Bifurcating Periodic Orbits And Hidden Attractor Of Nonlinear Dynamic Systems

Bifurcation problem is one of the most important subjects in nonlinear dy-namical systems. The main research object of bifurcation theory is to investigate the structural unstable systems and the phenomenon of the topological structures changing with the parameters of the systems varying. Bifurcating periodic solution is a very important kind of these bifurcation phenomena, and it correspond to the self-excited vibration in dynamical systems. Recently, some researcher find that there are not only self-excited vibration in dynamical systems, but also "hidden attractors" as well, and which is not easy to find and locate. This kind of vibration phenomenon can also cause unnecessary loss in our life and productive activity. Therefore, to further understand the dynamical behavior, we need to investigate the nature of hidden attractor and to locate hidden attractor.This paper deals with the issues of Hopf bifurcation and hidden attractors of some nonlinear dynamical systems, and the main theoretical structure of the paper is stated as follows.Taking delay τ as the bifurcation parameter, we derive the conditions under which the equilibrium point lose its stability when τ increase from zero and be-come larger gradually. The main way is to discuss the distribution of the roots of characteristic equation of the linearized system. Based on the center manifold theory of abstract differential equations, we calculate the bifurcation direction and the stability of the bifurcating periodic solutions in the following way:Firstly, by projecting the retarded functional differential equation into the center manifold and utilizing the normal form methods, we can derive the conditions to determine the bifurcation direction, the stability, amplitude, period of the bifurcating peri-odic solutions, and the formula of the projection in the center manifold. Further, based on the existence theorem of global periodic solutions, we derive the suffi-cient conditions for the bifurcating periodic solutions to occur in the large scale of the parameter, and determine the lower limit of the number of the possible global bifurcating periodic solutions. When a dynamic systems is symmetric, the pure imaginary characteristic roots are not simple. To deal with this case, Wu et al. extended such equivariant Hopf bifurcation theory to DDEs, and their work is the theoretic foundation for studying the dynamic systems with symmetric.Applying the harmonic linearization and describing function methods modify-ing by Leonov and Kuznetsov et al., and combined the method of small parameter, we can obtain the existence conditions and the initial conditions of periodic so-lution. Further more, using a special analytical-iteration algorithm-iteration pro-cedure and taking the point of harmonic periodic orbit at the first step, we can locate the hidden periodic oscillators or hidden chaotic attractors.Based on the instruction of the above theories, the main creativities in this thesis are as follows:1. We investigate the bifurcation phenomena of a ratio-dependent predator-prey model with two time delays. By analyzing the corresponding characteristic equation, the local stability of the positive equilibrium and the semi-trivial equi-librium was discussed. The existence of Hopf bifurcation for this system at the positive equilibrium was also established, and further we may determine the di-rection of Hopf bifurcation and the stability of periodic solutions bifurcating from the positive equilibrium at the critical point. Furthermore, we show that the local Hopf bifurcation implies the global Hopf bifurcation after the second critical value of delay. The global attractiveness of positive equilibrium are also obtained.2. An SIR epidemic model incorporating media coverage with time delay is proposed. The positivity and boundedness was studied firstly. The golbal asymp-totically stability of the disease free equilibrium and the local stability of endemic equilibrium is studied in succession. And then, the conditions on which periodic orbits bifurcating are given. Further more, we show that the local Hopf bifurcation implies the global Hopf bifurcation after the second critical value of delay. The obtained results show that the time delay in media coverage can not affect the stability of the disease-free equilibrium. However, when R0>1, the stability of the endemic equilibrium will be affected by the time delay, there will be a family of periodic orbits bifurcating from the endemic equilibrium when the time delay in-creasing through a critical value. Finally, some examples for numerical simulations are also included.3. An autonomy system with time-delayed feedback is studied by using the theory of functional differential equation and Hassard’s method; the conditions on which zero equilibrium exists andHopf bifurcation occurs are given, the quali-ties of the Hopf bifurcation are also studied. Finally, several numerical simulations are given; which indicate that when the delay passes through certain critical val-ues, chaotic oscillation is converted into a stable state or a stable periodic orbit. Illustrating with numerical simulations, we show that delayed feedback controller (DFC) is an effective method for chaos control.4. A general autonomous Van der Pol-Duffing oscillator is studied. Several issues, such as periodic bifurcations and the dynamical structures of the system are investigated either analytically or numerically. Especially, a phenomenon of hidden attractors is noticed and an algorithm for location of hidden attractors is given. The obtained results show that hidden attractors exist around the chaotic attractors, periodic attractors and stable equilibria.5. A coupled Duffing equation with time delay is studied. By means of char-acteristic roots technique, sufficient conditions are obtained for Hopf bifurcation occurring. And the spatio-temporal patterns of the bifurcating periodic solutions are also obtained. Some examples are given to demonstrate the theoretical analysis. Especially, in our numerical simulations, we find that there are hidden attractors in this coupled system, it’s the first time to find hidden attractor in dynamic system with time delay up to now.