Basic Theory For Dynamics Of Fractional Birkhoff Systems

Fractional order calculus has developed more than300years. Mathematicians have proposed three highly recognized definitions of fractional derivatives; these definitions include the Riemann-Liouville, the Caputo and the Riesz fractional derivatives. In the end of the1970s, Mandelbrot discovered a fact that a large number of fractional dimension examples existed in nature. Since then the study of the fractional dynamics has become a hot topic and attracted a lot of scientists to the study of the theory of differential-integral. On later, fractional differential equations won wide development in applications, which include dynamics, Non-Newton hydromechanics, biomechanics, biophysics, etc.In recent20years, scientists have established fractional Lagrange mechanics, fractional Hamilton mechanics, fractional generalized mechanics, fractional dynamics of nonholonomic system and fractional generalized Hamiltonian mechanics. The research on Birkhoffian systems plays a fundamental role in engineering science. However, fractional Birkhoffian systems still stay in integer order calculus level, as one of significant classical mechanical system. Therefore, fractional Birkhoffian equations, gradient representation, integral theory, the symmetry theory, the theory of equilibrium stability and the theory of movement stability are subjects for continued research.In this paper, we establish basic theory of fractional Birkhoffian systems, which include fractional Pfaff-Birkhoff principle, fractional Birkhoffian equation, gradient representation, algebraic structure, Poisson integral, constructor method of integral invariants, the theory of equilibrium stability and application of the practical problems.Section1explains briefly the fractional dynamical theory and the history and status of Birkhoffian dynamical theory, makes an induction of the problems of this paper.Section2describes briefly the definitions and the properties of the Riemann-Liouville fractional derivatives, the Caputo fractional derivatives, and the Riesz fractional derivatives, respectively. Based on three kinds of fractional derivatives, by using the variational method of Birkhoffian systems, we give the fractional Pfaff-Birkhoff principle of Birkhoffian systems and the fractional Birkhoffian equation. Then we construct three fractional dynamical models by using the method of this paper.In Sect.3based on the definition of the Riesz fractional derivatives, we explore the conditions that fractional Birkhoffian system is a gradient system, and give the gradient representation of fractional Birkhoffian systems. As a special case of fractional Birkhoffian system, a fractional Birkhoffian system can be reduced to a Birkhoffian system. Then, we can obtain the gradient representation of Birkhoffian systems, and study whether the three fractional Birkhoffian systems meet the conditions of gradient system.In Sect.4based on the definition of the Riesz fractional derivatives, we study algebraic structure and Poisson integral. We discover that fractional autonomous Birkhoffian system possesses consistent algebraic structure and Lie algebraic structure, and then we give the Poisson integral theorems of the fractional autonomous Birkhoffian system. When αâ†'1, the Poisson integral theorems of the fractional autonomous Birkhoffian system can be reduced to the theorems of the autonomous Birkhoffian system. Finally, three practical models are given to illustrate the method and results of this section.Section5studies the variational equations and integral invariants of fractional autonomous Birkhoffian system. By using the variational equations and first integrals, we present a class of integral invariants of fractional autonomous Birkhoffian system. Examples of a fractional dynamical system are given to illustrate the method and results of the application.Section6studies the stability of the equilibrium states of the fractional autonomous Birkhoffian system. By using the direct method and first approximate method, we judge the stability of the equilibrium states of the fractional autonomous Birkhoffian system.Section7concludes the research results of this paper, and gives some suggestions in many other aspects of fractional Birkhoffian system.