zbMATH — the first resource for mathematics

Melnikov formulas for nearly integrable Hamiltonian systems. (English) Zbl 0731.58037
Symplectic geometry, groupoids, and integrable systems, Sémin. Sud- Rhodan. Geom. VI, Berkeley/CA (USA) 1989, Math. Sci. Res. Inst. Publ. 20, 183-188 (1991).
[For the entire collection see Zbl 0722.00026.]
An intrinsic Melnikov vector valued function is given for nearly integrable Hamiltonian systems. It can be used to detect homoclinic orbits in Hamiltonian perturbations of completely integrable systems. As an example, it is shown that perturbation of the spherical pendulum on a rotating frame produce Silnikov’s spiralling chaos.

37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion