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Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain


Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

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On Coupled Delayed Van der Pol-Duffing Oscillators

Journal of Applied Nonlinear Dynamics 9(4) (2020) 567--574 | DOI:10.5890/JAND.2020.12.004

Ankan Pandey$^1$ , Mainak Mitra$^{2}$, A Ghose-Choudhury$^{3}$, Partha Guha$^{1}$

$^1$ SN Bose National Centre for Basic Sciences, JD Block, Sector III, Salt Lake, Kolkata 700106, India

$^2$ Ramakrishna Mission Residential College (Autonomous) Narendrapur, Kolkata-700103, West Bengal, India

$^3$ Department of Physics, Diamond Harbour Women's University, Sarisha, West Bengal 743368, India

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We investigate the dynamics of a delay differential coupled Duffing-Van der Pol oscillator equation. Using the Lindstedt's method, we derive the in-phase mode solutions and then obtain the slow flow equations governing the stability of the in-phase mode by employing the two variable perturbation method. We solve the slow flow equations using series expansion and obtain conditions for Hopf bifurcation and studied stability of the in-phase mode. Finally, we studied stability and bifurcations of the origin. Our interest in this system is due to the fact that it is related to the coupled laser oscillators.


  1. [1]  Wirkus, S.A. and Rand, R. (2002), {The dynamics of two coupled van der Pol oscillators with delay coupling}, Nonlinear Dynamics, 30(3), 205-221.
  2. [2]  Lynch, J.J. (1995), { Analysis and design of systems of coupled microwave oscillators}, PhD thesis, Department of Electrical and Computer Engineering, University of California at Santa Barbara.
  3. [3]  Lynch, J.J. and York, R.A. (1995), {Stability of m ode locked states of coupled oscillator arrays}, IEEE Trans. on Circuits and Systems, 42, 413-417.
  4. [4]  York, R.A. (1993), {Nonlinear analysis of phase re lationships in quasi-optical oscillator arrays}, IEEE Trans. on Microwave Theory and Tech., 41, 1799-1809.
  5. [5]  York, R.A. and Compton, R.C. (1991), { Quasi-optical power combining using mutually synchronized oscillator arrays}, IEEE Trans. on Microwave Theory and Tech., 39, 1000-1009.
  6. [6] Sargent III, M., Scully, M.O., and Lamb, Jr. W.E. (1974), { Laser Physics}, Addison-Wesley, Reading.
  7. [7]  Gluzman, M, and Rand, R. { Dynamics of two coupled Van der Pol oscillators with delay coupling revisited} arXiv:1705.03100v1[ math. DS].
  8. [8]  Wirkus, S.A. (1999), { The dynamics of two coupled Van der Pol oscillators with delay coupling}, PhD thesis, Cornell University.
  9. [9]  Cooke, K.L. and Grossman, Z. (1982), { Discrete delay, distributed delay and stability switches}, Journal of mathematical analysis and applications, 86(2), 592-627.
  10. [10]  Bhatt, S.J. and Hsu, C.S. (1966), {Stability Criteria for Second-Order Dynamical Systems With Time Lag}, ASME. J. Appl. Mech., 33(1), 113-118. DOI:10.1115/1.3624967.
  11. [11]  Rand, R.H., Cohen, A.H., and Holmes, P.J. (1988), { Systems of coupled oscillators as models of central pattern generators}, in A. H. Cohen, editor, Neural Control of Rhythmic Movements in Vertebrates. John Wiley.
  12. [12]  Rand, R.H. and Holmes, P.J. (1980), {Bifurcation of periodic motions in two weakly coupled van der Pol oscillators}, Int. J. Nonlinear Mechanics, 15, 387-399.
  13. [13]  Reddy, D.V.R., Sen, A., and Johnston, G.L. (1988), {Time delay induced death in coupled limit cycle oscillators}, Physical Review Letters, 80, 5109-5112.