Numerical exploration of Hamiltonian systems. (English) Zbl 0578.70019

Hénon, Michel

Chaotic behaviour of deterministic systems, Les Houches/Fr. 1981, Sess. 36, 53-170 (1983).

[For the entire collection see Zbl 0546.00032.]
The first part of this review article introduces, in an elementary manner, fundamental concepts like surface of section (Poincaré-map), periodic orbits, stability, integrable Hamiltonian systems (invariant tori) and ergodicity. The second part presenting several results of the qualitative exploration of dynamical systems is subdivided into paragraphs dealing with two-dimensional, three-dimensional, and many- dimensional systems, respectively. The extensive paragraph on systems with two degrees of freedom deals with the Kolmogoroff-Arnold-Moser theorem and the notion of chaos, illustrated by numerical investigations of the Hénon-Heiles system, the restricted three-body-problem and oval billiard. The paragraphs on systems with three degrees of freedom (introducing Arnold diffusion as a new possibility) and with many degrees of freedom (dealing in particular with n parallel self-gravitating sheets and with the Fermi-Pasta-Ulam system) are less extensive. Some remarks about mappings on a lattice (i.e. about discretization of the surface of section) conclude this highly useful survey, which should be recommended to everybody interested in qualitative dynamics, especially to students.

Reviewer: V.Perlick

MSC:

70Kxx	Nonlinear dynamics in mechanics
37C75	Stability theory for smooth dynamical systems
70H05	Hamilton’s equations
37D45	Strange attractors, chaotic dynamics of systems with hyperbolic behavior

Keywords:

mapping on lattices; Poincaré-map; invariant tori; many-dimensional systems; Kolmogoroff-Arnold-Moser theorem; Hénon-Heiles system; restricted three-body-problem; oval billiard; Arnold diffusion; n parallel self-gravitating sheets; Fermi-Pasta-Ulam system