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Numerical exploration of Hamiltonian systems. (English) Zbl 0578.70019
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

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