Dissertation > Excellent graduate degree dissertation topics show
Research on Bifurcation Dynamics of Coupled Timedelay Systems
Author: LiYanQiu
Tutor: JiangWeiHua
School: Harbin Institute of Technology
Course: Basic mathematics
Keywords: coupled system stability periodic solution normal form bifurcation
CLC: O19
Type: PhD thesis
Year: 2012
Downloads: 71
Quote: 0
Read: Download Dissertation
Abstract
Coupled systems widely present in many scientific fields and have valuable significance. The coupling increases the complexity of the system, and makes the system produce 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 using 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 theorem to study the stability, global Hopf bifurcation, the existences and spatiotemporalpatterns of periodic solutions, and high codimensional bifurcations including BogdanovTakens and double Hopf bifurcations of coupled timedelay 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 timedelay 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 bifurcating periodic solutions. Linearize the systems at the corresponding equilibria and givethe distribution of eigenvalues corresponding to the transcendental characteristic equations. We get the local stability of equilibria in linearly coupled MackeyGlass system,timedelay coupled FitzHughNagumo neural system, etc., using the theoretical resultsprovided by Wei and Ruan, and RouthHurwitz criterion. When the stability of the equilibrium changes, the system will experience a bifurcation. For the coupled systemsabove, we study a kind of codimension1bifurcation, i.e., giving the existences of Hopfbifurcation; further, the property of Hopf bifurcation is given using Hale’s center manifold 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 criterion, we prove the global existence of Hopf bifurcation of linearly coupled MackeyGlasssystem. The global Hopf bifurcation theorem established by Wu states that if the connected 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 bifurcation 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 spatiotemporal patterns of bifurcating periodic solutions such assynchronization, antiphase synchronization, phaselocking, mirrorreflecting and standing, and the stability and direction of corresponding periodic solutions. Taking delaycoupled FitzHughNagumo neural system and delay complex oscillator network as subjects investigated, we reveal spatiotemporal patterns of symmetric Hopf bifurcating periodic solutions, such as synchronization and antiphase synchronization, using the generalized symmetric Hopf bifurcation theorem developed by Wu, Guo and Lamb. Especially,we obtain the coexistence of phaselocked, standing and mirrorreflecting waves of complex 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 given 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 threedimensional tori. The bifurcations mainly includeBogdanovTakens bifurcation of delay coupled FitzHughNagumo 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 codimension2bifurcations 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.

Related Dissertations
 Analysis and Study of Abutment Stability in Concrete High Arch Dam by ThreeDimensional Nonlinear Finite Element Method,TV642.4
 The Study on Structural Calculation and Analysis Method of Latticetype Crane with Variable Crosssection Boom,TH21
 Power System Dynamic Voltage Stability Simulation Study Based on Precise Integration Method,TM712
 Simulation and Analysis about Switched Reluctance Generator Power Supply System,TM31
 Research and Design for the Laser’s Power Control System of Laser Direct Writing,TN249
 Stability Analysis of Systems with Time Delays,TP13
 Research on InputToState Stability of DiscreteTime Nonlinear Systems,TP13
 The Study of Dynamic Simulation of the Passive Dynamic QuasiQuarupedal Walker,TP242.6
 Astudy on Samuel Huntington’s Theory of Political Stability,D09
 Stability Analysis of Roller Compacted Concrete Gravity Dam Based on Timehistory Method,TV642.2
 Thermal Stability of Complexes of Chitosan Quaternary Ammonium Salt and Metal Ion,O634
 Research on Antiperiodic Solutions of Delayed Cellular Neural Networks without Assuming Global Lipschitz Conditions,TP183
 The Study and Application of the Generalized Hamiltonian System with Spherical Foliation Structure,O175
 Evaluation on Stability of Threshing and Redrying Lamina Structure of Tobacco,TS443
 With Delay fishing items and based on the ratio of prey  predator model,O175
 Exact Observability of Stochastic TimeVaring Systems and Weak Stability of DiscrereTime Stochastic TimeInvariant Systems,O231
 The Reform and Study Ofsitedirected Mutagenesis of Acid Xylanase XynⅢ from Aspergillus Niger,TQ925
 The Oxygen Reduction Activity of the PtBased Catalyst: First Principles Study,TM911.4
 Study on Some LotkaVolterra Predatorprey Models,O175
 Qualitative Study of Almost Periodic Solutions of Some Competition System,O175
 The Electron Dynamics and Transport Properties in the Coupled DoubleQuantumDot System,O471.1
CLC: > Mathematical sciences and chemical > Mathematics > Dynamical systems theory
© 2012 www.DissertationTopic.Net Mobile
