Basic Theories For Dynamics Of Fractional Nambu Systems

The study of fractional differential-integral has developed more than300years. However,because of little practical application, fractional differential-integral has not been taken seriously.In the end of the1970s, Mandelbrot discovered a fact that a large number of fractional dimensionexamples existed in nature. Since then the study of the fractional dynamics has become a hottopic and won wide development in theories and applications. Nambu system is more generalthan generalized Hamiltonian system, which is a kind of basic dynamics system with science andengineering background. However, the Nambu system still stays in integer order calculus level,basic theory for dynamics of fractional Nambu system is a subject for continued research.In this paper, we establish basic theory of fractional Nambu system, which include thefractional Nambu equations, the algebraic structure, the Poisson integral, the method ofconstructing integral invariants, the stability theory for the manifolds of equilibrium states andthe applications of the practical problem.Section1explains briefly the history and status of fractional dynamics and Nambu systemdynamics, makes an induction of the problems of this paper.Section2describes briefly the definitions and the properties of the Riemann–Liouville, theCaputo, the Riesz–Riemann–Liouville and Riesz–Caputo fractional derivatives, respectively.Based on the definition of combined fractional derivative, using the method of Nambu–Poisson,four fractional Nambu equations are established, respectively. Then we construct three fractionaldynamical models by using the method of this paper.In Sect.3based on the definition of the Riesz–Riemann–Liouville fractional derivative, weexplore the relations among the fractional Nambu system, fractional generalized Hamiltonsystem, generalized Hamilton system, fractional Birkhoff system, Birkhoff system, fractionalHamilton system, Hamilton system, fractional Lagrangian system and Lagrangian system, andgive the transformation conditions.In Sect.4based on the definition of the Riesz–Riemann–Liouville fractional derivative, westudy algebraic structure and Poisson integral. We discover that fractional Nambu systempossesses Lie algebraic structure, and then we give the Poisson integral theorems of thefractional Nambu system. As special cases, the Poisson integral conclusions of Nambu system,fractional generalized Hamilton system, gneralized Hamilton system, fractional Hamilton system and Hamilton system are obtained. Finally, three models are given to illustrate the method andresults of this section.In Sect.5we study the variational equations and integral invariants of fractional Nambusystem with Riesz–Riemann–Liouville fractional derivative. By using the variational equationsand first integrals, we construct a class of integral invariants of fractional Nambu system. Asspecial cases, the variational equations and integral invariants of Nambu system, fractionalgeneralized Hamilton system, gneralized Hamilton system, fractional Hamilton system andHamilton system are obtained. Examples of fractional dynamical system are given to illustratethe method and results of the application.In Sect.6based on the definition of the Riesz–Riemann–Liouville fractional derivative, westudy the stability theory for the manifolds of equilibrium states of fractional Nambu system,which includes equilibrium equations, perturbation equations, first approximate equations andthree propositions on the stability of the manifolds of equilibrium states of the system. As specialcases, we give the conclusions on the stability of the manifolds of Nambu system, fractionalgeneralized Hamilton system, gneralized Hamilton system, fractional Hamilton system andHamilton system. Examples of practical models are given to illustrate the method and results ofthe application.Section7concludes the research results of this paper, and gives some suggestions in manyother aspects of fractional Nambu system.