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Fractional kinetics of Hamiltonian chaotic systems. (English) Zbl 1046.82025
Hilfer, R. (ed.), Applications of fractional calculus in physics. Singapore: World Scientific (ISBN 981-02-3457-0). 203-239 (2000).
This article gives a brief review of the problem of fractional kinetics and time evolution moments. It is shown that fractal characteristics arise from smooth dynamics. Various terms of mapping such as Poincaré map, Poincaré recurrences, exit times and zero map are described. Next a motivation for introducing a fractal support in space-time for chaotic trajectories is provided. One of the interesting and useful results involves a fractional generalization of the Fokker-Planck-Kolmogorov equations developed formally. Its solution is derived by the application of the well-known Laplace-Fourier transform in terms of the Mittag-Leffler function, which is an extension of the exponential function. A detailed account of the Mittag-Leffler function is available from the monograph written by A. Erdélyi et al. [Higher transcendental functions. Vol. 3, McGraw-Hill, New York (1955; Zbl 0542.33002)].
MSC:
82C31Stochastic methods in time-dependent statistical mechanics
37D45Strange attractors, chaotic dynamics
82C10Quantum dynamics and nonequilibrium statistical mechanics (general)
37N20Dynamical systems in other branches of physics
70K55Transition to stochasticity (chaotic behavior)
37J99Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems