Study On Approach And Application Of Limit Cycle Bifurcation In Planar Vector Fields

Bifurcation theory in dynamics systems is one of the primary branches of qualitative theory of differential equations. It is chiefly concerned with changing rules of global topological structure of trajectories of vector fields depending on parameters because of the charge of parameters. About bifurcation theory of plane vector fields, the study on bifurcations of the limit cycle has become an important and popular topic. Hilbert' 16th problem is to determine the minimum upper bound for the number of limit cycles of plane vector fields, and their relative location. This research has combined with bifurcation theory since 1980s.There are many mathematicians devoting themselves to studying the Hilbert 16th problem or its weak form, posed by Arnold in 1977. However, the problem is still open even for the quadratic Hamiltonian system with quadratic perturbations. The weak Hilbert 16th problem is to determine the number of zeros of Abelian integrals, which is related to the number of limit cycles bifurcated from a given Hamiltonian system. Because of difficulty for solution to high equations, the study of Abel integrals is difficult. By now, the study on the weak Hilbert 16th problem is still the one of popular topic.This thesis focuses on approach and application of limit cycle bifurcation of planar vector fields. The main content can be generalized as the following.1. Using the Picard-Fuchs equations, the property of elliptic integrals and analytic theory of ordinary differential equations, it is proved that three is minimum upper bound for the number of isolated zeros of Abelian integrals for the cubic perturbations of cubic Hamiltonian integral systems with double centers. It has improved the conclusion in paper [66].2. An algebraic method is derived for calculating the number of zeros of Abelian integrals in this paper. It is different from former methods. The calculation can progress by symbols computing systems and make the study of limit cycle bifurcation change from qualitative to quantitative. As an example, the cyclicity of period annulus of non- symmetric and non- degenerate quadratic Hamilton systems under quadratic perturbations is two, and this two limit cycles has of arbitrary position.3. Limit cycle bifurcation of planar quadratic non-Hamiltonian integral systems is discussed. First, cyclicity of a class of non- Hamiltonian integrals systems is researched in thereversible case Q₃^R and b > 2, on the basis work of Jiang Yu and Chengzhi Li. Then, a classof Poincare bifurcation for quadratic systems with a bound of hyperbola and a single center is solved using the method of power series expansion for Abel integrals for the first time. The method is more suitable for high polynomial systems. Last, some concrete methods constructed are proposed for Poincare bifurcation of quadratic systems with two centers and two unbounded heteroclinic loops, in which there are three limit cycles with(0, 3)distribution or a triple limit cycle.4. A kind of SIRS epidemic models with nonlinear incidence rates kIp-1Sqis studied according to Hopf bifurcation theory for Limit cycle bifurcations of planar vector fields, andan accurate expression of equilibrium is obtained initially in case of p>2,q>1. Thenumerical example and simulative result is also given.The method to simplify expression of equilibrium is fit for general condition, and makes the calculation of focus quantity be concise and feasible.SEER, epidemic models of SARS and its parameter identification systems with a latent period and nonpermanent immunity are also established in this paper. The main mathematicalproperties of such control systems, flow-invariance of system (S,f(y,v)) andweak-invariance of system (S, F(y, Vad)) are investigated and proved. The numericalsimulations are done based on the data announced by world Health Organization, China Health Department and Hong Kong Health Department. The results show that our model and algorithm are accurate and effective.