A new method of dynamical stability, i.e. fractional generalized Hamiltonian method, and its applications

In the paper, we present fractional generalized Hamiltonian method of dynamical stability, in terms of Riesz-Riemann-Liouville derivative, and study its applications. For an actual dynamical system, the fractional generalized Hamiltonian method of constructing a fractional dynamical model is given, and then the six criterions for fractional generalized Hamiltonian method of dynamical stability are presented. As applications, by using the fractional generalized Hamiltonian method, we construct five kinds of actual fractional dynamical models, which include a fractional Euler-Poinsot model of rigid body that rotates with respect to a fixed-point, a fractional Hojman-Urrutia model, a fractional Lorentz-Dirac model, a fractional Whittaker model and a fractional Robbins-Lorenz model, and we explore the dynamical stability of these models, respectively. This work provides a general method for studying the dynamical stability of an actual fractional dynamical system that is related to science and engineering.