Dynamic Analysis Of Several Classes Of High Dimensional Smooth And Non-smooth Systems

In this paper,bifurcation and chaotic behavior for two kinds of smooth and one kind of non-smooth dynamical systems are studied.In Chapter 2,we discuss a class of three-dimensional smooth systems.The stability and bifurcation of the nonlinear dynamic model of a new type of energy storage unit are analyzed.First,the asymptotic stability condition of the equilibrium point is obtained by Routh-Hurwitz criterion.Secondly,the parameters of the bifurcation are determined by Hopf bifurcation theorem.Then,through the first Lyapunov coefficient theorem,the supercritical and subcritical cases of Hopf bifurcation are obtained.At last,numerical simulation is carried out by Runge-Kutta method to verify the theoretical analysis results.In Chapter 3,we study a class of four-dimensional smooth systems.Nonlinear dynamic characteristics of the rectangular thin plate with concentrated mass under combined action of primary resonance and 1:3 internal resonance are discussed.Firstly,based on the modulation equation,a near integrable Hamiltonian system can be obtained by a transformation.Then,the global bifurcations and chaotic dynamics for the rectangular thin plate are analyzed with the energy-phase method which is proposed by Haller and Wiggins.Finally,the effects of the phase shift and dissipation factor on pulse sequence and layer radius are analyzed.The results show that in the case of resonance,there exist the Silnikov-type multi-pulse orbits homoclinic to certain invariant sets,and the homoclinic trees describing the repeated bifurcation of multi-pulse solutions are obtained.The theoretical analysis results are verified by numerical simulations.In Chapter 4,a class of six dimensional non-smooth systems is analyzed.A three-degree-of-freedom mechanical impact vibration system is studied.Firstly,the regular modal matrix of the system is obtained;the general solution of the system equation is obtained by the method of modal superposition.Secondly,the Poincaré mapping is established,and the characteristic polynomial is obtained by the Jacobi matrix of the Poincaré mapping at the fixed point.Then,the parameter values of the bifurcation are discriminated by the double period bifurcation theorem.Finally,the theoretical analysis results are confirmed by numerical simulations.