Mixed mode oscillations as well as the bifurcation mechanism in a Duffing's oscillator with two external periodic excitations

The main purpose of the paper is to present an approach to account for the mechanism of bursting oscillations occurring in the systems with multiple periodic excitations. Since the traditional slow-fast analysis method can be used only for the systems with two scales in time domain, when there exists an order gap between the exciting frequencies and the natural frequency, how to explore the mechanism of the complicated dynamics remains an open problem, especially for the case when two exciting terms exist. To explain our approach, a relative simple Duffing's oscillator with two external periodic excitations is introduced as an example. For the case when one exciting frequency is integer times of the other exciting frequency, by employing Moivre's equation, the two exciting terms can be transformed into the functions of one basic periodic exciting term. Regarding the basic periodic exciting term as a slow-varying parameter, the two exciting terms can be changed into the functions of the slow-varying parameter, based on which the whole model can be transformed into a generalized autonomous system with one slow-varying parameter. Equilibrium branches as well as the related bifurcations of the generalized autonomous system can be derived with the variations of the amplitudes of the two excitations, which describes the relationship between the state variables of the generalized autonomous system and the slow-varying parameter. Considering the slow-varying parameter as a generalized state variable, one may obtain the so-called transformed phase portraits, which present the relationship between the state variable and the slow-varying parameter. The bifurcation mechanism of the mixed oscillations can be obtained by overlapping the equilibrium branches and the transformed phase portraits. Upon the approach, different types of bursting oscillations are presented. It is pointed that when the trajectory moves almost strictly along the segments of the stable equilibrium branches to the fold bifurcation points, jumping phenomena to other stable equilibrium segments can be observed, leading to the repetitive spiking oscillations, which can be approximated by the transient procedure from the bifurcation points to the stable equilibrium segments. Furthermore, because of the distributions of stable segments on the equilibrium branches of the generalized autonomous system may vary with the exciting amplitudes, the forms of bursting oscillations may change. When more fold bifurcations involve in the bursting attractors, more forms of quiescent states as well as spiking states exist in the oscillations, leading to different types of bursting oscillations.