Basic theory of fractional Mei symmetrical perturbation and its applications

Abstract

In this paper, we present a new method of fractional dynamics, i.e., the fractional Mei symmetrical perturbation method of a disturbed system, and explore the adiabatic invariant directly led by the perturbation. For a dynamical system which is disturbed by small forces of perturbation, the disturbed fractional generalized Hamiltonian equation is investigated and, under a more general kind of fractional infinitesimal transformation of a Lie group, the fractional Mei symmetrical definition and determining equation of a disturbed dynamical system are given; then, the determining equation of fractional Mei symmetrical perturbation is obtained. In particular, we present the fractional Mei symmetrical perturbation method of a disturbed dynamical system and it is found that, using the new method, we can find a new kind of non-Noether adiabatic invariant directly led by the perturbation. As special cases, we obtain a new fractional Mei symmetrical conservation law of the undisturbed dynamical system and a Mei symmetrical perturbation theorem of the disturbed integer dynamical system. Also, as the fractional Mei symmetrical perturbation method’s applications, we find the adiabatic invariants of a disturbed fractional Duffing oscillator and a disturbed fractional Lotka biochemical oscillator. This work constructs a basic theoretical framework of fractional Mei symmetrical perturbation method and provides a general method of fractional dynamics.