Authors : Shashank S. Jha

Volume/Issue : Volume 11 - 2026, Issue 8 - August

Google Scholar : https://tinyurl.com/4zt376bp

Scribd : https://tinyurl.com/3mbc8pwc

DOI : https://doi.org/10.38124/ijisrt/26aug240

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Abstract : Dynamics of dispersive delay coupled system of two stable limit cycle oscillators is presented. The effect of dispersive delay coupling on synchronization dynamics of two phase oscillators having different frequencies is investigated. First, we determine fixed points, and then analyze their Lyapunov stability and phase space trajectories over practically realizable range of system parameters. It is shown that large number of phase locked solutions is obtained in contrast to constant delay coupled system. Both, the limit cycle and the chaotic dynamics are seen to occur in different regions of the parametric spaces. The motivation behind this study is whether this dispersive delay coupled oscillator system can be used as chaos generator for communication applications. This study contributes to a good understanding of the impact of dispersive time delay on interacting oscillators.

Keywords : Synchronization; Coupled Phase Oscillators; Dispersive Delay.

References :

1. Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence, Springer, Berlin, Heidelberg, NY, 1984.
2. P. Stuart and G. Stein, A Neurophysiological Study of Two Systems of Coupled Oscillators in Crayfish, Committee of Neurological Sciences, Stanford University, 1969.
3. A. Beuter, J. B´elair and C. Labrie, Feedback, delays, and dynamics of oscillatory behaviors in patients with neurological diseases, Bull. Math. Biol. 55, pp.  525–541, 1993.
4. D. Storti and R. H. Rand, A Simplified Model of Coupled Relaxation Oscillators, Int. J. Non-Linear Mechanics, vol. 22, No. 4, pp. 283-289, 1987.
5. M. F. Crowley and I. R. Epstein, Experimental and Theoretical Studies of a Coupled Chemical Oscillator: Phase Death, Multlstability, and In-Phase and Out-of-Phase Entrainment, J. Phys. Chem., vol. 93, pp. 2496-2502, 1989.
6. P. Woafo, H. B. Fotsin and J. C. Chedjou, Dynamics of Two Nonlinearly Coupled Oscillators, Physica Scripta. Vol. 57, pp. 195-200, 1998.
7. S. H. Strogatz, Nonlinear Dynamics and Chaos: with Application to Physics, Biology, Chemistry, and Engineering, Perseus Books Publishing, L.L.C., 1994.
8. Pikovsky, M. Rosenblum, J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University Press, Cambridge, 2003.
9. L. Zonghua and C. Shigang, General method of synchronization, Phy. Rev. E, vol. 55, no. 6, pp. 6651- 6655, 1997.
10. G. Schmidt and A. A. Chernikov, General form of coupling leading to synchronization of oscillating dynamical systems, Phy. Rev. E, vol. 60, no. 3, pp. 2767- 2770, 1999.
11. S. H. Strogatz, From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators, Physica D, vol. 143, pp. 1–20, 2000.
12. S. Coombes, and G. J. Lord, Desynchronization of pulse-coupled integrate-and-fire neurons, Phys. Rev. E, vol. 55, pp. 2104–2107, 1997.
13. H. Fujisaka and T. Yamada, Stability Theory of Synchronized Motion in Coupled – Oscillator Systems, Progress of Theoretical Physics, vol. 69, no. 1, pp. 32-47, 1993.
14. M. G. Rosenblum, and A. S. Pikovsky, Controlling synchronization in an ensemble of globally coupled oscillators, Phys. Rev. Lett., vol. 92, no. 11, pp. 114102(1-4), 2004.
15. N. MacDonald, Time Lags in Biological Models, Lecture Notes in iomathematics, Springer, Berlin, vol. 27, 1978.
16. J. M. Cushing, Integrodifferential Equations and Delay Models in Population Dynamics, Lecture Notes in Biomathematics, Springer-Verlag, New York, vol. 20, 1977.
17. E. Boe and H. C. Chang, Dynamics of Delayed Systems Under Feedback Control, Chemical Engineering Science, Vol. 44, No. 6, pp. 1281-1294, 1989.
18. J. E. Marshall, H. Gorecki, A. Korytowski and K. Walton, Time-Delay Systems : Stability and Performance Criteria with Applications, West Sussex, Eng. : Ellis Horwood Limited, 1992.
19. G. Schuster and P. Wagner, Mutual Entrainment of Two Limit Cycle Oscillators with Time Delayed Coupling,  Vol. 81, No. 5, pp. 939-945, 1989.
20. V. K. Jirsa, Dispersion and Time Delay Effects in Synchronized Spike-Burst networks, Cogn. Neurodyn., vol. 2, pp. 29-38, 2008.
21. F. M. Atay, Complex Time-Delay Systems: Theory and Applications, Springer Complexity, 2010.
22. S. S. Jha and R. D. S. Yadava, Synchronization Based SAW Sensor Using Delay Line Coupled Dual Oscillator Phase Dynamics, Sensors & Transducers Journal, Vol. 141, Issue 6, pp. 71-91, 2012.
23. S. Kim, S. H. Park and C. S. Ryu, Multistability in Coupled Oscillator Systems with Time Delay, Phys. Rev. Lett., vol. 79, no. 15, pp. 2911- 2914, 1997.
24. S. H. Park, S. Kim, H. B. Pyo and S. Lee, Effects of time-delayed interactions on dynamic patterns in a coupled phase oscillator system, Phy. Rev. E, vol. 60, no. 4, pp. 4962-4965, 1999.
25. E. M. Izhikevich, Phase models with explicit time delays, Phy. Rev. E, vol. 58, no. 1, pp. 905 – 908, 1998.
26. M. Lakshmanan and D. V. Sensthilkumar, Dynamics of Nonlinear Time-Delay Systems, Springer, 2010.
27. S. I. Niculescu and K. Gu, Advances in Time-Delay Systems, Lecture Notes in Computational Science and Engineering, Springer, 2004.
28. F. M. Atay, Van der Pol’s Oscillator Under Delayed Feedback, Journal of Sound and Vibration, vol. 218, no. 2, pp. 333-339, 1998.
29. D. V. R. Reddy, A. Sen and G. L. Johnston, Time Delay Induced Death in Coupled Limit Cycle Oscillators, Phy. Rev. Lett., vol. 80, no. 23, pp. 5109- 5112, 1998.
30. D. V. R. Reddy, A. Sen and G. L. Johnston, Time delay effects on coupled limit cycle oscillators at Hopf bifurcation, Physica D, vol. 129, pp 15–34, 1999.
31. M. K. Stephen Yeung and S. H. Strogatz, Time Delay in the Kuramoto Model of Coupled Oscillators, Phy. Rev. Lett., vol. 82, no. 3, pp. 648- 651, 1999.
32. D. V. R. Reddy, A. Sen and G. L. Johnston, Dynamics of a limit cycle oscillator under time delayed linear and nonlinear feedbacks, Physica D, vol. 144, pp. 335–357, 2000.
33. M. S. Mahmoud, Switched Time-Delay Systems: Stability and Control, Springer Science + Business Media, LLC 2010.
34. O. V. Popovych, C. Hauptmann and P. A. Tass, Impact of Nonlinear Delayed Feedback on Synchronized Oscillators, J Biol Phys, vol. 34, pp. 367–379, 2008.
35. M. K. Stephen Yeung and S. H. Strogatz, Time Delay in the Kuramoto Model of Coupled Oscillators, Phys. Rev. Lett., vol. 82, no. 3, pp. 648–651, 1999.
36. O. V. Popovych, V. Krachkovskyi and P. A. Tass, Two Fold Impact of Delayed Feedback on Coupled Oscillators, International Journal of Bifurcation and Chaos, vol. 17, no. 7, pp. 2517–2530, 2007.
37. V. S. Dolat and R. C. Williamson, A Continuously Variable Delay – Line System, IEEE Ultrasonics Symposium Proceedings, 1976.
38. P. C. Bressloff and S. Coombes, Symmetry and Phase-Locking in a Ring of Pulse-coupled Oscillators with Distributed Delays, Physics D, vol. 126, pp. 99-122,  1999.
39. Fatihcan M. Atay, Distributed Delays Facilitate Amplitude Death of Coupled Oscillators, Phy. Rev. Lett., vol. 91, no. 9, pp. 094101 (1-4), 2003.
40. P. Hartemann, Examples of Manufactured dispersive Delay Lines Using Acoustic Surface Waves, The Radio and Electronic Engineer, vol. 46, no. 5, pp. 229-236, 1976.
41. M. R. Daniel, J. D. Adam and P. R. Emtage, Dispersive Delay at Gigahertz Frequencies using Magnetostatic Waves, Circuits Systems Signal Process, vol. 4, no. I-2, pp. 115-135, 1985.
42. R.D.S. Yadava, Enhancing mass sensitivity of SAW delay line sensors by chirping transducers, Sensors and Actuators B, vol. 114, pp. 127–131, 2006.
43. R. Bellman and K.L. Cooke, Differential-Difference Equations, Academic Press Inc. (London) Ltd., pp. 300-390, 1963.
44. W. Michiels and S. I. Niculescu, Stability and Stabilization of Time-Delay Systems: An Eigenvalue-Based Approach, Advances in design and control, 2007.
45. B. Balachandran, T. K. Nagy and D. E. Gilsinn, Delay Differential Equations: Recent Advances and New Directions, Springer Science + Business Media, LLC 2009.
46. C. Campbell, Surface Acoustic Wave Devices for Mobile and Wireless Communications, Academic Press, 1998.
47. D. P. Morgan, Surface-wave devices for signal processing, Elsevier, 1985.
48. D. P. Morgan, Surface Acoustic Wave Filters With Applications to Electronic Communications and Signal Processing, Academic Press, 2007.
49. T. E. Parker and G. K. Montress, Precision surface-acoustic-wave (SAW) oscillators, IEEE Trans. Ultrason. Ferroelec. Freq. Control, vol. 35, no. 3, pp. 342-364, 1988.
50. L. Reindl, G. Scholl, T. Ostertag, C. C. W. Ruppel, W. E. Bulst and F. Seifert, SAW Devices as Wireless Passive Sensors, Proc. IEEE Ultrasonics Symp, 1996.
51. D. S. Ballantine, R. M. White, S. J. Martin, A. J. Ricco, E. T. Zellers, G. C. Frye and H. Wohltjen, Acoustic Wave Sensors: Theory, Design, & Physico-Chemical Applications, Academic Press, 1996.
52. L. M. Pecora, Thomas L. Carroll, Gregg A. Johnson, Douglas J. Mar and James F. Heagy, Fundamentals of synchronization in chaotic systems, concepts, and applications, Chaos, vol. 7, no. 4, pp. 520-543, 1997.
53. T. L. Carroll and L. M. Pecora, Nonlinear dynamics in circuits, World Scientific Publishing Co. Pte. Ltd., 1995.
54. L. E. Larson, J. M. Li and L. S. Tsimring, Editors, Digital Communications Using Chaos and Nonlinear Dynamics, Springer Science + Business Media, New York, 2006.