Research On Basic Theory And Method Of Symmetrical Perturbation Of Fractional Dynamical Systems

Symmetry perturbation and adiabatic invariants of dynamical systems is mathematical mechanics and mathematical physics discipline frontier research subject,in many fields of natural science and engineering science plays an important role.But the methods for symmetrical perturbation of dynamical systems still need to be explored.This paper,under the fourth kind of fractional infinitesimal transformation of Lie group,based on analytical mechanical representation of five kinds of fractional dynamic systems proposed by Luo Shaokai,gives symmetrical perturbation basic theory and method of the for disturbed fractional dynamical system respectively,and finds the adiabatic invariants directly led by the perturbation.As new theory and method's applications,symmetry perturbation and adiabatic invariants of more than ten fractional dynamics models with practical background are studied respectively.Section 1 explains briefly the history and status of symmetrical perturbation of dynamical systems and fractional dynamics,and presents an important problem to be solved in this paper: symmetrical perturbation and adiabatic invariant of fractional dynamical systems.In Sect.2,summarizes mainly the method of four kinds of methods of fractional infinitesimal transformation of Lie group proposed by Luo Shaokai,gives the fractional infinitesimal generator vector and recursive formula of generator function for undisturbed dynamical system respectively,and gives the fractional infinitesimal generator vector and expansion formula of generator function for disturbed dynamical system.This is the mathematical basis of the research work of this paper.In Sect.3,based on disturbed Lagrange representation of fractional dynamics system,study the Mei symmetrical perturbation and Lie symmetrical perturbation of disturbed fractional Lagrange system,and finds the adiabatic invariants directly led by these two kinds of the perturbation.As fractional Lagrange system symmetricalperturbation method's applications,we respectively study disturbed fractional Hénon-Heiles model and disturbed fractional Kepler model,and find these models' adiabatic invariants directly led by the symmetrical perturbation.In Sect.4,based on disturbed Hamilton representation of fractional dynamics system,study the Mei symmetrical perturbation and Lie symmetrical perturbation of disturbed fractional Hamilton system,and finds the adiabatic invariants directly led by these two kinds of the perturbation.As fractional Hamilton system symmetrical perturbation method's applications,we respectively study disturbed fractional Emden model and disturbed fractional Hénon-Heiles model,and find these modeld's adiabatic invariants directly led by the symmetrical perturbation.In Sect.5,based on disturbed generalized Hamilton representation of fractional dynamics system,study the Mei symmetrical perturbation and Lie symmetrical perturbation of disturbed fractional generalized Hamilton system,and finds the adiabatic invariants directly led by these two kinds of the perturbation.As fractional generalized Hamilton system symmetrical perturbation method's applications,we respectively study disturbed fractional Duffing oscillator model,disturbed fractional Lotka biochemical oscillator model and disturbed fractional Whittaker model,and find these models' adiabatic invariants directly led by the symmetrical perturbation.In Sect.6,based on disturbed Nambu representation of fractional dynamics system,study the Mei symmetrical perturbation and Lie symmetrical perturbation of disturbed fractional Nambu system,and finds the adiabatic invariants directly led by these two kinds of the perturbation.As disturbed fractional Nambu system symmetrical perturbation method's applications,we respectively study disturbed fractional relativistic Yamaleev oscillator model,disturbed fractional Euler–Poinsot model and disturbed fractional general relativistic Buchduhl model,and find these models' adiabatic invariants directly led by the symmetrical perturbation.In Sect.7,based on disturbed Birkhoff representation of fractional dynamics system,study the Mei symmetrical perturbation and Lie symmetrical perturbation of disturbed fractional Birkhoff system,and finds the adiabatic invariants directly led by these two kinds of the perturbation.As disturbed fractional Birkhoff systemsymmetrical perturbation method's applications,we respectively study disturbed fractional Emden model,disturbed fractional general relativistic Buchduhl model and disturbed fractional Lotka biochemical oscillator model,and find these models' adiabatic invariants directly led by the symmetrical perturbation.Section 8 concludes the innovative work in this paper,and gives some suggestions for follow-up research on symmetrical perturbation of fractional dynamical systems.