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.

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 |