Some Bifurcation Problems Of The Plane Dynamic System And Applications

This paper consists of three parts.Firstly,on the basis of the series references [22]-[33], we summarize the sufficient and necessary conditions such that the homoclinic cycle of the qudratic system is de-fined by the cubic,quartic,quintic,sextic invariant algebraic curve,respectively. When the parameters of the homoclinic cycle of the qudratic system change to the bifurcation values of the generation and disappearance of the homoclinic, we get the eight ways of the homoclinic which disappears and may have the high-order bifur-cation qudratic system which arises the homoclinic cycle. Then, by perturbing the homoclinic bifurcation system properly, we obtain the conditions for the perturbed system arising limit cycle by using the rotation vector field theorem. Finally,the homoclinic bifurcation system considers the periodic perturbance, we obtain the conditions for the perturbed system arising chaotic behavior by using the method of Melnikov function.Secondly,on the basis of the references [12], consider the system with a true center of its linear part, using the implicit function theorem,we obtain a sufficient condition of the periodic solution which is generated by the bifurcation of a family of the closed trajectories. Further, constructing the suitable example which the original system has a family of closed trajectories.Then,by selecting the appropriated disturbance,we get the n limit cycles produced in a neighborhood of the closed trajectories.Thirdly,we propse a delayed Leslie-Gower predator-prey system with a single delay. The linear stability of the system is investigated and Hopf bifurcations are demonstrated. In particular, the formulae determining the direction of the bifurca-tions and the stability of the bifurcating periodic solutions are given by using the normal form theory and center manifold theorem. Finally, numerical simulations supporting our theoretical results are also included.