Fractional Kinetics of Chaotic Dynamics of Particles in Complex Systems

摘要

The illness and death of G. M. Zaslavsky caused significant shifts in the work carried out under this grant. Nonetheless it was possible for the remaining research scientists originally intended to be supported by the grant, Dr. Mark Edelman and Dr. Vasily Tarasov, to work in the broad directions laid out in the proposal. In three papers, which are attached, they were able to set up fractional differential equation to represent the evolution of the particle distribution function for systems of physical interest and relevance. The first paper dealt with a variation of the standard map, the second of a system with memory, and the third with a dissipative system. In each case it was possible to identify solutions of the continuous fractional kinetic equation with the properties of a discrete map. This identification provided, on the one hand, a reasonable solution algorithm for the fractional kinetic equation in question, and on the other hand a connection to discrete dynamical systems with potentially chaotic dynamics. In every case stochastic webs were identified and attractor basins were found. The detailed results are in the papers, the first published in Physics Letters A, the second in Journal of Physics, A, Math and Theoretical, while the third is under review. The work has been presented at a colloquium at Yeshiva University, and will be presented in the summer of 2010 at the Fourth International Conference 'Frontiers of Nonlinear Physics', Nizhny, Novgorod, Russia (http://www.fnp.sci-nnov.ru/). This work lends itself to substantial future developments.