Complicated Behaviors And Mechanism Analysis Of Some Nonlinear Dynamical Systems With Special Structures

As one of the significant branch of nonlinear dynamics, systems with special structures,which contain many particular nonlinear characteristics, have attracted much attention in recent years and become one of the hot topic in the fields of nonlinear dynamics, In this dissertation, the dynamical behaviors and bifurcation mechanism of the systems with special structures, such as switching, multiple time scales, delay etc., have been investigated by using bifurcation theory of nonlinear dynamics, non-smooth dynamics analysis and numerical simulations. Meanwhile, based on the theory of Poincare mapping, Floquet multiplier and methods of fast-slow dynamic analysis, the dynamical evolution process and the road to complex motion are discussed with the changing of parameters.Complicated behaviors of the model in parameter periodic switching systems or periodic switching systems between two non-identical subsystems are investigated in details by constructing concrete models. Upon the local analysis, the critical conditions such as fold bifurcation and Hopf bifurcation are derived to explore the bifurcations of the compound systems with different stable solutions such as focus or stable cycle in the two subsystems. Different types of non-smooth bifurcations occur in this switching system, which may result in chaotic oscillations. Using improved Poincare mapping analysis method, the Lyapunov exponents and the Floquet multiplies of this switching system is calculated. Based on the conditions when the Floquet multiplies of corresponding fixed point associated with the periodic solution pass the unit circle, parameter bifurcation sets of the switched system can be obtained, dividing the parameter space into several regions corresponding to different types of attractors. The system could undergo the period-doubling bifurcation, saddle-note bifurcation, symmetry-breaking bifurcation and so forth to chaos, as the control parameter was set on some certain intervals.We present the bursting patterns in delayed Duffing oscillator, including symmetric Fold-fold bursting and symmetric Hopf-Hopf bursting when periodic forcing changes slowly. We make an analysis of the system bifurcations and dynamics as a function of the delayed feedback and the periodic forcing. We calculate the conditions of Fold bifurcation and Hopf bifurcation as well as its stability related to external forcing and delay. We also identify two regimes of bursting depending on the magnitude of the delay itself and the strength of time delayed coupling in the model. When A< 1, the symmetric Fold/fold burster appears and is caused by the fold bifurcation. When A> 1 and r satisfies the critical condition of Hopf bifurcation, symmetric Hopf/Hopf burster can be observed, in which two limit cycles caused by the Hopf bifurcation interact with each other. Delay can be used as a control parameter to change the dynamical behaviors in different bursters in our research.The mechanism for the action of time delay in a non-autonomous system with two time scales is investigated. The original mathematical model under consideration is a shape memory alloy oscillator with external forcing. Due to the system in a Fold bifurcation and Hopf bifurcation of the coexistence of bifurcation model, oscillator appears complex busting oscillations. Due to the stable limit cycle and stable equilibrium points, the system trajectories jump and transit from the quiescent state to the excited state. System dynamics behavior exist two kinds of movement trend, one kind is to the limit with divergence trend, the other is a trend of jumping movement along the equilibrium curve. Rich bursting patterns can be presented with properly chosen delay and gain in the delayed feedback path. Applying a time delayed feedback control may be one of the best approaches to control or create complicated bursting dynamical motions.At last, this dissertation addresses the modified function projective bursting synchronization (MFPBS) for a class of multiple-time-scaled systems with uncertainties and external disturbances. An active control law is established to guarantee uncertainties and disturbances rejection and realize the MFPBS. By using an active sliding mode control method (SMC), bursting synchronization can occur between two different chaotic systems with different time scales. The sufficient condition is drawn for the asymptotical stability of error dynamics. More complex bursting oscillations such as chaotic bursting, as well as chaotic or periodic spiking can be obtained in slave systems with free scales when the MFPBS is maintained.