Mixed-mode Oscillations And Their Dynamical Mechanisms In A Class Of Slow Time-varying Nonlinear System

The problem of mixed-mode oscillations in nonlinear systems with multiple time scale is one of the leading subjects and hot intrests in nonlinear science.It has wide application background, and involved in various fields of engineering and technology.In this dissertation, a class of special nonlinear system with multiple time scale have been concerned, i. e. modified Van der Pol-Duffing system(MADVP) with slow time-varying. This dissertation discussed the generation of mixed-mode oscillations in systems with multiple time scales resulting from the effects of slowly changing excitations. It will analyze the effect of the amplitude and frequency of slowly changing excitations on mixed-mode oscillations in detail. The phenomenon of mixed-mode oscillations have been classified. The main contents are as follows:Firstly, This paper investigates complex dynamical behaviors in the MADVP system with a slowly varying external forcing. Two-time scale mixed-mode oscillations(MMOs), resulted from the presence of slowly changing external forcing,are revealed.When the equilibrium of the fast subsystem loses its stability, two different dynamical characteristics can be presented in the fast subsystem, i.e., the so-called “mono-stability” and “bi-stability”. MMOs related to the “mono-stability”and “bi-stability” are studied: when equilibrium point attractor instability the trajectory of system only to the unique trajectory attractor, it is so called that“mono-stability”; when equilibrium point attractor instability the trajectory of system can trace the limit circle attractor or equilibrium point attractor, it is so called that“bi-stability”. Three different kinds of MMOs are obtained, i.e., the sym-metric MMOs of “delayed supHopf/delayed supHopf” type,“subHopf /fold cycle” type and“subHopf/subHopf” type.Secondly, This dissertation introduced a method of studying the mixed-mode oscillations for nonlinear system with two slow time-varying excitation. When the ratio of frequency between the two slow time-varying excitation satisfied certain conditions, we may introduce a new variable, so that the two slow time-varying excitation can be represented by the slow variable, and its expression has beendetermined. Thus, the system with two slow time-varying excitation can be transformed to a new fast-slow system, where the two slowly varying external forcing can be represented by a new slowly varying external forcing. Thereafter, we reveal the dynamical mechanisms of mixed-mode oscillations in system with tow slowly varying external forcing by analyzing the fast subsystem of fast-slow system. This paper cased in MADVP with two slowly varying external forcing, verifing the effectiveness of the method. We analyzed the mechanisms of mixed-mode oscillations from the point of different resonance mode and amplitudes.Finally, the work of graduate stage have been summarized. Some problems have been pionted out. The author read some relevant literature, so that the content of study can be improved in the future.