Bifurcation Analysis Of Nonlinear System With Time Delay

In this Ph.D.thesis,the center manifold theoren,normal form theory,method of multiple scales,frequency domain approach,perturbation method and global Hopf b-ifurcation theorem are used to investigate the bifurcations of several types of delayed nonlinear models with strongly practical background.The main results are summarized as follows:1.Bifurcations of a simple discrete model with time delay,i.e.x(n+1)=(1-?)x(n)+?x(n-?)-x3(n-?)is studied.By discussing root distributions in a special polynomial of the form ??(?-(1-?))-?,we determine the necessary and sufficient conditions to keep the model stable under different time delay(?=1,2,3).Moreover,the critical conditions under which a pair of complex conjugate eigenvalues of the polynomial with ?=2/3 lie on the unit circle(Hopf bifurcation may occur)are proposed.We should mention that above analysis can also be applied to discover the existence of codimension-two bifurcations with 1:1 strong resonance and 1:2 strong res-onance in this model.Further analysis shows that the model and its linearized system is equivariant when ?,? takes a particular value.A comparison is made between a simple discrete model x(n+1)=(1-?)x(n)+?x(n-1)-x3(n-3)and its perturbation with de-layed feedback x(n+1)=(1-?)x(n)+?x(n-1)-x(n-3)+?(x(n-3)-x(n-1)).There exist rich dynamics in the perturbed system,including the coexistence of multiple oscil-lation patterns(p2 orbits,p4 orbits,p8 orbits),limit cycles by Hopf bifurcations,chaos and even hyperchaotic behavior.Therefore,the introduction of delayed feedback can alter the special topological structure of the original system,which leads to complexity.2.Bifurcation of a Cournot duopoly game with time delay is analyzed.Based on Cournot duopoly game,we propose a Cournot duopoly continuous game with two delays.By stability analysis,sufficient conditions under which Nash equilibrium is asymptotically stable are derived.According to Hopf bifurcation theorem,we give the critical value of the parameter to ensure that Hopf bifurcation occurs.Explicit formu-lae for determining the direction and the stability of Hopf bifurcation and periods are obtained.Numerical simulations are carried out to illustrate our main results and the influence of system parameters on stability.3.Bifurcation of a Goodwin gene expression model with three delays is studied.By using the frequency domain approach,we choose the sum of all delays as bifurca-tion parameter to investigate the characteristic equation and obtain the critical value of the parameter to ensure that Hopf bifurcation occurs.The direction and the stability of bifurcating periodic solutions are determined by the Nyquist criterion and the graphi-cal Hopf bifurcation theorem.By using the Bendixson's criterion for high-dimensional ordinary differential equations and global Hopf bifurcation theorem,the global exis-tence of periodic solutions is established.Numerical results are also included to give a verification test of the theoretical analysis.4.Bifurcation of a gene expression model with with state-dependent delay is taken into account.We introduce the state-dependent delay into a gene expression model and propose a gene expression model with with state-dependent delay.We analyze the local stability of steady-state solution.Then,conditions to ensure that Hopf bifurcation occurs are determined.Furthermore,different to traditional procedure,perturbation method is employed to determine the direction of Hopf bifurcation and the change of the period of the bifurcating periodic solutions.In order to illustrate our theoretical results and reveal the global existence of periodic solutions,several numerical examples of state-dependent delay are also included in the end.5.Bifurcation of a Kaldor-Kalecki business cycle model with time delay is investi-gated.Frequency domain approach is used to obtain the conditions to ensure that Hopf bifurcation occurs and the Nyquist criterion and the graphical Hopf bifurcation theorem are applied to determine the direction and the stability of Hopf bifurcation.Then Hopf bifurcation of this model is controlled based on a time-delayed feedback controller.We find that when control parameter reaches a certain value,Hopf bifurcation may be post-poned and even disappers.At last,we introduce a distributed delay into this model and build a Kaldor-Kalecki business cycle model with both discrete and distributed delays.By using the method of multiple scales,formulae for determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are derived.Numerical simulations are carried out to illustrate the influence of system parameters on stability.