Chapter 1 Bifurcation, Limit Cycle and Chaos of Nonlinear Dynamical Systems

In this chapter, we shall consider bifurcations of dynamical systems which are described by ordinary differential equations, difference equations and time delayed differential equations. Particular attention is given to limit cycles and chaos, Hopf bifurcation control, chaos control and chaos synchronization. Both mathematical and practical engineering problems are considered. The materials presented in this chapter are based on the research results obtained recently by the author and his co-workers.

[1]  Xinghuo Yu,et al.  Time Delayed Repetitive Learning Control for Chaotic Systems , 2002, Int. J. Bifurc. Chaos.

[2]  Ali H. Nayfeh,et al.  BIFURCATIONS IN A POWER SYSTEM MODEL , 1996 .

[3]  Carroll,et al.  Synchronization in chaotic systems. , 1990, Physical review letters.

[4]  V. I. Arnol'd,et al.  Loss of stability of self-oscillations close to resonance and versal deformations of equivariant vector fields , 1977 .

[5]  Christopher Essex,et al.  Estimation of chaotic parameter regimes via generalized competitive mode approach , 2002 .

[6]  Ian Stewart,et al.  Mathematics: The Lorenz attractor exists , 2000, Nature.

[7]  Guanrong Chen,et al.  Estimating the bounds for the Lorenz family of chaotic systems , 2005 .

[8]  Chen Hai-bo,et al.  Linear recursion formulas of quantities of singular point and applications , 2004, Appl. Math. Comput..

[9]  Guanrong Chen,et al.  Hopf bifurcation Control Using Nonlinear Feedback with Polynomial Functions , 2004, Int. J. Bifurc. Chaos.

[10]  Jian Xu,et al.  Study of double Hopf bifurcation and chaos for an oscillator with time delayed feedback , 2002 .

[11]  Joshua E. S. Socolar,et al.  Stability of periodic orbits controlled by time-delay feedback , 1995, chao-dyn/9510019.

[12]  Guanrong Chen,et al.  Hopf bifurcation in an Internet congestion control model , 2004 .

[13]  Pei Yu,et al.  Globally Attractive and Positive Invariant Set of the Lorenz System , 2006, Int. J. Bifurc. Chaos.

[14]  L. Chua,et al.  The double scroll family , 1986 .

[15]  Gerold Baier,et al.  A Chaotic hierarchy , 1991 .

[16]  Stephen Lynch,et al.  Dynamical Systems with Applications using Maple , 2000 .

[17]  G. Leonov,et al.  Attraktorlokalisierung des Lorenz-Systems , 1987 .

[18]  Jianhong Wu,et al.  Introduction to Neural Dynamics and Signal Transmission Delay , 2001 .

[19]  O. Rössler.  An equation for continuous chaos , 1976 .

[20]  H. S. Y. Chan,et al.  Some lower bounds forH(n) in Hilbert’s 16th problem , 2002 .

[21]  Julien Clinton Sprott,et al.  Simplest dissipative chaotic flow , 1997 .

[22]  P. Yu,et al.  SYMBOLIC COMPUTATION OF NORMAL FORMS FOR RESONANT DOUBLE HOPF BIFURCATIONS USING A PERTURBATION TECHNIQUE , 2001 .

[23]  H. Zoladek,et al.  Eleven small limit cycles in a cubic vector field , 1995 .

[24]  Liviu Librescu,et al.  Flutter, Postflutter, and Control of a Supersonic Wing Section , 2002 .

[25]  L. O. Chua,et al.  The double scroll family. I: Rigorous of chaos. II: Rigorous analysis of bifurcation phenomena , 1986 .

[26]  Guanrong Chen,et al.  On feedback-controlled synchronization of chaotic systems , 2003, Int. J. Syst. Sci..

[27]  Alan V. Oppenheim,et al.  Circuit implementation of synchronized chaos with applications to communications. , 1993, Physical review letters.

[28]  Henri Poincaré,et al.  méthodes nouvelles de la mécanique céleste , 1892 .

[29]  X. Liao,et al.  Absolute Stability of Nonlinear Control Systems , 2008 .

[30]  Frank Kelly,et al.  Rate control for communication networks: shadow prices, proportional fairness and stability , 1998, J. Oper. Res. Soc..

[31]  O. Rössler.  CONTINUOUS CHAOS—FOUR PROTOTYPE EQUATIONS , 1979 .

[32]  W. Steeb,et al.  The Rikitake Two-Disk Dynamo System and Domains with Periodic Orbits , 1999 .

[33]  Pei Yu,et al.  COMPUTATION OF NORMAL FORMS VIA A PERTURBATION TECHNIQUE , 1998 .

[34]  Yuan Yuan,et al.  Aeroelasticity of Time-Delayed Feedback Control of Two-Dimensional Supersonic Lifting Surfaces , 2004 .

[35]  Yirong Liu,et al.  A cubic system with twelve small amplitude limit cycles , 2005 .

[36]  Pei Yu,et al.  New Estimations for Globally Attractive and Positive Invariant Set of the Family of the Lorenz Systems , 2006, Int. J. Bifurc. Chaos.

[37]  Richard W. J. Neufeld,et al.  Dynamic differentials of stress and coping , 1999 .

[38]  Guanrong Chen,et al.  Bifurcation Control: Theories, Methods, and Applications , 2000, Int. J. Bifurc. Chaos.

[39]  Pei Yu,et al.  Bifurcation of limit cycles in a quintic Hamiltonian system under a sixth-order perturbation , 2005 .

[40]  Celso Grebogi,et al.  Erratum: ``Controlling chaos'' [Phys. Rev. Lett. 64, 1196 (1990)] , 1990 .

[41]  Lu Jinhu,et al.  Controlling the Chen attractor using linear feedback based on parameter identification , 2002 .

[42]  Guanrong Chen,et al.  Chaos stabilization: an inverse optimal control approach , 2002 .

[43]  Ying-Cheng Lai,et al.  Controlling chaos , 1994 .

[44]  Grégoire Nicolis,et al.  Self-Organization in nonequilibrium systems , 1977 .

[45]  Hua O. Wang,et al.  Bifurcation control of a chaotic system , 1995, Autom..

[46]  Carroll,et al.  Driving systems with chaotic signals. , 1991, Physical review. A, Atomic, molecular, and optical physics.

[47]  Jibin Li,et al.  Bifurcation set and limit cycles forming compound eyes in a perturbed Hamiltonian system , 1991 .

[48]  Guanrong Chen,et al.  Bifurcation Dynamics in Control Systems , 1999 .

[49]  Pei Yu,et al.  Twelve Limit Cycles in a cubic Case of the 16TH Hilbert Problem , 2005, Int. J. Bifurc. Chaos.

[50]  P. Yu,et al.  Twelve limit cycles in a cubic order planar system with $Z_2$- symmetry , 2004 .

[51]  B. Hassard,et al.  Theory and applications of Hopf bifurcation , 1981 .

[52]  韩茂安.  LIAPUNOV CONSTANTS AND HOPF CYCLICITY OF LIENARD SYSTEMS , 1999 .

[53]  C. Essex,et al.  Delayed stochastic differential model for quiet standing. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[54]  Balth. van der Pol Jun..  LXXXVIII. On “relaxation-oscillations” , 1926 .

[55]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[56]  Bernd Krauskopf,et al.  Bifurcation analysis of an inverted pendulum with delayed feedback control near a triple-zero eigenvalue singularity , 2003 .

[57]  Xinghuo Yu,et al.  Bifurcation Control: Theory and Applications , 2003 .

[58]  Pei Yu,et al.  Controlling and anti-controlling Hopf bifurcations in discrete maps using polynomial functions , 2005 .

[59]  Guanrong Chen,et al.  Controlling oscillation amplitudes via Feedback , 2000, Int. J. Bifurc. Chaos.

[60]  Liviu Librescu,et al.  Elastostatics and Kinetics of Anisotropic and Heterogeneous Shell-Type Structures , 1975 .

[61]  H. Haken.  Synergetics: an Introduction, Nonequilibrium Phase Transitions and Self-organization in Physics, Chemistry, and Biology , 1977 .

[62]  Pei Yu,et al.  Small limit cycles bifurcating from fine focus points in cubic order Z2-equivariant vector fields , 2005 .

[63]  Frank Kelly,et al.  Models for a self–managed Internet , 2000, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[64]  S. Wiggins.  Introduction to Applied Nonlinear Dynamical Systems and Chaos , 1989 .

[65]  E. Lorenz.  Deterministic nonperiodic flow , 1963 .

[66]  Christopher Essex,et al.  Competitive Modes and their Application , 2006, Int. J. Bifurc. Chaos.

[67]  N. N. Bautin,et al.  On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type , 1954 .

[68]  E. Abed,et al.  Local feedback stabilization and bifurcation control, I. Hopf bifurcation , 1986 .

[69]  N. G. Lloyd,et al.  A Cubic System with Eight Small-Amplitude Limit Cycles , 1991 .

[70]  C. Pierre,et al.  A NEW GALERKIN-BASED APPROACH FOR ACCURATE NON-LINEAR NORMAL MODES THROUGH INVARIANT MANIFOLDS , 2002 .

[71]  Guanrong Chen,et al.  Chaos Synchronization of General Lur'e Systems via Time-Delay Feedback Control , 2003, Int. J. Bifurc. Chaos.

[72]  Wei Zhang,et al.  Vibration analysis on a thin plate with the aid of computation of normal forms , 2001 .

[73]  Julien Clinton Sprott,et al.  Algebraically Simple Chaotic Flows , 2000 .

[74]  N. G. Lloyd,et al.  Some cubic systems with several limit cycles , 1988 .

[75]  R. Srikant,et al.  Global stability of congestion controllers for the Internet , 2003, IEEE Trans. Autom. Control..

[76]  Guanrong Chen,et al.  Effective chaotic orbit tracker: a prediction-based digital redesign approach , 2000 .

[77]  Jibin Li,et al.  Hilbert's 16th Problem and bifurcations of Planar Polynomial Vector Fields , 2003, Int. J. Bifurc. Chaos.

[78]  Gennady A. Leonov,et al.  Attraktoreingrenzung für nichtlineare Systeme , 1987 .

[79]  Takashi Matsumoto,et al.  A chaotic attractor from Chua's circuit , 1984 .

[80]  Christophe Pierre,et al.  Normal Modes for Non-Linear Vibratory Systems , 1993 .

[81]  Zhen Chen,et al.  Hopf bifurcation Control for an Internet Congestion Model , 2005, Int. J. Bifurc. Chaos.

[82]  E. Lorenz,et al.  The essence of chaos , 1993 .

[83]  Akio Tsuneda,et al.  A Gallery of attractors from Smooth Chua's equation , 2005, Int. J. Bifurc. Chaos.

[84]  Pei Yu,et al.  A Method for Computing Center Manifold and Normal Forms , 2000 .

[85]  R. M. Noyes,et al.  Oscillations in chemical systems. IV. Limit cycle behavior in a model of a real chemical reaction , 1974 .

[86]  Pei Yu,et al.  Analysis on the Globally Exponent Synchronization of Chua's Circuit Using Absolute Stability Theory , 2005, Int. J. Bifurc. Chaos.

[87]  Ramesh Johari,et al.  End-to-end congestion control for the internet: delays and stability , 2001, TNET.

[88]  Guanrong Chen,et al.  LINEAR TIME-DELAY FEEDBACK CONTROL OF A PATHOLOGICAL RHYTHM IN A CARDIAC CONDUCTION MODEL , 1997 .

[89]  S. Smale.  Mathematical problems for the next century , 1998 .