The Problems Of Bifurcation Computation And Stability Of Several Class Differential Equations

There is abundant theory about the limit cycles bifurcation problem of smooth autonomoussystems on the plane. Such as Hopf bifurcation, Poincare′bifurcation, Homoclinic, Heteroclinicbifurcation etc. And the research results derive well applications in concrete equations. Whenthe origin of the linearized systems of the study systems is a focus, by computation the Liapunovconstants of the systems, one can discuss the problems of center-focus and limit cycle bifurcation.When considering the bifurcation problem of the near-Hamiltonian systems, it will use its firstorder Melnikov function.The research about limit cycles bifurcation on non-smooth systems on the plane is still at aninitial stage. When the origin of the linearized systems of general non-smooth systems is focus-focus type, some authors gave the first several Liapunov constants of the non-smooth systems.Recently, the problem of the number of limit cycles which bifurcate from the perturbed periodicorbits is considered in [33].In this thesis, we mainly considered two aspects content, one hand, we considered the limitcycle bifurcation problem about non-smooth dynamical systems on the plane, it is presented inChapter 2 and 3; on the other hand, we discussed the stability, bifurcation problems about severaldelayed predator-prey systems and did numerical simulations, one can see Chapter 4-7.In Chapter 2, the problem of the number of limit cycles bifurcated from the origin of a classnon-smooth Lie′nard systems on the plane is researched, by incorporating some transformationsand constructing the Poincare′return maps to compute the Liapunov constants of the correspondingsystems. By referring [21] and [23], we give some new techniques and methods, then some newtheory are obtained, and some concrete examples are presented. In Chapter 3, similar to using thefirst order Melnikov function of the perturbed smooth Hamiltonian systems on the plane, one candiscuss the number of bifurcating limit cycles of the systems. We use new techniques to deducethe first order Melnikov function of a class perturbed piecewise Hamiltonian systems. By usingthe Melnikov function to some concrete examples, we know the number of bifurcating limit cyclesof the systems.In Chapter 4, a class predator-prey system with harvesting and stage structure is consideredby [42] and [43], respectively, the authors only considered the properties of some special case ofthe system. We discussed the local and global stability problems of the positive equilibrium of thegeneral case of the system. In Chapter 5 and 6, by constructing appropriate Liapunov functionsand functionals, the stability problems of two class predator-prey systems are researched. A delayed predator-prey model with prey dispersal and refuge is considered in Chapter 7,by using Routh-Hurwitz theorem, we analyzed the conditions of the local stability and bifurcatedperiodic solution of the positive equilibrium. By transforming the normal form and using thecenter manifold theorem, we gave the expressions of determining the Hopf bifurcation directionand the stability of the bifurcating periodic solution. Finally, by applying the numerical methods,we considered several bifurcation phenomena of this system under the impulsive perturbation.The main innovation of the thesis: 1. present the method to determine the order of the finefocus and the Hopf cyclicity of non-smooth Lie′nard systems on the plane; 2. deduce the first orderMelnikov function expression of the perturbed piecewise Hamiltonian systems; 3. give stabilityanalysis and numerical simulations on some modified biological mathematical models.