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Nonlinear Dynamics

, Volume 49, Issue 1–2, pp 117–129 | Cite as

Stability, bifurcation and chaos of a delayed oscillator with negative damping and delayed feedback control

  • X. Xu
  • H. Y. Hu
  • H. L. Wang
Original Article

Abstract

This paper presents a detailed analysis on the dynamics of a delayed oscillator with negative damping and delayed feedback control. Firstly, a linear stability analysis for the trivial equilibrium is given. Then, the direction of Hopf bifurcation and stability of periodic solutions bifurcating from trivial equilibrium are determined by using the normal form theory and center manifold theorem. It shows that with properly chosen delay and gain in the delayed feedback path, this controlled delayed system may have stable equilibrium, or periodic solutions, or quasi-periodic solutions, or coexisting stable solutions. In addition, the controlled system may exhibit period-doubling bifurcation which eventually leads to chaos. Finally, some new interesting phenomena, such as the coexistence of periodic orbits and chaotic attractors, have been observed. The results indicate that delayed feedback control can make systems with state delay produce more complicated dynamics.

Keywords

Time delay Delayed spring force Hopf bifurcation Chaos Period-doubling bifurcation 

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References

  1. 1.
    Hu, H.Y., Wang, Z.H.: Dynamics of Controlled Mechanical Systems with Delayed Feedback. Springer-Verlag, Heidelberg (2002)zbMATHGoogle Scholar
  2. 2.
    Reddy, D.V.R., Sen, A., Johnston, G.L.: Dynamics of a limit cycle oscillator under time delayed linear and nonlinear feedbacks. Physica D 144, 335–357 2000zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Lu, S.P., Ge, W.G.: Existence of positive periodic solutions for neutral population model with multiple delays. J. Comput. Appl. Math. 166, 371–383 2004zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Shayer, L.P., Campbell, S.A.: Stability, bifurcation, and multistability in a system of two coupled neurons with multiple time delays. Soc. Ind. Appl. Math. 61, 673–700 2000zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Hale, J.K.: Theory of Functional Differential Equations. Springer-Verlag, New York (1977)zbMATHGoogle Scholar
  6. 6.
    Diekmann, O.: Delay Equations, Functional-, Complex-, and Nonlinear Analysis. Springer, New York (1995)zbMATHGoogle Scholar
  7. 7.
    Stepan, G.: Retarded Dynamical Systems: Stability and Characteristic Functions. Longman Scientific and Technical, Essex (1989)zbMATHGoogle Scholar
  8. 8.
    Kuang, Y.: Delay Differential Equations with Applications to Population Dynamics. Academic Press, New York (1993)Google Scholar
  9. 9.
    Qin, Y.X. et al.: Stability of Dynamic Systems with Delays. Science Press, Beijing (1989)Google Scholar
  10. 10.
    Campbell, S.A.: Stability and bifurcation of a simple neural network with multiple time delays. Fields Inst. Commun. 21, 65–79 1999Google Scholar
  11. 11.
    Sipahi, R., Olgac, N.: Complete stability robustness of third-order LTI multiple time-delay systems. Automatica 41, 1413–1422 2005zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Pyragas, K.: Continuous control of chaos by self-controlling feedback. Phys. Lett. A 170, 421–428 1992CrossRefGoogle Scholar
  13. 13.
    Hu, H.Y.: Using delayed state feedback to stabilize periodic motions of an oscillator. J. Sound Vibrat. 275, 1009–1025 2004CrossRefGoogle Scholar
  14. 14.
    Xu, J., Chung, K.W.: Effects of time delayed position feedback on a van der Pol-Duffing oscillator. Physica D 180, 17–39 2003zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Liao, X.F., Chen, G.R.: Local stability, Hopf and resonant codimension-two bifurcation in a harmonic oscillator with two time delays. Int. J. Bifurcat. Chaos 11, 2105–2121 2001zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Moiola, J.L., Chiacchiarini, H.G., Ddeages, A.C.: Bifurcation and Hopf degeneracies in nonlinear feedback systems with the time-delay. Int. J. Bifurcat. Chaos 6, 661–672 1996zbMATHCrossRefGoogle Scholar
  17. 17.
    Wei, J.J., Jiang, W.H.: Stability and bifurcation analysis in Van der Pol's oscillator with delayed feedback. J. Sound Vibrat. 283, 801–809 2005CrossRefMathSciNetGoogle Scholar
  18. 18.
    Kakmeni, F.M.M., Bowong, S., Tchawoua, C. et al.: Chaos control and synchronization of a Phi(6)-van der Pol oscillator. Phys. Lett. A 322, 305–323 2004)CrossRefMathSciNetGoogle Scholar
  19. 19.
    De Oliveira, J.C.F.: Oscillations in a van der Pol equation with delayed argument. J. Math. Anal. Appl. 275, 789–803 2002zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Atay, F.M.: Van der Pol's oscillator under delayed feedback. J. Sound Vibrat. 218, 333–339 1998CrossRefMathSciNetGoogle Scholar
  21. 21.
    Tian, Y.P., Yu, X.H., Chua, L.O.: Time-delayed impulsive control of chaotic hybrid systems. Int. J. Bifurcat. Chaos 14, 1091–1104 2004zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Atay, F.M.: Delayed-feedback control of oscillations in nonlinear planar systems. Int. J. Control 75, 297–304 2002zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Campbell, S.A., Belair, J., Ohira, T. et al.: Complex dynamics and multistability in a damped harmonic oscillator with delayed negative feedback. Chaos 5, 640–645 1995zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Minorsky, N.: Nonlinear Oscillations. D. Van Nostrand Company, Inc. Princeton, NJ (1962)zbMATHGoogle Scholar
  25. 25.
    Yoshitake, Y., Inoue, J., Sueoka, A.: Vibration of a forced self-excited system with time delay. Trans. JSME Ser. C 49, 298–305 1984Google Scholar
  26. 26.
    Xu, J., Lu, Q.S.: Hopf bifurcation of time-delay lienard equations. Int. J. Bifurcat. Chaos 9, 939–951 1999zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Wang, Z.H., Hu, H.Y.: An energy analysis of the local dynamics of a delayed oscillator near a hopf bifurcation. Nonlinear Dyn. 46, 149–159 (2006)Google Scholar
  28. 28.
    Das, S.L., Chatterjee, A.: Multiple scales without center manifold reductions for delay differential equations near Hopf bifurcations. Nonlinear Dyn. 30, 323–335 2002zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Raghothama, A., Narayanan, S.: Periodic response and chaos in nonlinear systems with parametric excitation and time delay. Nonlinear Dyn. 27, 341–365 2003CrossRefMathSciNetGoogle Scholar
  30. 30.
    Plaut, R.H., Hsieh, J.C.: Chaos in a mechanism with time delays under parametric and external excitation. J. Sound Vibrat. 114, 73–90 1987MathSciNetGoogle Scholar
  31. 31.
    Rabotnov, Y.N.: Creep Problems in Structural Members. North-Holland Publishing Company-Amsterdam, London (1966)Google Scholar
  32. 32.
    Rabotnov, Y.N.: Elements of Hereditary Solid Mechanics. MIR Publishers, Moscow (1980)zbMATHGoogle Scholar
  33. 33.
    Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. John Wiley & Sons, New York (1979)zbMATHGoogle Scholar
  34. 34.
    Stepan, G., Szabo, Z.: Impact induced internal fatigue cracks. In: Proceedings ASME DETC 17th Biennial Conference on Mechanical Vibration and Noise, Las Vegas, DETC99/VIB-8351, pp. 1–7 (1999)Google Scholar
  35. 35.
    Rocard, Y.: General Dynamics of Vibrations. Crosby Lichwood & Son Ltd., London (1960)Google Scholar
  36. 36.
    Kalmar-Nagy, T., Stepan, G., Moon, F.C.: Subcritical Hopf bifurcation in the delay equation model for machine tool vibrations. Nonlinear Dyn. 26, 121–142 2001zbMATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Leine, R.I., van Campen, D.H., van de Vrande, B.L.: Bifurcations in nonlinear discontinuous systems. Nonlinear Dyn. 23, 105–164 2000zbMATHCrossRefGoogle Scholar
  38. 38.
    Stepan, G., Kalmar-Nagy, T.: Nonlinear regenerative machine tool vibration. In: Proceedings ASME DETC 17th Biennial Conference on Mechanical Vibration and Noise, Sacramento, DETC97/VIB-4021, pp. 1–11 (1997)Google Scholar
  39. 39.
    Cooke, K.L., Grossman, Z.: Discrete delay, distributed delay and stability switches. J. Math. Anal. Appl. 86, 592–627 1982zbMATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    Wang, Z.H., Hu, H.Y.: Stability switches of time-delayed dynamic systems with unknown parameters. J. Sound Vibrat. 233, 215–233 2000CrossRefMathSciNetGoogle Scholar
  41. 41.
    Hassard, B.D., Kazarinoff, N.D., Wan, Y.H.: Theory and Applications of Hopf Bifurcation. Cambridge University Press, Cambridge (1981)zbMATHGoogle Scholar
  42. 42.
    Ermentrout, B.: XPPAUT 3.0—The Differential Equations Tool. University of Pittsburgh, Pittsburgh 1997Google Scholar
  43. 43.
    Wahi, P., Chatterjee, A.: Regenerative tool chatter near a codimension 2 Hopf point using multiple scales. Nonlinear Dyn. 40, 323–338 2005zbMATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    Stepan, G., Haller, G.: Quasiperiodic oscillations in robot dynamics. Nonlinear Dyn. 8, 513–528 1995MathSciNetGoogle Scholar
  45. 45.
    Wang, H.L., Hu, H.Y., Wang, Z.H.: Global dynamics of a Duffing oscillator with delayed displacement feedback. Int. J. Bifurcat. Chaos 14, 2753–2775 2004zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.College of MathematicsJilin UniversityChangchunP.R. China
  2. 2.Institute of Vibration Engineering ResearchNanjing University of Aeronautics and AstronauticsNanjingP.R. China

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