A Study About Several Novel Routes To Mixed-mode Oscillations

The problem of mixed-mode oscillations(MMOs)with multiple time scales has extensive engineering background,and the problem that exploring various possible underlying mechanisms of MMOs and classifying types of MMOs is one of the leading subjects and hot interests in nonlinear science.Based on classical nonlinear systems such as the Rayleigh system,the Duffing system and the van der Pol-Duffing system,various new routes to MMOs,i.e.,the "speed escaping" mechanism of attractors,the tortuous mechanism of hysteresis curve and the delayed bifurcation mechanism,are revealed by using bifurcation theory,frequency-transform fast-slow analysis,numerical simulations,and so on.The basic contents of this dissertation are given as follows:In chapter 1,the development history of nonlinear dynamics,current research status of multiple time scales bifurcation types and research methods as well as the main contents of this are introduced.In chapter 2,the underlying mechanisms of MMOs caused by speed escape of attractors in the Rayleigh equation with multiple-frequency excitations are revealed.The fast subsystem exhibits two critical points,which bound the region of a periodic attractor or an equilibrium point attractor and outside which are divergent regions.Both the periodic attractor and equilibrium point attractor can rapidly move far away from the original place when the control parameter reaches the critical values.Based on this,the novel mechanism to MMOs which we call speed escape of attractors is revealed;Besides two different types of MMOs,i.e.,MMOs of point-point type and MMOs of cycle-cycle type,are obtained.In chapter 3,we investigate complex dynamical behaviors of the Duffing system with multiple-frequency external forcing,in which the turnover-of-hysteresis-induced MMOs patterns can be observed.It is shown that the equilibrium hysteresis curve of the fast subsystem becomes the one with twists and turns,which leads to distinct oscillations in the quasi-static processes of these MMOs patterns.And this forms a new route to MMOs,which we call turnover of hysteresis.Besides,the effects of forcingfrequencies and amplitudes on the turnover-of-hysteresis-induced MMOs are explored.Our study shows that the three frequency components observed in the MMOs are decided by the forcing frequencies,while the transition of the MMOs depends on the forcing amplitudes.In chapter 4,based on Hopf bifurcation delay,New routes to MMOs are presented and two novel MMOs patterns are obtained.These MMOs patterns are classified as compound "delayed supHopf/fold cycle"-"subHopf/supHopf" type and MMOs of "subHopf/supHopf" type via "delayed supHopf/supHopf" hysteresis loop,respectively.Our results show that Hopf bifurcation delay plays a decisive role in the generation of these MMOs patterns,which enriches the routes to MMOs and deepens the understanding of underlying mechanisms of MMOs.In chapter 5,some meaningful results are summarized.Also some existing problem s as well as the future work are pointed out.