Limit Cycle Bifurcations And Global Exponential Stability For Several Classes Of Differential Systems

In this paper, we mainly investigate limit cycle bifurcations for several classes of ifferential systems, together with the existence and global exponential stability of anti-ieriodic solutions of a biological model. This paper is mainly divided into the following our parts.In chapter 1, we first introduce the research background and significance concerning he limit cycle bifurcations and anti-periodic solutions, then present our main work.In chapter 2, we deal with the problem of limit cycle bifurcations near a 2-polycycle or-polycycle for a class of integrable systems by using the first order Melnikov function. We rst get the formal expansion of the Melnikov function corresponding to the heteroclinic oop and then give some computational formulas for the first several coefficients of the xpansion. Based on the coefficients, we obtain a lower bound for the maximal number of mit cycles near the polycycle. As an application of our main results, we consider quadratic ntegrable polynomial systems, obtaining at least two limit cycles.In chapter 3, we study limit cycle bifurcations of a kind of piecewise polynomial differ-ntial systems by perturbing a piecewise linear Hamiltonian system with multiple switching nes. We give the expression of the first order Melnikov function when the unperturbed piecewise Hamiltonian system has a compound global center, a compound homoclinic loop, i compound 2-polycycle, a compound 3-polycycle or a compound 4-polycycle respectively. 3y using the method of the first order Melnikov function, we obtain lower bounds of the aximal number of limit cycles in the above five different cases. Further, we derive upper ounds of the number of limit cycles in later four different cases.In chapter 4, we use the method of coincidence degree and construct suitable Lyapunov unctional to investigate the existence and global exponential stability of anti-periodic solu-ions of impulsive Cohen-Grossberg neural networks with delays on time scales. Finally, an example is given to illustrate our feasible results.