Nonlinear Dynamics

, Volume 75, Issue 1–2, pp 35–47 | Cite as

Stability and bifurcation analysis of delay coupled Van der Pol–Duffing oscillators

  • Hong Zang
  • Tonghua Zhang
  • Yanduo Zhang
Original Paper


In this paper, we aim to investigate the dynamics of a system of Van der Pol–Duffing oscillators with delay coupling. First, taking the time delay as a bifurcation parameter, the stability of the equilibrium, and the existence of Hopf bifurcation are investigated. Then using the center manifold reduction technique and normal form theory, we give the direction of the Hopf bifurcation. And then by means of the symmetric bifurcation theory for delay differential equations and the representation theory of groups, we claim the bifurcation periodic solution induced by time delay is antiphase locked oscillation. Finally, at the end of the paper, numerical simulations are carried out to support our theoretical analysis.


Coupled oscillators Phase locked oscillation Time delay 



The authors thank the referees for their valuable suggestions and comments concerning improving the work. The authors would like to acknowledge the support from National Natural Science Foundation of China (11101318 and 11001212).


  1. 1.
    Belykh, V.N., Pankratova, E.V.: Chaotic dynamics of two Van der Pol–Duffing oscillators with Huygens coupling. Regul. Chaotic Dyn. 15, 274–284 (2010) CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Benford, J., Sze, H., Woo, W., Smith, R.R., Harteneck, B.: Phase locking of relativistic magnetrons. Phys. Rev. Lett. 62, 969 (1989) CrossRefGoogle Scholar
  3. 3.
    Bi, Q.S.: Dynamical analysis of two coupled parametrically excited Van der Pol oscillators. Int. J. Non-Linear Mech. 39, 33–54 (2004) CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Chow, C., Mallet-Paret, J.: Integral averaging and bifurcation. J. Differ. Equ. 26, 112–159 (1977) CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Collins, J.J., Stewart, I.N.: Coupled nonlinear oscillators and the symmetries of animal gaits. J. Nonlinear Sci. 3, 349 (1993) CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Daido, H.: Multibranch entrainment and scaling in large populations of coupled oscillators. Phys. Rev. Lett. 77, 1406 (1996) CrossRefGoogle Scholar
  7. 7.
    Golubitsky, M., Stewart, I., Schaeffer, D.: Singularities and Groups in Bifurcation Theory, Vol. II. Springer, New York (1988) CrossRefMATHGoogle Scholar
  8. 8.
    Hale, J.K., Lunel, S.M.V.: Introduction to Functional Differential Equations. Springer, New York (1993) CrossRefMATHGoogle Scholar
  9. 9.
    Hassard, B.D., Kazarinoff, N.D., Wan, Y.H.: Theory and Applications of Hopf Bifurcation. Cambridge University Press, Cambridge (1981) MATHGoogle Scholar
  10. 10.
    Ioos, G., Joseph, D.: Elementary Stability and Bifurcation Theory. Springer, New York (1980) CrossRefGoogle Scholar
  11. 11.
    Ji, J., Zhang, N.: Additive resonances of a controlled Van der Pol–Duffing oscillator. J. Sound Vib. 315, 22–33 (2008) CrossRefGoogle Scholar
  12. 12.
    Kuznetsov, A.P., Stankevich, N.V., Turukina, L.V.: Coupled Van der Pol–Duffing oscillators: phase dynamics and structure of synchronization tongues. Physica D 238, 1203–1215 (2009) CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Maccari, A.: Vibration amplitude control for a Van der Pol–Duffing oscillator with time delay. J. Sound Vib. 317, 20–29 (2008) CrossRefGoogle Scholar
  14. 14.
    Nakajima, K., Sawada, Y.: Experimental studies on the weak coupling of oscillatory chemical reaction systems. J. Chem. Phys. 72, 2231 (1980) CrossRefGoogle Scholar
  15. 15.
    Nayfeh, A.H.: The Method of Normal Forms Second, Updated and Enlarged Edition. Wiley-VCH, Boschstr (2011) CrossRefGoogle Scholar
  16. 16.
    Nayfeh, A.H.: Order reduction of retarded nonlinear systems—the method of multiple scales versus center-manifold reduction. Nonlinear Dyn. 51, 483–500 (2008) CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Njah, A.N.: Synchronization and anti-synchronization of double hump Duffing–Van der Pol oscillators via active control. J. Inf. Comput. Sci. 4(4), 243–250 (2009) Google Scholar
  18. 18.
    Njah, A.N., Vincent, U.E.: Chaos synchronization between single and double wells Duffing–Van der Pol oscillators using active control. Chaos Solitons Fractals 37, 1356–1361 (2008) CrossRefMATHGoogle Scholar
  19. 19.
    Norimichi, H., Slawomir, R.: Existence of limit cycles for coupled Van der Pol equations. J. Differ. Equ. 195, 194–209 (2003) CrossRefMATHGoogle Scholar
  20. 20.
    Pastor, I., Pérez-García, V.M., Encinas, F., GuerraI, J.M.: Ordered and chaotic behaviour of two coupled Van der Pol oscillators. Phys. Rev. E 48, 171–182 (1993) CrossRefGoogle Scholar
  21. 21.
    Pecora, L.M.: Synchronization conditions and desynchronizing patterns in coupled limit cycle and chaotic systems. Phys. Rev. E 58, 347 (1998) CrossRefMathSciNetGoogle Scholar
  22. 22.
    Reddy, D.V.R., Sen, A., Johnston, G.L.: Time delay induced death in coupled limit cycle oscillators. Phys. Rev. Lett. 82, 648–672 (1999) CrossRefGoogle Scholar
  23. 23.
    Rompala, K., Rand, R., Howland, H.: Dynamics of three coupled Van der Pol oscillators with application to circadian rhythms. Commun. Nonlinear Sci. Numer. Simul. 12, 794803 (2007) MathSciNetGoogle Scholar
  24. 24.
    Shiino, M., Frankowicz, M.: Synchronization of infinitely many coupled limit-cycle oscillators. Phys. Lett. A 136, 103 (1989) CrossRefMathSciNetGoogle Scholar
  25. 25.
    Song, Y.L.: Hopf bifurcation and spatio-temporal patterns in delay-coupled Van der Pol oscillators. Nonlinear Dyn. 63, 223–237 (2011) CrossRefMATHGoogle Scholar
  26. 26.
    Song, Y.L., Wei, J., Yuan, Y.: Stability switches and Hopf bifurcation in a pair of delay-coupled oscillations. J. Nonlinear Sci. 17, 145–166 (2007) CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, New York (2010) Google Scholar
  28. 28.
    Wirkus, S., Rand, R.: The dynamics of two coupled Van der Pol oscillators with delay coupling. Nonlinear Dyn. 30, 205–210 (2002) CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Wu, J.: Symmetric functional-differential equations and neural networks with memory. Trans. Am. Math. Soc. 350, 4799–4838 (1998) CrossRefMATHGoogle Scholar
  30. 30.
    Yamapi, R., Filatrella, G.: Strange attractors and synchronization dynamics of coupled Van der Pol–Duffing oscillators. Commun. Nonlinear Sci. Numer. Simul. 13, 1121–1130 (2008) CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Zhang, J., Gu, X.: Stability and bifurcation analysis in the delay-coupled Van der Pol oscillators. Appl. Math. Model. 34, 2291–2299 (2009) CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Hubei Key Lab of Intelligent RobotWuhan Institute of TechnologyWuhanChina
  2. 2.Swinburne University of TechnologyHawthornAustralia

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