
eBook Packages: Springer Book Archive