Fractional Birkhoffian method for equilibrium stability of dynamical systems

• Fractional Birkhoffian method for equilibrium stability of a dynamical system is presented.
• Seven criterions for new method of equilibrium stability are given.
• The new method׳s applications to dynamical systems are studied.
• Four kinds of actual fractional dynamical models are constructed.
• Equilibrium stability of four actual fractional dynamical models is explored, respectively.

In this paper, we present a new method, i.e. fractional Birkhoffian method, for stability of equilibrium positions of dynamical systems, in terms of Riesz derivatives, and study its applications. For an actual dynamical system, the fractional Birkhoffian method of constructing a fractional dynamical model is given, and then the seven criterions for fractional Birkhoffian method of equilibrium stability are established. As applications, by using the fractional Birkhoffian method, we construct four kinds of actual fractional dynamical models, which include a fractional Duffing oscillator model, a fractional Whittaker model, a fractional Emden model and a fractional Hojman–Urrutia model, and we explore the equilibrium stability of these models respectively. This work provides a general method for studying the equilibrium stability of an actual fractional dynamical system that is related to science and engineering.