Coupling Effects Of Two Time Scales In Modified Chua's Oscillator Under Periodic Excitations

The complicated behaviors as well as the bifurcation mechanism of the dynamical systems with different time scales have become one of the frontier and hot subjects at home and abroad.In recent years,lots of fruitful works have been developed around the coupling systems of two time scales,however,due to the characteristics of such systems,which still remains many problems to be solved.Based on the trend of two-time scale system,the related works of this paper is presented and the specific contents are as follows:To deeply reveal the effect of different time scales of the nonlinear systems with two or more periodic excitations,based on a modified four-dimensional Chua's oscillator,by introducing two periodically changed electric sources,a dynamical model with scales under two excitations with frequency ratio at 1:2 is established.When the two exciting frequencies are strictly resonant,both of which are much smaller than the natural frequency of the system,the combination of the two different exciting terms can be transformed as a function of a periodic term with single frequency,which can be regarded as a slow-varying parameter.The distribution of the fast subsystem equilibrium branches as well as the bifurcation behaviors with the different amplitudes of the excitations are given,three typical cases corresponding to the different situations of the exciting amplitudes are considered,in which different forms of bursting oscillations are observed.Furthermore,it is pointed out that the more fold bifurcation points involve the behaviors,the more complicated bursting oscillations may appear.In order to explore the mechanism of the two time scales of nonlinear systems with multiple types of excitation couplings,the system with two excitations which combine parametric excitation and external periodic excitation is taken as an example.When an order gap between the exciting frequency and the natural frequency exists,the obvious slow-fast effects in such systems can be derived.It is found that with the different excitation amplitude,the range of slow-varying parameter will also change,which corresponding to the fast subsystem presents different types of bifurcations and bursting oscillations,which is the main reason for the system exhibiting varied forms of bursting oscillation modes.Furthermore the relevant bifurcation analysis and transformation phase portraits are employed to account for the mechanism of the bursting oscillations.In addition,we found that the increase of the external exciting amplitude leads to the decrease of the temporal intervals when the trajectory passes different bifurcation points,respectively,which results in different transition forms between the quiescent states and the spiking states.Many nonlinear dynamical systems including numerous non-smooth factors in engineering,such as the switching in the electronic circuits and the collision motions in the mechanical systems,etc.Non-smooth systems exist many particularities,for instance,strong nonlinear and singularity of systems caused by the non-differentiability or discontinuity of vector fields,which reduces few researches on the bursting behaviors as well as the mechanism of the non-smooth systems with the coupling effects of two-time scales in frequency domain.In order to investigate the complicated behaviors as well as the bifurcation mechanism of the dynamical systems with different time scales and the non-smooth factors,by introducing a voltage-based steering switch and a periodically changed electric source,a non-smooth dynamical system of modified chua's oscillator with two time scales which in frequency domain is established.Due to the existence of non-smooth boundary,the trajectory is switched between different subsystems,and when the trajectory of the system passes across the non-smooth boundary,the behavior of sliding occurs.By choosing suitable parameter values,two different types of bursting oscillations are given,and based on the transformation phase portraits and bifurcation analysis,the evolution of dynamical behaviors corresponding to relative bifurcations is investigated.Finally,the influence of the amplitude and the frequency of external excitation on the distribution of bursting oscillations is mainly discussed and it is pointed out that the choice of exciting frequency possesses rationality in this paper.