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ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com

Distributed Delay Effects on Coupled van der Pol Oscillators, and a Chaotic van der Pol-Rayleigh System

Discontinuity, Nonlinearity, and Complexity 10(1) (2021) 87--115 | DOI:10.5890/DNC.2021.03.007

Ryan Roopnarain, S. Roy Choudhury

Department of Mathematics, University of Central Florida, 4393 Andromeda Loop N Orlando, FL 32816, USA

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Abstract

Distributed delays modeled by `weak generic kernels' are introduced in two well-known coupled van der Pol systems, as well as a chaotic van der Pol-Rayleigh system with parametric forcing. The systems are closed via the `linear chain trick'. Linear stability analysis of the systems and conditions for Hopf bifurcation that initiates oscillations are investigated, including deriving the normal form at bifurcation, and deducing the stability of the resulting limit cycle attractor. The value of the delay parameter $a = a_{Hopf}$ at Hopf bifurcation picks out the onset of Amplitude Death(AD) in all three systems, with oscillations at larger values (corresponding to weaker delay). In both van der Pol systems, the Hopf-generated limit cycles for $a > a_{Hopf}$ turn out to be remarkably stable under very large variations of all other system parameters beyond the Hopf bifurcation point, and do not undergo further symmetry breaking, cyclic-fold, flip, transcritical or Neimark-Sacker bifurcations. Numerical simulations reveal strong distortion and rotation of the limit cycles in phase space as the parameters are pushed far into the post-Hopf regime, and also reveal other features, such as how the oscillation amplitudes and time periods of the physical variables on the limit cycle attractor change as the delay and other parameters are varied. For the chaotic system, very strong delays may still lead to the cessation of oscillations and the onset of AD. Varying of the other important system parameter, the parametric excitation, leads to a rich sequence of dynamical behaviors, with the bifurcations leading from one regime (or type of attractor) into the next being carefully tracked.

References

  1. [1]  Saxena, G., Prasad, A., and Ramaswamy, R. (2012), Amplitude death: The emergence of stationarity in coupled nonlinear systems, Phys. Rep., 521(5), 205-28. https://doi.org/10.1016/j.physrep.2012.09.003.
  2. [2]  Koseska, A., Volkov, E., and Kurths, J. (2013), Oscillation quenching mechanisms: Amplitude vs. oscillation death, Phys. Rep., 531(4), 173-99. {https://doi.org/10.1016/j.physrep.2013.06.001}.
  3. [3]  Crowley, M.F. and Field, R.J. (1981), Electrically coupled Belousov-Zhabotinsky oscillators: A potential chaos generator. In: Vidal C. \& Pacault A. editors. Nonlinear Phenomena in Chemical Dynamics, Berlin, Heidelberg: Springer; 147-53. {https://doi.org/10.1007/978-3-642-81778-6\_21}.
  4. [4]  Bar-Eli, K. (1984), Coupling of chemical oscillators, J. Phys. Chem., 88(16), 3616-22. {https://doi.org/10.1016/0025-5564(94)90081-7}.
  5. [5]  Bar-Eli, K. (2011), Oscillations death revisited; coupling of identical chemical oscillators, Phys. Chem. Chem. Phys., 13(24), 11606-14. {https://doi.org/10.1039/C0CP02750B}.
  6. [6]  Kumar, V.R., Jayaraman, V.K., Kulkarni, B.D., and Doraiswamy, L.K. (1983), Dynamic behaviour of coupled CSTRs operating under different conditions, Chem. Eng. Sci., 38(5), 673-86. {https://doi.org/10.1016/0009-2509(83)80180-8}.
  7. [7]  Koseska, A., Volkov, E., and Kurths, J. (2010), Parameter mismatches and oscillation death in coupled oscillators, Chaos, 20(2), 023132. {https://doi.org/10.1063/1.3456937}.
  8. [8]  Reddy, D.R., Sen, A., and Johnston, G.L. (1998), Time delay induced death in coupled limit cycle oscillators, Phys. Rev. Lett., 80(23), 5109. {https://doi.org/10.1103/PhysRevLett.80.5109}.
  9. [9]  Reddy, D.R., Sen, A., and Johnston, G.L. (1999), Time delay effects on coupled limit cycle oscillators at Hopf bifurcation, Physica D, 129(1-2), 15-34. {https://doi.org/10.1016/S0167-2789(99)00004-4}.
  10. [10]  Reddy, D.R., Sen, A., and Johnston, G.L. (2000), Experimental evidence of time-delay-induced death in coupled limit-cycle oscillators, Phys. Rev. Lett., 85(16), 3381. {https://doi.org/10.1103/PhysRevLett.85.3381}.
  11. [11]  Reddy, D.R., Sen, A., and Johnston, G.L. (2000), Dynamics of a limit cycle oscillator under time delayed linear and nonlinear feedbacks, Physica D, 144(3-4), 335-57. {https://doi.org/10.1016/S0167-2789(00)00086-5}.
  12. [12]  Senthilkumar, D.V. and Kurths, J. (2010), Dynamics of Nonlinear Time-Delay Systems, 1st ed. Berlin: Springer; {https://doi.org/10.1007/978-3-642-14938-2}
  13. [13]  Atay, F.M. (2003), Distributed delays facilitate amplitude death of coupled oscillators, Phys. Rev. Lett., 91(9), 094101. {https://doi.org/10.1103/PhysRevLett.91.094101}.
  14. [14]  Saxena, G., Prasad, A., and Ramaswamy, R. (2010), Dynamical effects of integrative time-delay coupling, Phys. Rev. E, 82(1), 017201. {https://doi.org/10.1103/PhysRevE.82.017201}.
  15. [15]  Saxena, G., Prasad, A., and Ramaswamy, R. (2011), The effect of finite response-time in coupled dynamical systems, Pramana-J. Phys., 77(5), 865-71. {https://doi.org/10.1007/s12043-011-0179-z}.
  16. [16]  Kim, M.Y. (2005), Delay induced instabilities in coupled semiconductor lasers and Mackey-Glass electronic circuits. (Doctoral dissertation). {http://hdl.handle.net/1903/2722}. % may not be the correct citation but no example of how to cite theses.
  17. [17]  Kim, M.Y., Roy, R., Aron, J.L., Carr, T.W., and Schwartz, I.B. (2005), Scaling behavior of laser population dynamics with time-delayed coupling: theory and experiment, Phys. Rev. Lett., 94(8), 088101. {https://doi.org/10.1103/ PhysRevLett.94.088101}.
  18. [18]  Karnatak, R., Punetha, N., Prasad, A., and Ramaswamy, R. (2010), Nature of the phase-flip transition in the synchronized approach to amplitude death, Phys. Rev. E, 82(4), 046219. https://doi.org/10.1103/ PhysRevE.82.046219.
  19. [19]  Karnatak, R., Ramaswamy, R., and Prasad, A. (2009), Synchronization regimes in conjugate coupled chaotic oscillators, Chaos, 19(3), 033143. {https://doi.org/10.1063/1.3236385}.
  20. [20]  Zhang, X., Wu, Y., and Peng, J. (2011), Analytical conditions for amplitude death induced by conjugate variable couplings, Int. J. Bif. Chaos, 21(01), 225-35. {https://doi.org/10.1142/S0218127411028386}.d
  21. [21]  Konishi, K. (2003), Amplitude death induced by dynamic coupling, Phys. Rev. E, 68(6), 067202. https://doi.org/ 10.1103/PhysRevE.68.067202.
  22. [22]  Prasad, A., Dhamala, M., Adhikari, B.M., and Ramaswamy, R. (2010), Amplitude death in nonlinear oscillators with nonlinear coupling, Phys. Rev. E, 81(2), 027201. {https://doi.org/10.1103/PhysRevE.81.027201}.
  23. [23]  Prasad, A., Lai, Y.C., Gavrielides, A., and Kovanis, V. (2003), Amplitude modulation in a pair of time-delay coupled external-cavity semiconductor lasers, Phys. Lett. A, 318(1-2), 71-7. {https://doi.org/10.1016/j.physleta.2003.08.072}.
  24. [24]  Sharma, P.R., Sharma, A., Shrimali, M.D., and Prasad, A. (2011), Targeting fixed-point solutions in nonlinear oscillators through linear augmentation, Phys. Rev. E, 83(6), 067201. {https://doi.org/10.1103/PhysRevE.83.067201}.
  25. [25]  Resmi, V., Ambika, G., and Amritkar, R.E. (2010), Synchronized states in chaotic systems coupled indirectly through a dynamic environment, Phys. Rev. E, 81(4), 046216. {https://doi.org/10.1103/PhysRevE.81.046216} %%%%% VDP REF
  26. [26]  Krise, S. and Choudhury, S.R. (2003), Bifurcations and Chaos in a Predator-Prey Model with Delay and a Laser-Diode System with Self-Sustained Pulsations, Chaos, Solitons \& Fractals, 16(1), 59-77. {https://doi.org/10.1016/S0960-0779(02)00199-6}
  27. [27]  Cushing, J.M. (1977), Integrodifferential equations and delay models in population dynamics. Berlin, Heidelberg: Springer; {https://doi.org/10.1007/978-3-642-93073-7}.
  28. [28]  MacDonald, N. (1978), Time Lage in Biological Models, Lecture Notes in Biomathematics. Berlin, Heidelberg: Springer; {https://doi.org/10.1007/978-3-642-93107-9}.
  29. [29]  Warminski, J. (2012), Regular and chaotic vibrations of van der Pol and Rayleigh oscillators driven by parametric excitation, Procedia IUTAM, 5, 78-87. {https://doi.org/10.1016/j.piutam.2012.06.011}.
  30. [30]  Atay, F.M. (2003), Total and partial amplitude death in networks of diffusively coupled oscillators, Physica D, 183(1-2), 1-18.

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