Bogdanov–Takens bifurcation in an oscillator with negative damping and delayed position feedback

Abstract In this paper, Bogdanov–Takens bifurcation occurring in an oscillator with negative damping and delayed position feedback is investigated. By using center manifold reduction and normal form theory, dynamical classification near Bogdanov–Takens point can be completely figured out in terms of the second and third derivatives of delayed feedback term evaluated at the zero equilibrium. The obtained normal form and numerical simulations show that multistability, heteroclinic orbits, stable double homoclinic orbits, large amplitude periodic oscillation, and subcritical Hopf bifurcation occur in an oscillator with negative damping and delayed position feedback. The results indicate that negative damping and delayed position feedback can make the system produce more complicated dynamics.

[1]  Y. Rocard.  General Dynamics of Vibrations , 1960 .

[2]  Yuan Yuan,et al.  Zero singularities of codimension two and three in delay differential equations , 2008 .

[3]  M. Tadé,et al.  Stability and multiple bifurcations of a damped harmonic oscillator with delayed feedback near zero eigenvalue singularity. , 2008, Chaos.

[4]  L. Magalhães,et al.  Normal Forms for Retarded Functional Differential Equations and Applications to Bogdanov-Takens Singularity , 1995 .

[5]  Ioannis Minis,et al.  Complex dynamics in metal cutting , 1993 .

[6]  Yingxiang Xu,et al.  Homoclinic orbits and Hopf bifurcations in delay differential systems with T–B singularity , 2008 .

[7]  Hsueh-Chia Chang,et al.  Transition to Chaos from a Two-Torus in a Delayed Feedback System , 1991 .

[8]  André Longtin,et al.  Bifurcation analysis of a class of first-order nonlinear delay-differential equations with reflectional symmetry , 2002 .

[9]  Ferdinand Verhulst,et al.  Suppressing Flow-Induced Vibrations by Parametric Excitation , 2003 .

[10]  Xiaofeng Liao,et al.  Local stability, Hopf and resonant codimension-Two bifurcation in a harmonic oscillator with Two Time delays , 2001, Int. J. Bifurc. Chaos.

[11]  Sue Ann Campbell,et al.  Complex dynamics and multistability in a damped harmonic oscillator with delayed negative feedback. , 1995, Chaos.

[12]  Chun Liu,et al.  An Analytical Model of Cutting Dynamics. Part 1: Model Building , 1985 .

[13]  A Beuter,et al.  Feedback and delays in neurological diseases: a modeling study using dynamical systems. , 1993, Bulletin of mathematical biology.

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

[15]  G. Schmidt,et al.  Tondl, A., On the Interaction between Self-Excited and Parametric Vibrations. Prague 1978. Monographs and Memoranda 25. Special Printing of the National Research Institute for Machine Design, Bechovice. 127 S , 1980 .

[16]  John G. Milton,et al.  Limit cycles, tori, and complex dynamics in a second-order differential equation with delayed negative feedback , 1995 .

[17]  C. Hsu,et al.  Stability Criteria for Second-Order Dynamical Systems With Time Lag , 1966 .

[18]  Haiyan Hu,et al.  Stability, bifurcation and chaos of a delayed oscillator with negative damping and delayed feedback control , 2007 .

[19]  Kestutis Pyragas.  Continuous control of chaos by self-controlling feedback , 1992 .

[20]  Aleš Tondl.  TO THE PROBLEM OF QUENCHING SELF-EXCITED VIBRATIONS , 1998 .

[21]  Sue Ann Campbell,et al.  Resonant Hopf–Hopf Interactions in Delay Differential Equations , 1998 .

[22]  Edgar Knobloch,et al.  An unfolding of the Takens-Bogdanov singularity , 1991 .

[23]  Junjie Wei,et al.  Singularity Analysis on a Planar System with Multiple Delays , 2007 .

[24]  D. W. Wu,et al.  An Analytical Model of Cutting Dynamics. Part 2: Verification , 1985 .

[25]  L. Magalhães,et al.  Normal Forms for Retarded Functional Differential Equations with Parameters and Applications to Hopf Bifurcation , 1995 .

[26]  Tamás Kalmár-Nagy,et al.  Subcritical Hopf Bifurcation in the Delay Equation Model for Machine Tool Vibrations , 2001 .

[27]  Hsueh-Chia Chang,et al.  Dynamics of delayed systems under feedback control , 1989 .

[28]  H. Hu,et al.  Dynamics of Controlled Mechanical Systems with Delayed Feedback , 2002 .

[29]  Yongli Song,et al.  Steady-State, Hopf and steady-State-Hopf bifurcations in Delay differential equations with Applications to a damped harmonic oscillator with Delay Feedback , 2012, Int. J. Bifurc. Chaos.

[30]  Weihua Jiang,et al.  Bogdanov–Takens singularity in Van der Pol’s oscillator with delayed feedback , 2007 .

[31]  R. Leine,et al.  Bifurcations in Nonlinear Discontinuous Systems , 2000 .