Double-Hopf Bifurcation In Delay Differential System And Its Application

Delay’s generation is because people can not react to some external changes(such as seeing lightning,hearing thunder and feeling rain)in a timely way,which imperceptibly delays the correct time when people receive information.In research of many actual models,it is necessary to consider the impact of delay.In this thesis,The KorenFeingold(KF)cloud-rain model,a typical delay differential system and the main research object,was primitively studied as the problem of predator and prey.So later,some scholars began to research its bifurcation phenomena and dynamic behaviors near the bifurcation point,these behaviors including the existence of periodic and quasiperiodic solutions,and so on.For KF model,we can use the Garlerkin method,which takes orthogonal polynomial as the basis function,to get its approximate system and solve it,so as to reduce the difficulty of research,and Koornwinder polynomial is selected in this thesis.The main research of this thesis includes the following three points: First of all,based on the research idea of Hopf bifurcation of GK(Galerkin-Koornwinder)approximation system,taking the time delay and time scale α as the double-Hopf bifurcation parameters.The system’s normal form and its coefficient expression are obtained by using the center manifold theorem to take effectively reduction.The conditions of diagnosing the type of approximation system’s double-Hopf bifurcation are presented.Thus,The above states are presented as a decision theorem and given the corresponding proof.Secondly,when the double-Hopf bifurcation is studied,unfold of normal form contains at least the fifth-order resonance terms.Then we find that the leading-order approximation of the center manifold function cannot satisfy the coefficient representation of the fifth order terms,so the expression of the higher-order approximation of the center manifold function need to be further calculated by Taylor series.Thirdly,the double-Hopf bifurcation decision theorem and the higher order approximation of the central manifold function are applied to the KF cloud-rain model.Taking two sets of different parameters to analysis the two types of double-Hopf bifurcation,which including the ”simple” case and the ”difficult” case.And the stability of the system’s equilibrium points near each bifurcation point are proved theoretically.