Research On Bifurcation Dynamics Of Coupled Time-delay Systems

Coupled systems widely present in many scientific fields and have valuable signifi-cance. The coupling increases the complexity of the system, and makes the system pro-duce richer and more complex dynamics. The research about coupled systems is mainlyto discuss their dynamics, controls and synchronization, and these studies are based onthe qualitative theory of nonlinear dynamical systems. For example, we can obtain theconditions of amplitude death and the existences of various synchronization solutions us-ing Lyapunov stability theory. In addition, the system has abundant dynamic behavior atthe boundary of amplitude death. By discussing the bifurcations, we can get a variety ofdiferent topologies in the vector field.In this paper, we utilize the eigenvalue method, the center manifold theorem, thenormal form method, global bifurcation theorem and the symmetric Hopf bifurcation the-orem to study the stability, global Hopf bifurcation, the existences and spatio-temporalpatterns of periodic solutions, and high codimensional bifurcations including Bogdanov-Takens and double Hopf bifurcations of coupled time-delay systems. The details are asfollows:1. By taking the time delay as a parameter, and analyzing the characteristic equationsat the equilibria and the normal form for the bifurcation of coupled time-delay systems,we obtain the locally asymptotically stable regions of the equilibria, the condition of Hopfbifurcation, the direction of bifurcation and the orbitally asymptotical stability of bifur-cating periodic solutions. Linearize the systems at the corresponding equilibria and givethe distribution of eigenvalues corresponding to the transcendental characteristic equa-tions. We get the local stability of equilibria in linearly coupled Mackey-Glass system,time-delay coupled FitzHugh-Nagumo neural system, etc., using the theoretical resultsprovided by Wei and Ruan, and Routh-Hurwitz criterion. When the stability of the e-quilibrium changes, the system will experience a bifurcation. For the coupled systemsabove, we study a kind of codimension-1bifurcation, i.e., giving the existences of Hopfbifurcation; further, the property of Hopf bifurcation is given using Hale’s center mani-fold theorem, the normal form method provided by Hassard et al. and the algorithm ofcomputing normal forms of delay diferential equations raised by Wei. 2. Utilizing global Hopf bifurcation theorem and high dimensional Bendixson crite-rion, we prove the global existence of Hopf bifurcation of linearly coupled Mackey-Glasssystem. The global Hopf bifurcation theorem established by Wu states that if the connect-ed component of an isolated center is unbounded, we has proven that the correspondingperiodic solution and its period are bounded combining the Bendixson criterion of highdimensional ordinary diferential equations brought by Li and Muldowney, so the bifur-cation parameter must be unbounded. Thus, we get the global Hopf bifurcation.3. For some coupled systems with symmetry, we get the existence of symmetricHopf bifurcation, the spatio-temporal patterns of bifurcating periodic solutions such assynchronization, anti-phase synchronization, phase-locking, mirror-reflecting and stand-ing, and the stability and direction of corresponding periodic solutions. Taking delay-coupled FitzHugh-Nagumo neural system and delay complex oscillator network as sub-jects investigated, we reveal spatio-temporal patterns of symmetric Hopf bifurcating peri-odic solutions, such as synchronization and anti-phase synchronization, using the gener-alized symmetric Hopf bifurcation theorem developed by Wu, Guo and Lamb. Especially,we obtain the coexistence of phase-locked, standing and mirror-reflecting waves of com-plex oscillator network with delay, and further, we find the stability and the bifurcationdirection of the various forms of periodic solutions above combining the computationalmethods of center manifold and normal form about functional diferential equations giv-en by Faria with the results about the property of symmetric Hopf bifurcating periodicsolutions for ordinary diferential equations in the monograph written by Golubitsky et al.4. Finally, we give the existences of two types of high codimension bifurcations andthe changes of the local topologies caused by bifurcations, and find the existences of limitcycles, homoclinic orbits and three-dimensional tori. The bifurcations mainly includeBogdanov-Takens bifurcation of delay coupled FitzHugh-Nagumo neural system and thedouble Hopf bifurcation of the coupled limit cycle oscillator system with time delay. Bydiscussing the eigenvalues of corresponding linear parts, the critical values of the twokinds of codimension-2bifurcations are received. Referring to the method and process ofcomputing center manifold and normal form provided by Faria, we find the normal format the singular point and the bifurcation set near it, and give the complete classificationsof the local topologies.