Stability for manifolds of the equilibrium state of fractional Birkhoffian systems

Abstract

In the paper, we present a new stability theory of fractional dynamics, i.e., the stability for manifolds of equilibrium state of a fractional Birkhoffian system, in terms of Riesz derivatives, and explore its applications. For an autonomous fractional Birkhoffian system, the gradient representation and second-order gradient representation are studied, and the conditions under which the system can be considered as a gradient system and a second-order gradient system are given, respectively. Then, equilibrium equations, perturbation equations and first approximate equations are obtained, and the stability theorem for manifolds of equilibrium state of the general autonomous system is used to a fractional Birkhoffian system, and three criterions on the stability for manifolds of the equilibrium state of the system are investigated. As special cases of the fractional Birkhoffian method, the three criterions can also be applied to a traditional Birkhoffian system, a fractional Hamiltonian system and a classical Hamiltonian system, respectively. As applications, by using the fractional Birkhoffian method of this paper, we construct three kinds of fractional dynamical models, which include a four-dimensional fractional dynamical model, a fractional Henon–Heiles model and a fractional Lorentz–Dirac model, and we explore the stability for manifolds of the equilibrium state of these models, respectively. This work provides a general method for studying the dynamic stability of an actual fractional dynamical model that is related to science and engineering.