Estimation of the amplitude of resonance in the general standard map. (English) Zbl 1010.70013

Olvera, Arturo

Exp. Math. 10, No. 3, 401-418 (2001).

The author studies stability of two-dimensional Hamiltonian systems and gives conjectures about the resonance amplitude in the general standard map under periodic perturbations. A relation between a perturbation parameter and the amplitude of resonance is obtained for arbitrary rotation number, and the lower bound is found for the resonance amplitude. The author also shows that some perturbation functions lead to collapse, i.e. the corresponding amplitude goes to zero. Finally, selected examples of perturbations are calculated.

Reviewer: Václav Burjan (Praha)

MSC:

70H15	Canonical and symplectic transformations for problems in Hamiltonian and Lagrangian mechanics
37N05	Dynamical systems in classical and celestial mechanics

Keywords:

stability; two-dimensional Hamiltonian systems; resonance amplitude; general standard map; periodic perturbations; rotation number; collapse

References:

[1]	Abramowitz M., Handbook of mathematical functions with formulas, graphs, and mathematical tables (1972) · Zbl 0543.33001
[2]	Chirikov B. V., Phys. Rep. 52 (5) pp 264– (1979)
[3]	Escande D. F., Phys. D 6 (1) pp 119– (1982) · doi:10.1016/0167-2789(82)90010-0
[4]	DOI: 10.1063/1.524170 · doi:10.1063/1.524170
[5]	Lichtenberg A. J., Regular and stochastic motion (1983) · Zbl 0506.70016
[6]	Mather J. N., Topology 21 pp 457– (1982) · Zbl 0506.58032 · doi:10.1016/0040-9383(82)90023-4
[7]	Olvera A., Phys. D 26 (1) pp 181– (1987) · Zbl 0612.58039 · doi:10.1016/0167-2789(87)90222-3
[8]	Olvera A., European Conference on Iteration Theory (ECIT 87) (Caldes de Malavella, Spain, 1987) pp 438– (1989)
[9]	Simó C., Modern methods in celestial mechanics (Goutelas, France, 1989) pp 285– (1990)
[10]	Veerman P., ”Dynamical systems and twist maps” (1993)