Modified slow-fast analysis method for slow-fast dynamical systems with two scales in frequency domain

A modified slow-fast analysis method is presented for the periodically excited non-autonomous dynamical system with an order gap between the exciting frequency and the natural frequency. By regarding the exciting term as a slow-varying parameter, a generalized autonomous fast subsystem can be defined, the equilibrium branches as well as the bifurcations of which can be employed to account for the mechanism of the bursting oscillations by combining the transformed phase portrait introduced. As an example, a typical periodically excited Hartley model is used to demonstrate the validness of the method, in which the exciting frequency is far less than the natural frequency. The equilibrium branches and their bifurcations of the fast subsystem with the variation of the slow-varying parameter are presented. Bursting oscillations for two typical cases are considered, which reveals that, fold bifurcation may cause the the trajectory to jump between different equilibrium branches, while Hopf bifurcation may cause the trajectory to oscillate around the stable limit cycle.