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Order and chaos in dynamical systems. (English) Zbl 1205.37051

Summary: We consider the characteristics of order and chaos in dynamical systems, with emphasis on the orbits in astronomical systems. Celestial mechanics deals with orbits in the solar system, which are mainly ordered. On the other hand the orbits of stars in galaxies were considered to be chaotic. However numerical experiments have shown that in general a system contains both ordered and chaotic orbits. Thus a new classification of dynamical systems has been established. We describe ordered and chaotic orbits in galaxies and in mappings. Some ordered orbits appear even in strongly perturbed systems. The transition from order to chaos is due to resonance overlapping. Then we describe some recent developments concerning order and chaos in the solar system and in galaxies. The outer spiral arms in strong barred galaxies are composed mainly of sticky chaotic orbits. Ordered and chaotic orbits appear also in Bohmian quantum mechanics. If the initial probability \(p\) is not equal to the square of the wave function \(|\psi |^{2}\), then in the case of ordered orbits \(p\) never approaches \(|\psi |^{2}\), while in the case of chaotic orbits \(p \rightarrow |\psi |^{2}\) after a time interval called “quantum Nekhoroshev time”.

MSC:

37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
81Q50 Quantum chaos
37N05 Dynamical systems in classical and celestial mechanics
70F15 Celestial mechanics
70K55 Transition to stochasticity (chaotic behavior) for nonlinear problems in mechanics
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