几类非线性时滞微分方程的稳定性与分支分析
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摘要
时滞微分方程因其对客观现象的描述和刻画比常微分方程更加准确和合理而得到广泛关注和研究，并被应用到众多领域。而分支理论研究的是结构不稳定的系统随参数变化时，当参数经过某些临界值时解的拓扑结构发生变化，分支现象也普遍存在于现实生活中。因此我们在本文中研究时滞微分方程的分支问题，主要是研究了不动点分支和Hopf分支。
不动点分支和Hopf分支都是比较常见的分支现象。不动点分支是指当参数经过临界值时系统的平衡点个数或者是稳定性发生变化。Hopf分支是指当参数经过临界值时系统平衡点的稳定性发生反转，并在平衡点附近产生小振幅周期解。通常情况下，不动点分支和Hopf分支的产生总是伴随着系统平衡点稳定性的变化，因此在研究分支问题时我们首先讨论系统平衡点的稳定性，然后研究具体的分支性质，包括不动点分支的类型、Hopf分支的分支方向和从Hopf分支值分支出的周期解的稳定性等。
本文研究四类具有实际背景的时滞微分方程，分别是具有一般形式非线性项的单向耦合系统、时滞Rosenzweig-MacArthur型的带有食饵移入项的捕食被捕食模型、带有Mach-Zehnder光电调制器的光电反馈环路系统以及耦合Lang-Kobayashi速率方程。通过分析系统的线性化方程的特征根的分布并结合极限方程的渐近半流的方法，讨论了平衡点的局部稳定性以及产生不动点分支和Hopf分支的条件，并利用Lyapunov泛函和Lassel不变集原理讨论了平衡点的全局稳定性。根据Hopf分支定理和重合度的延展定理证明了周期解的存在性。此外利用Faria和Hassard的规范型方法分别计算了不动点分支和Hopf分支在中心流形上的规范型，进而讨论了分支性质，并根据吴建宏的全局Hopf分支定理证明了分支周期解的全局存在性。
Delayed diferential equations are widely used in many fields since it can describethe natural phenomena much better than ordinary diferential equations. Meanwhile, bi-furcation means that for a system with unstable structure, when a certain parameter varies,the topological structure of the solutions changes, which is very common in our dailylives. So here we study the bifurcation of delayed diferential equations.
Fixed point bifurcation means the number or the stability of the equilibria change asthe parameter passes through critical value. Hopf bifurcation means that a small ampli-tude periodic solution comes out with the changes of stability for equilibrium point as acertain parameter varies and passes through a critical value. Since the appearance of theHopf bifurcation always accompanies with the change of the stability of the equilibria,so it is inevitable to discuss the stability of the equilibria of the equation when we studybifurcations firstly, and then we study the properties of the bifurcations including the pat-terns of the fixed point bifurcation, the direction of Hopf bifurcation and the stability ofthe bifurcating periodic solutions.
We mainly study four practical problems, including a unidirectionally coupled sys-tem with a nonlinear function of general form, delayed Rosenweig-MacArthur modelwith constant rate prey immigration, a delayed-feedback optical loop system with Mach-Zehnder modulator and coupled Lang-Kobayashi rate equation. By analysing the dis-tribution of the characteristic roots of the linearized equation and using the method ofasymptotic semi-flow of limit equation, we study the local stability of the equilibria andget the conditions under which fixed point bifurcation and Hopf bifurcation occurs. Then,using Lyapunov function and Lassel invariant set principle we investigate the global sta-bility of the equilibria. Basing on the Hopf bifurcation theorem and the coincide degreecontinuation theorem, we prove the existence of the periodic solutions. Furthermore, wecalculate the normal form of the fixed point bifurcation and the Hopf bifurcation on thecenter manifold by the method from Faria and Hassard. Finally, we verify the globalexistence of the periodic solutions by global Hopf bifurcation theorem of Wu.
不动点分支和Hopf分支都是比较常见的分支现象。不动点分支是指当参数经过临界值时系统的平衡点个数或者是稳定性发生变化。Hopf分支是指当参数经过临界值时系统平衡点的稳定性发生反转，并在平衡点附近产生小振幅周期解。通常情况下，不动点分支和Hopf分支的产生总是伴随着系统平衡点稳定性的变化，因此在研究分支问题时我们首先讨论系统平衡点的稳定性，然后研究具体的分支性质，包括不动点分支的类型、Hopf分支的分支方向和从Hopf分支值分支出的周期解的稳定性等。
本文研究四类具有实际背景的时滞微分方程，分别是具有一般形式非线性项的单向耦合系统、时滞Rosenzweig-MacArthur型的带有食饵移入项的捕食被捕食模型、带有Mach-Zehnder光电调制器的光电反馈环路系统以及耦合Lang-Kobayashi速率方程。通过分析系统的线性化方程的特征根的分布并结合极限方程的渐近半流的方法，讨论了平衡点的局部稳定性以及产生不动点分支和Hopf分支的条件，并利用Lyapunov泛函和Lassel不变集原理讨论了平衡点的全局稳定性。根据Hopf分支定理和重合度的延展定理证明了周期解的存在性。此外利用Faria和Hassard的规范型方法分别计算了不动点分支和Hopf分支在中心流形上的规范型，进而讨论了分支性质，并根据吴建宏的全局Hopf分支定理证明了分支周期解的全局存在性。
Delayed diferential equations are widely used in many fields since it can describethe natural phenomena much better than ordinary diferential equations. Meanwhile, bi-furcation means that for a system with unstable structure, when a certain parameter varies,the topological structure of the solutions changes, which is very common in our dailylives. So here we study the bifurcation of delayed diferential equations.
Fixed point bifurcation means the number or the stability of the equilibria change asthe parameter passes through critical value. Hopf bifurcation means that a small ampli-tude periodic solution comes out with the changes of stability for equilibrium point as acertain parameter varies and passes through a critical value. Since the appearance of theHopf bifurcation always accompanies with the change of the stability of the equilibria,so it is inevitable to discuss the stability of the equilibria of the equation when we studybifurcations firstly, and then we study the properties of the bifurcations including the pat-terns of the fixed point bifurcation, the direction of Hopf bifurcation and the stability ofthe bifurcating periodic solutions.
We mainly study four practical problems, including a unidirectionally coupled sys-tem with a nonlinear function of general form, delayed Rosenweig-MacArthur modelwith constant rate prey immigration, a delayed-feedback optical loop system with Mach-Zehnder modulator and coupled Lang-Kobayashi rate equation. By analysing the dis-tribution of the characteristic roots of the linearized equation and using the method ofasymptotic semi-flow of limit equation, we study the local stability of the equilibria andget the conditions under which fixed point bifurcation and Hopf bifurcation occurs. Then,using Lyapunov function and Lassel invariant set principle we investigate the global sta-bility of the equilibria. Basing on the Hopf bifurcation theorem and the coincide degreecontinuation theorem, we prove the existence of the periodic solutions. Furthermore, wecalculate the normal form of the fixed point bifurcation and the Hopf bifurcation on thecenter manifold by the method from Faria and Hassard. Finally, we verify the globalexistence of the periodic solutions by global Hopf bifurcation theorem of Wu.
引文
[1]郑祖庥.泛函微分方程理论[M].合肥:安徽教育出版社,1994.
[2] Kuang Y. Delay Diferential Equations with Application in Population Dynamic-s[M]. New York: Academic Press,1993.
[3] Smith H. An Introduction to Delay Diferential Equations with Sciences Applica-tions to the Life[M]. New York: Springer-Verlag,2010.
[4] Hale J. Theory of Functional Diferential Equation[M]. New York: Springer-Verlag,1977.
[5] Hale J, Lunnel S M V. Introduction to Functional Diferential Equation[M]. NewYork: Springer-Verlag,1993.
[6] Diekmann O, Stephan A. Delay Diferential Diference Equation[M]. New York:Academic Press,1963.
[7] Wiggins S. Introduction to Applied Nonlinear Dynamical Systems and Chaos[M].New York: Springer-Verlag,1990.
[8]张芷芬,李承志,郑志明,等.向量场的分岔理论基础[M].北京:高等教育出版社,1997.
[9]张锦炎,冯贝叶.常微分方程几何理论与分支问题[M].北京:北京大学出版社,2000.
[10] Chow S, Hale J. Method of Bifurcation Theory[M]. New York: Springer-Verlag,1982.
[11] Guckenheimer J, Holmes P. Nonlinear Oscillations, Dynamical Systems and Bi-furcations of Vector Fields[M]. New York: Springer-Verlag,1983.
[12] Kuznetsov Y. Elements of Applied Bifurcation Theory[M]. New York: Springer-Verlag,1998.
[13] Han M. Bifurcation Theory of Limit Cycles[M]. Beijing: Sciense Press,2013.
[14] McCluskey C C. Complete Global Stability for an SIR Epidemic Model with Delay—Distributed or Discrete[J]. Nonlinear Analysis: Real World Applications,2010,11(1):55–59.
[15] Kar T, Batabyal A. Stability and Bifurcation of a Prey–Predator Model with TimeDelay[J]. Comptes Rendus Biologies,2009,332(7):642–651.
[16] Erzgra¨ber H, Wille E, Krauskopf B, et al. Amplitude–Phase Dynamics Near theLocking Region of Two Delay-Coupled Semiconductor Lasers[J]. Nonlinearity,2009,22(3):585.
[17] Lakshmanan M, Senthilkumar D D V. Dynamics of Nonlinear Time-Telay System-s[M]. Berlin: Springer-Verlag,2010.
[18] Senthilkumar D, Lakshmanan M. Bifurcations and Chaos in Time Delayed Piece-wise Linear Dynamical Systems[J]. International Journal of Bifurcation and Chaos,2005,15(09):2895–2912.
[19] He X Z, Li K, Wei J, et al. Market Stability Switches in a Continuous-TimeFinancial Market with Heterogeneous Beliefs[J]. Economic Modelling,2009,26(6):1432–1442.
[20] Hassan T S. Oscillation of Third Order Nonlinear Delay Dynamic Equations onTime Scales[J]. Mathematical and Computer Modelling,2009,49(7):1573–1586.
[21]魏俊杰,王洪斌,蒋卫华.时滞微分方程的分支理论及应用[M].北京:科学出版社,2012.
[22] Ruan S, Wei J. On the Zeros of Transcendental Functions with Applications toStability of Delay Diferential Equations with Two Delays[J]. Dynamics of Con-tinuous, Discrete&Impulsive Systems. Series A. Mathematical Analysis,2003,10:863–874.
[23] Carr J. Application of Center Manifold Theory[M]. NewYork: Springer-Verlag,1981.
[24] Hassard B, Kazarinof N, Wan Y. Theory and Applications of Hopf Bifurcation[M].Cambridge: Cambridge Univ. Press,1981.
[25] Faria T, Magalha es L T. Normal Form for Retarded Functional Diferential Equa-tions with Parameters and Applications to Hopf Bifurcation[J]. Journal of Difer-ential Equations,1995,122:181–200.
[26] Faria T, Magalha es L T. Normal Form for Retarded Functional Diferential Equa-tions and Applications to Bogdanov-Takens Singularity[J]. Journal of DiferentialEquations,1995,122:201–224.
[27] Wu J. Symmetric Functional Diferential Equations and Neural Networks withMemory[J]. Transactions of the American Mathematical Society,1998,350:4799–4838.
[28] Zhao X. Asymptotic Behavior for Asymptotically Periodic Semiflow with Appli-cations[J]. Communications on Applied Nonlinear Analysis,1996,3:43–66.
[29] Mischaikow K, Smith H L, Thieme H. Asymptotically Autonomous Semiflow:Chain Recurrence and Liapunov Functions,[J]. Transactions of the AmericanMathematical Society,1995,347:1669–1685.
[30] Zhao X. Convergence in Asymptotically Periodic Tridiagonal Competitive-Cooperative Systems[J]. Canadian Applied Mathematics Quarterly,1998,6(3).
[31] Gaines R, Mawhin J. Coincidence Degree and Nonlinear Diferential Equation-s[M]. Berlin: Springer-Verlag,1977.
[32] Mackey M C, Glass L. Oscillation and Chaos in Physiological Control Systems[J].Science,1977,197:287–289.
[33] Lin W, Ma H. Synchronization Between Adaptively Coupled Systems with Dis-crete and Distributed Time-Delays[J]. Automatic Control, IEEE Transactions on,2010,55(4):819–830.
[34] Sano S, Uchida A, Yoshimori S, et al. Dual Synchronization of Chaos in Mackey-Glass Electronic Circuits with Time-Delayed Feedback[J]. Physical Review E,2007,75(1):016207.
[35] Zunino L, Soriano M C, Fischer I, et al. Permutation-Information-Theory Ap-proach to Unveil Delay Dynamics from Time-Series Analysis[J]. Physical ReviewE,2010,82(4):046212.
[36] Pyragas K. Synchronization of Coupled Time-Delay Systems: Analytical Estima-tions[J]. Physical Review E,1998,58:3067–3071.
[37] Voss H U. Anticipating Chaotic Synchronization[J]. Physical Review E,2000,61:5115–5119.
[38] Shahverdiev E M, Shore K A. Generalized Synchronization in Time-Delayed Sys-tems[J]. Physical Review E,2005,71(016201).
[39] Zhou S B, Li H. Synchronization Threshold of a Coupled Time-delay System[J].Physical Review E,2007,75(037203).
[40] Senthilkumar D V, Kurths J, Lakshmanan M. Stability of Synchronization in Cou-pled Time-Delay Systems Using Krasovskii-Lyapunov Theory[J]. Physical ReviewE,2009,79(066208).
[41] Wei J. Bifurcation Analysis in Scalar Delay Diferential Equations[J]. Nonlinearity,2007,20:2483–2498.
[42] Jiang W, Wei J. Bifurcation Analysis in Van Der Pol’s Oscillator with DelayedFeedback[J]. Journal of Computational and Applied Mathematics,2008,213:604–615.
[43] Jiang W, Yuan Y. Bogdanov-Takens Singularity in van der Pol’s oscillator withdelayed feedback[J]. Physica D,2007,227:149–161.
[44] Castillo-Chavez C, Thieme H. Asymptotically Autonomous Epidemic Models[J].Mathematical Population Dynamics: Analysis of Heterogeneity,1995,1:33–50.
[45] Thieme H. Convergence Results and a Poincare′-Bendixson Trichotomy forAsymptotically autonomous Diferential Equations[J]. Journal of MathematicalBiology,1992,30:755–763.
[46] Qu Y, Wei J, Ruan S. Stability and Bifurcation Analysis in A Hematopoietic StemCell Dynamics with Multiple Delays[J]. Physica D,2010,239:2011–2044.
[47] Fan M, Wang K. Global Existence of Positive Periodic Solutions of Predator-Prey System with Infinite Delays[J]. Journal of Mathematics Analysis Application,2001,262:1–11.
[48]杜波,葛渭高.时滞Rayleigh方程周期解问题[J].数学的实践与认识,2008,38(2):144–149.
[49] Edelstein-Keshet L. Mathematical Models in Biology[M]. New York: McGraw-Hill Inc.,1977.
[50] Murray J D. Lectures on Nonlinear Diferential Equation Models in Biology[M].Oxford: Oxford University press,1977.
[51] Murray J D. Mathematical Biology[M]. Berlin: Springer-Verlag,1993.
[52] Gurney W S C, Nesbet R M. Ecological Dynamics[M]. Oxford: Oxford Universitypress,1998.
[53] Milgram A. The Stability of the Boubaker Polynomials Expansion Scheme(BPES)-Based Solution to Lotka–Volterra Problem[J]. Journal of Theoretical Bi-ology,2011,271(1):157–158.
[54] Meng X, Chen L. Permanence and Global Stability in an Impulsive Lotka–VolterraN-Species Competitive System with both Discrete Delays and Continuous De-lays[J]. International Journal of Biomathematics,2008,1(02):179–196.
[55] Jansen V A A. The Dynamics of Two Difusively Coupled Predator-Prey Popula-tions[J]. Theoretical Population Biology,2001,45:119–131.
[56] Liu X, Chen L. Complex Dynamics of Holling Type II Lotka-Volterra Predator-Prey System with Impulsive Perturbations on the Predator[J]. Chaos, Solitons&Fractals,2003,16:311–320.
[57] Xu R, Chaplain M A J, Davidson F A. Periodic Solutons for a Predator-Prey Modelwith Holling Type II Functional Response and Time Delays[J]. Applied Mathemat-ics and Computation,2005,161:637–654.
[58] Feng W, Rock B, Hinson J. On A New Model of Two-Patch Predator-Prey Systemwith Migration of Both Species[J]. Journal of Applied Analysis and Computation,2011,1:193–203.
[59] Ruan S. On Nonlinear Dynamics of Predator-Prey Models with Discrete Delay[J].Mathematical Modelling of Natural Phenomena,2009,4(02):140–188.
[60] Hsu S B. On Global Stability of A Predator-Prey System[J]. Mathematical Bio-sciences,1978,39:1–10.
[61] Hsu S B, Waltman P. Competing Predators[J]. SIAM Journal on MathematicalAnalysis,1978,35:617–625.
[62] Hsu S B, Hwang T W, Kuang Y. Global Analysis of the Michaelis-Menten-Type Ratio-Dependent Predator-Prey System[J]. Journal of Mathematical Biology,1978,42:489–506.
[63] Conway E D, Smoller J A. Difusion and the Predator-Prey Interaction[J]. SIAMJournal on Applied Mathematics,1977,33(4):673–686.
[64] Leard B, Rebaza J. Analysis of Predator–Prey Models with Continuous ThresholdHarvesting[J]. Applied Mathematics and Computation,2011,217(12):5265–5278.
[65] Cheng K. Uniqueness of A Limit Cycle for A Predator-Prey System[J]. SIAMJournal on Mathematical Analysis,1981,12:541–548.
[66] Kuang Y, Freedman H I. Uniqueness of Limit Cycles in Gause-Type Models ofPredator-Prey Systems[J]. Mathematical Biosciences,1988,88:67–84.
[67] Xiao D, Ruan S. On the Uniqueness and Nonexistence of Limit Cycles forPredator-Prey System[J]. Nonlinearity,2003,16:1–17.
[68] Brauer F, Soudack A C. Stability Regions and Transition Phenomea for HarvestedPredator-Prey Systems[J]. Journal of Mathematical Biology,1979,7:319–337.
[69] Brauer F, Soudack A C. Constant-Rate Stocking of Predator-Prey Systems[J]. Jour-nal of Mathematical Biology,1981,11:1–14.
[70] Sugie J, Saito Y. Uniqueness of Limit Cycles In a Rosenzweig-Macarthur Modelwith Prey Immigration[J]. SIAM Journal on Applied Mathematics,2012,72:299–316.
[71] Lang R, Kobayashi K. External Optical Feedback Efects on Semiconductor Injec-tion Laser Properties[J]. IEEE Journal of Quantum Electronics,1980,16:347–355.
[72] Stepan G. Retarded Dynamical Systems: Stability and Characteristic Function-s[M]. London: Longman Scientific and Technical,1989.
[73] Heil T, Fischer I, Elsa¨ber W, et al. Dynamics of Semiconductor Lasers Subject toDelayed Optical Feedback: The Short Cavity Regime[J]. Physical Review Letters,2001,87(243901).
[74] Reddy D V R, Sen A, Johnston G L. Time Delay Induced Death in Coupled LimitCycle Oscillators[J]. Physical Review Letters,1998,80:5109–5112.
[75] Kouomou Y C, Colet P, Larger L, et al. Chaotic Breathers in Delayed Electro-Optical Systems,[J]. Physical Review Letters,2005,95(203903).
[76] Pan S, Yao J. Switchable UWB Pulse Generation Using a Phase Modulator and aReconfigurable Asymmetric Mach-Zehnder interferometer[J]. Optics letters,2009,34(2):160–162.
[77] Lavrov R, Peil M, Jacquot M, et al. Electro-Optic Delay Oscillator with NonlocalNonlinearity: Optical Phase Dynamics, Chaos, and Synchronization[J]. PhysicalReview E,2009,80(2):026207.
[78] Chembo Kouomou Y, Colet P, Larger L, et al. Chaotic Breathers in DelayedElectro-Optical Systems[J]. Physical Review Letters,2005,95(20):203903.
[79] Ravoori B, Cohen A B, Setty A V, et al. Adaptive Synchronization of CoupledChaotic Oscillators[J]. Physical Review E,2009,80(056205).
[80] Pecora L M, Carroll T L. Master Stability Functions for Synchronized CoupledSystems[J]. Physical Review Letters,1998,80:2109–2112.
[81] Cuomo K M, Oppenhein A V. Circuit Implementation of Synchronized Chaos withApplications to Communications[J]. Physical Review Letters,1993,71:65–68.
[82] Goedgebuer J P, Larger L, Porte H. Optical Cryptosystem Based on Synchroniza-tion of Hyperchaos Generated by a Delayed Feedback Tunable Laser Diode[J].Physical Review Letters,1998,80:2249–2252.
[83] Arenas A, Diaz-Guilera A, Kurths J, et al. Synchronization in Complex Network-s[J]. Physics Reports,2008,80:93–153.
[84] Heil T, Fischer I, Elsa¨sser W, et al. Chaos Synchronization and SpontaneousSymmetry-Breaking in Symmetrically Delay-Coupled Semiconductor Lasers[J].Physical Review Letters,2001,86:795–798.
[85] Fischer I, Liu Y, Davis P. Synchronization of Chaotic Semiconductor Laser Dy-namics on Subnanosecond Time Scales and Its Potential For Chaos Communica-tion[J]. Physical Review A,2000,62(011801).
[86] Cohen A B, Ravoori B, Mruphy T E, et al. Using Synchronization for Predic-tion of High-Dimensional Chaotic Dynamics[J]. Physical Review Letters,2008,101(154102).
[87] Hill M T, de. Waardt H, Khoe G D, et al. All-Optical Flip-Flop Based on CoupledLaser Diodes[J]. IEEE Journal of Quantum Electronics,2001,37:405–413.
[88] Manfra E F, Caldas I L, Viana R L, et al. Type-I Intermittency and Crisis-InducedIntermittency in a Semiconductor Laser Under Injection Current Modulation[J].Nonlinear Dynamics,2002,27:185–195.
[89] Erzgra¨ber H, Krauskopf B, Lenstra D. Compound Laser Modes of Mutually Delay-Coupled Lasers[J]. SIAM Journal on Applied Dynamical Systems,2006,5:30–65.
[90] Erzgra¨ber H, Lenstra D, Krauskopf B, et al. Mmutually Delay-Coupled Semi-conductor Lasers: Mode Bifurcation Scenarios[J]. Optics Communications,2005,255:286–296.
[91] Erzgra¨ber H, Wieczorek S, Krauskopf B. Dynamics of Two Semiconductor LasersCoupled by a Passive Resonator[J]. Physical Review E,2010,81(5):056201.
[92] Erzgra¨ber H, Krauskopf B, Lenstra D. Bifurcation Analysis of a SemiconductorLaser with Filtered Optical Feedback[J]. SIAM Journal on Applied DynamicalSystems,2007,6(1):1–28.
[93] Dahms T, Ho¨vel P, Scho¨ll E. Stabilizing Continuous-Wave Output in Semiconduc-tor Lasers by Time-Delayed Feedback[J]. Physical Review E,2008,78(5):056213.
[94] Fiedler B, Yanchuk S, Flunkert V, et al. Delay Stabilization of Rotating Wavesnear Fold Bifurcation and Application to All-Optical Control of a SemiconductorLaser[J]. Physical Review E,2008,77(6):066207.
[95] Erzgra¨ber H, Just W. Global View on a Nonlinear Oscillator Subject toTime-Delayed Feedback Control[J]. Physica D: Nonlinear Phenomena,2009,238(16):1680–1687.
[96] Fiedler B, Flunkert V, Ho¨vel P, et al. Delay Stabilization of Periodic Orbits inCoupled Oscillator Systems[J]. Philosophical Transactions of the Royal SocietyA: Mathematical, Physical and Engineering Sciences,2010,368(1911):319–341.
[97] Blackbeard N, Erzgra¨ber H, Wieczorek S. Shear-Induced Bifurcations and Chaosin Models of Three Coupled Lasers[J]. SIAM Journal on Applied Dynamical Sys-tems,2011,10(2):469–509.
[98] Antonelli C, Mecozzi A. Periodic Locking of Chaos in Semiconductor Lasers withOptical Feedback[J]. Optics Communications,2009,282(14):2917–2920.
[99] Rontani D, Locquet A, Sciamanna M, et al. Time-Delay Identification in a Chaot-ic Semiconductor Laser with Optical Feedback: a Dynamical Point of View[J].Quantum Electronics, IEEE Journal of,2009,45(7):879–1891.
[100] Nixon M, Friedman M, Ronen E, et al. Synchronized Cluster Formation in CoupledLaser Networks[J]. Physical Review Letters,2011,106(223901).
[101] Liu X, Gao Q, Niu L. A Revisit to Synchronization of Lurie Systems with Time-Delay Feedback Control[J]. Nonlinear Dynamics,2010,59:297–307.
[102] Feng C F. Projective Synchronization Between Two Diferent Time-DelayedChaotic Systems Using Active Control Approach[J]. Nonlinear Dynamics,2010,62:453–459.
[103] Huang H, Feng G, Cao J. Exponential Synchronization of Chaotic Lur’e Systemswith Delayed Feedback Control[J]. Nonlinear Dynamics,2009,57:441–453.
[104] Lu J, Cao J. Adaptive Synchronization of Uncertain Dynamical Networks withDelayed Coupling[J]. Nonlinear Dynamics,2008,53:107–115.
[105] Senthilkumar D, Kurths J. Characteristics and Synchronization of Time-DelaySystems Driven by a Common Noise[J]. The European Physical Journal SpecialTopics,2010,187(1):87–93.
[106] Lang R, Kobayashi K. External Optical Feedback Efects on Semiconductor Injec-tion Laser Properties[J]. IEEE Journal of Quantum Electronics,1980,16:347–355.
[107] Alsing P M, Kovanis V, Gavrielides A, et al. Lang and Kobayashi Phase Equa-tion[J]. Physical Review A,1996,53:4429–4434.
[108] Farmer J D. Chaotic Attractors of an Infinite-Dimensional Dynamical System[J].Physica D,1982,4:366–393.
[109] Flunkert V, D’Huys O, Danckaert J, et al. Bubbling in Delay-Coupled Lases[J].Physical Review E,2009,79(065201).
[110] Green K. Stability Near Threshold in a Semiconductor Laser Subject to OpticalFeedback: a Bifurcation Analysis of the Lang-Kobayashi Equations[J]. PhysicalReview E,2009,79(3):036210.
[111] Yanchuk S, Wolfrum M. A Multiple Time Scale Approach to the Stability of Ex-ternal Cavity Modes in the Lang-Kobayashi System Using the Limit of Large De-lay[J]. SIAM Journal on Applied Dynamical Systems,2010,9(2):519–535.
[2] Kuang Y. Delay Diferential Equations with Application in Population Dynamic-s[M]. New York: Academic Press,1993.
[3] Smith H. An Introduction to Delay Diferential Equations with Sciences Applica-tions to the Life[M]. New York: Springer-Verlag,2010.
[4] Hale J. Theory of Functional Diferential Equation[M]. New York: Springer-Verlag,1977.
[5] Hale J, Lunnel S M V. Introduction to Functional Diferential Equation[M]. NewYork: Springer-Verlag,1993.
[6] Diekmann O, Stephan A. Delay Diferential Diference Equation[M]. New York:Academic Press,1963.
[7] Wiggins S. Introduction to Applied Nonlinear Dynamical Systems and Chaos[M].New York: Springer-Verlag,1990.
[8]张芷芬,李承志,郑志明,等.向量场的分岔理论基础[M].北京:高等教育出版社,1997.
[9]张锦炎,冯贝叶.常微分方程几何理论与分支问题[M].北京:北京大学出版社,2000.
[10] Chow S, Hale J. Method of Bifurcation Theory[M]. New York: Springer-Verlag,1982.
[11] Guckenheimer J, Holmes P. Nonlinear Oscillations, Dynamical Systems and Bi-furcations of Vector Fields[M]. New York: Springer-Verlag,1983.
[12] Kuznetsov Y. Elements of Applied Bifurcation Theory[M]. New York: Springer-Verlag,1998.
[13] Han M. Bifurcation Theory of Limit Cycles[M]. Beijing: Sciense Press,2013.
[14] McCluskey C C. Complete Global Stability for an SIR Epidemic Model with Delay—Distributed or Discrete[J]. Nonlinear Analysis: Real World Applications,2010,11(1):55–59.
[15] Kar T, Batabyal A. Stability and Bifurcation of a Prey–Predator Model with TimeDelay[J]. Comptes Rendus Biologies,2009,332(7):642–651.
[16] Erzgra¨ber H, Wille E, Krauskopf B, et al. Amplitude–Phase Dynamics Near theLocking Region of Two Delay-Coupled Semiconductor Lasers[J]. Nonlinearity,2009,22(3):585.
[17] Lakshmanan M, Senthilkumar D D V. Dynamics of Nonlinear Time-Telay System-s[M]. Berlin: Springer-Verlag,2010.
[18] Senthilkumar D, Lakshmanan M. Bifurcations and Chaos in Time Delayed Piece-wise Linear Dynamical Systems[J]. International Journal of Bifurcation and Chaos,2005,15(09):2895–2912.
[19] He X Z, Li K, Wei J, et al. Market Stability Switches in a Continuous-TimeFinancial Market with Heterogeneous Beliefs[J]. Economic Modelling,2009,26(6):1432–1442.
[20] Hassan T S. Oscillation of Third Order Nonlinear Delay Dynamic Equations onTime Scales[J]. Mathematical and Computer Modelling,2009,49(7):1573–1586.
[21]魏俊杰,王洪斌,蒋卫华.时滞微分方程的分支理论及应用[M].北京:科学出版社,2012.
[22] Ruan S, Wei J. On the Zeros of Transcendental Functions with Applications toStability of Delay Diferential Equations with Two Delays[J]. Dynamics of Con-tinuous, Discrete&Impulsive Systems. Series A. Mathematical Analysis,2003,10:863–874.
[23] Carr J. Application of Center Manifold Theory[M]. NewYork: Springer-Verlag,1981.
[24] Hassard B, Kazarinof N, Wan Y. Theory and Applications of Hopf Bifurcation[M].Cambridge: Cambridge Univ. Press,1981.
[25] Faria T, Magalha es L T. Normal Form for Retarded Functional Diferential Equa-tions with Parameters and Applications to Hopf Bifurcation[J]. Journal of Difer-ential Equations,1995,122:181–200.
[26] Faria T, Magalha es L T. Normal Form for Retarded Functional Diferential Equa-tions and Applications to Bogdanov-Takens Singularity[J]. Journal of DiferentialEquations,1995,122:201–224.
[27] Wu J. Symmetric Functional Diferential Equations and Neural Networks withMemory[J]. Transactions of the American Mathematical Society,1998,350:4799–4838.
[28] Zhao X. Asymptotic Behavior for Asymptotically Periodic Semiflow with Appli-cations[J]. Communications on Applied Nonlinear Analysis,1996,3:43–66.
[29] Mischaikow K, Smith H L, Thieme H. Asymptotically Autonomous Semiflow:Chain Recurrence and Liapunov Functions,[J]. Transactions of the AmericanMathematical Society,1995,347:1669–1685.
[30] Zhao X. Convergence in Asymptotically Periodic Tridiagonal Competitive-Cooperative Systems[J]. Canadian Applied Mathematics Quarterly,1998,6(3).
[31] Gaines R, Mawhin J. Coincidence Degree and Nonlinear Diferential Equation-s[M]. Berlin: Springer-Verlag,1977.
[32] Mackey M C, Glass L. Oscillation and Chaos in Physiological Control Systems[J].Science,1977,197:287–289.
[33] Lin W, Ma H. Synchronization Between Adaptively Coupled Systems with Dis-crete and Distributed Time-Delays[J]. Automatic Control, IEEE Transactions on,2010,55(4):819–830.
[34] Sano S, Uchida A, Yoshimori S, et al. Dual Synchronization of Chaos in Mackey-Glass Electronic Circuits with Time-Delayed Feedback[J]. Physical Review E,2007,75(1):016207.
[35] Zunino L, Soriano M C, Fischer I, et al. Permutation-Information-Theory Ap-proach to Unveil Delay Dynamics from Time-Series Analysis[J]. Physical ReviewE,2010,82(4):046212.
[36] Pyragas K. Synchronization of Coupled Time-Delay Systems: Analytical Estima-tions[J]. Physical Review E,1998,58:3067–3071.
[37] Voss H U. Anticipating Chaotic Synchronization[J]. Physical Review E,2000,61:5115–5119.
[38] Shahverdiev E M, Shore K A. Generalized Synchronization in Time-Delayed Sys-tems[J]. Physical Review E,2005,71(016201).
[39] Zhou S B, Li H. Synchronization Threshold of a Coupled Time-delay System[J].Physical Review E,2007,75(037203).
[40] Senthilkumar D V, Kurths J, Lakshmanan M. Stability of Synchronization in Cou-pled Time-Delay Systems Using Krasovskii-Lyapunov Theory[J]. Physical ReviewE,2009,79(066208).
[41] Wei J. Bifurcation Analysis in Scalar Delay Diferential Equations[J]. Nonlinearity,2007,20:2483–2498.
[42] Jiang W, Wei J. Bifurcation Analysis in Van Der Pol’s Oscillator with DelayedFeedback[J]. Journal of Computational and Applied Mathematics,2008,213:604–615.
[43] Jiang W, Yuan Y. Bogdanov-Takens Singularity in van der Pol’s oscillator withdelayed feedback[J]. Physica D,2007,227:149–161.
[44] Castillo-Chavez C, Thieme H. Asymptotically Autonomous Epidemic Models[J].Mathematical Population Dynamics: Analysis of Heterogeneity,1995,1:33–50.
[45] Thieme H. Convergence Results and a Poincare′-Bendixson Trichotomy forAsymptotically autonomous Diferential Equations[J]. Journal of MathematicalBiology,1992,30:755–763.
[46] Qu Y, Wei J, Ruan S. Stability and Bifurcation Analysis in A Hematopoietic StemCell Dynamics with Multiple Delays[J]. Physica D,2010,239:2011–2044.
[47] Fan M, Wang K. Global Existence of Positive Periodic Solutions of Predator-Prey System with Infinite Delays[J]. Journal of Mathematics Analysis Application,2001,262:1–11.
[48]杜波,葛渭高.时滞Rayleigh方程周期解问题[J].数学的实践与认识,2008,38(2):144–149.
[49] Edelstein-Keshet L. Mathematical Models in Biology[M]. New York: McGraw-Hill Inc.,1977.
[50] Murray J D. Lectures on Nonlinear Diferential Equation Models in Biology[M].Oxford: Oxford University press,1977.
[51] Murray J D. Mathematical Biology[M]. Berlin: Springer-Verlag,1993.
[52] Gurney W S C, Nesbet R M. Ecological Dynamics[M]. Oxford: Oxford Universitypress,1998.
[53] Milgram A. The Stability of the Boubaker Polynomials Expansion Scheme(BPES)-Based Solution to Lotka–Volterra Problem[J]. Journal of Theoretical Bi-ology,2011,271(1):157–158.
[54] Meng X, Chen L. Permanence and Global Stability in an Impulsive Lotka–VolterraN-Species Competitive System with both Discrete Delays and Continuous De-lays[J]. International Journal of Biomathematics,2008,1(02):179–196.
[55] Jansen V A A. The Dynamics of Two Difusively Coupled Predator-Prey Popula-tions[J]. Theoretical Population Biology,2001,45:119–131.
[56] Liu X, Chen L. Complex Dynamics of Holling Type II Lotka-Volterra Predator-Prey System with Impulsive Perturbations on the Predator[J]. Chaos, Solitons&Fractals,2003,16:311–320.
[57] Xu R, Chaplain M A J, Davidson F A. Periodic Solutons for a Predator-Prey Modelwith Holling Type II Functional Response and Time Delays[J]. Applied Mathemat-ics and Computation,2005,161:637–654.
[58] Feng W, Rock B, Hinson J. On A New Model of Two-Patch Predator-Prey Systemwith Migration of Both Species[J]. Journal of Applied Analysis and Computation,2011,1:193–203.
[59] Ruan S. On Nonlinear Dynamics of Predator-Prey Models with Discrete Delay[J].Mathematical Modelling of Natural Phenomena,2009,4(02):140–188.
[60] Hsu S B. On Global Stability of A Predator-Prey System[J]. Mathematical Bio-sciences,1978,39:1–10.
[61] Hsu S B, Waltman P. Competing Predators[J]. SIAM Journal on MathematicalAnalysis,1978,35:617–625.
[62] Hsu S B, Hwang T W, Kuang Y. Global Analysis of the Michaelis-Menten-Type Ratio-Dependent Predator-Prey System[J]. Journal of Mathematical Biology,1978,42:489–506.
[63] Conway E D, Smoller J A. Difusion and the Predator-Prey Interaction[J]. SIAMJournal on Applied Mathematics,1977,33(4):673–686.
[64] Leard B, Rebaza J. Analysis of Predator–Prey Models with Continuous ThresholdHarvesting[J]. Applied Mathematics and Computation,2011,217(12):5265–5278.
[65] Cheng K. Uniqueness of A Limit Cycle for A Predator-Prey System[J]. SIAMJournal on Mathematical Analysis,1981,12:541–548.
[66] Kuang Y, Freedman H I. Uniqueness of Limit Cycles in Gause-Type Models ofPredator-Prey Systems[J]. Mathematical Biosciences,1988,88:67–84.
[67] Xiao D, Ruan S. On the Uniqueness and Nonexistence of Limit Cycles forPredator-Prey System[J]. Nonlinearity,2003,16:1–17.
[68] Brauer F, Soudack A C. Stability Regions and Transition Phenomea for HarvestedPredator-Prey Systems[J]. Journal of Mathematical Biology,1979,7:319–337.
[69] Brauer F, Soudack A C. Constant-Rate Stocking of Predator-Prey Systems[J]. Jour-nal of Mathematical Biology,1981,11:1–14.
[70] Sugie J, Saito Y. Uniqueness of Limit Cycles In a Rosenzweig-Macarthur Modelwith Prey Immigration[J]. SIAM Journal on Applied Mathematics,2012,72:299–316.
[71] Lang R, Kobayashi K. External Optical Feedback Efects on Semiconductor Injec-tion Laser Properties[J]. IEEE Journal of Quantum Electronics,1980,16:347–355.
[72] Stepan G. Retarded Dynamical Systems: Stability and Characteristic Function-s[M]. London: Longman Scientific and Technical,1989.
[73] Heil T, Fischer I, Elsa¨ber W, et al. Dynamics of Semiconductor Lasers Subject toDelayed Optical Feedback: The Short Cavity Regime[J]. Physical Review Letters,2001,87(243901).
[74] Reddy D V R, Sen A, Johnston G L. Time Delay Induced Death in Coupled LimitCycle Oscillators[J]. Physical Review Letters,1998,80:5109–5112.
[75] Kouomou Y C, Colet P, Larger L, et al. Chaotic Breathers in Delayed Electro-Optical Systems,[J]. Physical Review Letters,2005,95(203903).
[76] Pan S, Yao J. Switchable UWB Pulse Generation Using a Phase Modulator and aReconfigurable Asymmetric Mach-Zehnder interferometer[J]. Optics letters,2009,34(2):160–162.
[77] Lavrov R, Peil M, Jacquot M, et al. Electro-Optic Delay Oscillator with NonlocalNonlinearity: Optical Phase Dynamics, Chaos, and Synchronization[J]. PhysicalReview E,2009,80(2):026207.
[78] Chembo Kouomou Y, Colet P, Larger L, et al. Chaotic Breathers in DelayedElectro-Optical Systems[J]. Physical Review Letters,2005,95(20):203903.
[79] Ravoori B, Cohen A B, Setty A V, et al. Adaptive Synchronization of CoupledChaotic Oscillators[J]. Physical Review E,2009,80(056205).
[80] Pecora L M, Carroll T L. Master Stability Functions for Synchronized CoupledSystems[J]. Physical Review Letters,1998,80:2109–2112.
[81] Cuomo K M, Oppenhein A V. Circuit Implementation of Synchronized Chaos withApplications to Communications[J]. Physical Review Letters,1993,71:65–68.
[82] Goedgebuer J P, Larger L, Porte H. Optical Cryptosystem Based on Synchroniza-tion of Hyperchaos Generated by a Delayed Feedback Tunable Laser Diode[J].Physical Review Letters,1998,80:2249–2252.
[83] Arenas A, Diaz-Guilera A, Kurths J, et al. Synchronization in Complex Network-s[J]. Physics Reports,2008,80:93–153.
[84] Heil T, Fischer I, Elsa¨sser W, et al. Chaos Synchronization and SpontaneousSymmetry-Breaking in Symmetrically Delay-Coupled Semiconductor Lasers[J].Physical Review Letters,2001,86:795–798.
[85] Fischer I, Liu Y, Davis P. Synchronization of Chaotic Semiconductor Laser Dy-namics on Subnanosecond Time Scales and Its Potential For Chaos Communica-tion[J]. Physical Review A,2000,62(011801).
[86] Cohen A B, Ravoori B, Mruphy T E, et al. Using Synchronization for Predic-tion of High-Dimensional Chaotic Dynamics[J]. Physical Review Letters,2008,101(154102).
[87] Hill M T, de. Waardt H, Khoe G D, et al. All-Optical Flip-Flop Based on CoupledLaser Diodes[J]. IEEE Journal of Quantum Electronics,2001,37:405–413.
[88] Manfra E F, Caldas I L, Viana R L, et al. Type-I Intermittency and Crisis-InducedIntermittency in a Semiconductor Laser Under Injection Current Modulation[J].Nonlinear Dynamics,2002,27:185–195.
[89] Erzgra¨ber H, Krauskopf B, Lenstra D. Compound Laser Modes of Mutually Delay-Coupled Lasers[J]. SIAM Journal on Applied Dynamical Systems,2006,5:30–65.
[90] Erzgra¨ber H, Lenstra D, Krauskopf B, et al. Mmutually Delay-Coupled Semi-conductor Lasers: Mode Bifurcation Scenarios[J]. Optics Communications,2005,255:286–296.
[91] Erzgra¨ber H, Wieczorek S, Krauskopf B. Dynamics of Two Semiconductor LasersCoupled by a Passive Resonator[J]. Physical Review E,2010,81(5):056201.
[92] Erzgra¨ber H, Krauskopf B, Lenstra D. Bifurcation Analysis of a SemiconductorLaser with Filtered Optical Feedback[J]. SIAM Journal on Applied DynamicalSystems,2007,6(1):1–28.
[93] Dahms T, Ho¨vel P, Scho¨ll E. Stabilizing Continuous-Wave Output in Semiconduc-tor Lasers by Time-Delayed Feedback[J]. Physical Review E,2008,78(5):056213.
[94] Fiedler B, Yanchuk S, Flunkert V, et al. Delay Stabilization of Rotating Wavesnear Fold Bifurcation and Application to All-Optical Control of a SemiconductorLaser[J]. Physical Review E,2008,77(6):066207.
[95] Erzgra¨ber H, Just W. Global View on a Nonlinear Oscillator Subject toTime-Delayed Feedback Control[J]. Physica D: Nonlinear Phenomena,2009,238(16):1680–1687.
[96] Fiedler B, Flunkert V, Ho¨vel P, et al. Delay Stabilization of Periodic Orbits inCoupled Oscillator Systems[J]. Philosophical Transactions of the Royal SocietyA: Mathematical, Physical and Engineering Sciences,2010,368(1911):319–341.
[97] Blackbeard N, Erzgra¨ber H, Wieczorek S. Shear-Induced Bifurcations and Chaosin Models of Three Coupled Lasers[J]. SIAM Journal on Applied Dynamical Sys-tems,2011,10(2):469–509.
[98] Antonelli C, Mecozzi A. Periodic Locking of Chaos in Semiconductor Lasers withOptical Feedback[J]. Optics Communications,2009,282(14):2917–2920.
[99] Rontani D, Locquet A, Sciamanna M, et al. Time-Delay Identification in a Chaot-ic Semiconductor Laser with Optical Feedback: a Dynamical Point of View[J].Quantum Electronics, IEEE Journal of,2009,45(7):879–1891.
[100] Nixon M, Friedman M, Ronen E, et al. Synchronized Cluster Formation in CoupledLaser Networks[J]. Physical Review Letters,2011,106(223901).
[101] Liu X, Gao Q, Niu L. A Revisit to Synchronization of Lurie Systems with Time-Delay Feedback Control[J]. Nonlinear Dynamics,2010,59:297–307.
[102] Feng C F. Projective Synchronization Between Two Diferent Time-DelayedChaotic Systems Using Active Control Approach[J]. Nonlinear Dynamics,2010,62:453–459.
[103] Huang H, Feng G, Cao J. Exponential Synchronization of Chaotic Lur’e Systemswith Delayed Feedback Control[J]. Nonlinear Dynamics,2009,57:441–453.
[104] Lu J, Cao J. Adaptive Synchronization of Uncertain Dynamical Networks withDelayed Coupling[J]. Nonlinear Dynamics,2008,53:107–115.
[105] Senthilkumar D, Kurths J. Characteristics and Synchronization of Time-DelaySystems Driven by a Common Noise[J]. The European Physical Journal SpecialTopics,2010,187(1):87–93.
[106] Lang R, Kobayashi K. External Optical Feedback Efects on Semiconductor Injec-tion Laser Properties[J]. IEEE Journal of Quantum Electronics,1980,16:347–355.
[107] Alsing P M, Kovanis V, Gavrielides A, et al. Lang and Kobayashi Phase Equa-tion[J]. Physical Review A,1996,53:4429–4434.
[108] Farmer J D. Chaotic Attractors of an Infinite-Dimensional Dynamical System[J].Physica D,1982,4:366–393.
[109] Flunkert V, D’Huys O, Danckaert J, et al. Bubbling in Delay-Coupled Lases[J].Physical Review E,2009,79(065201).
[110] Green K. Stability Near Threshold in a Semiconductor Laser Subject to OpticalFeedback: a Bifurcation Analysis of the Lang-Kobayashi Equations[J]. PhysicalReview E,2009,79(3):036210.
[111] Yanchuk S, Wolfrum M. A Multiple Time Scale Approach to the Stability of Ex-ternal Cavity Modes in the Lang-Kobayashi System Using the Limit of Large De-lay[J]. SIAM Journal on Applied Dynamical Systems,2010,9(2):519–535.