Dynamic Behavior for a System of Four Coupled Nonlinear Oscillators with Delays

Gail Sellers

doi:https://doi.org/10.14445/22315373/IJMTT-V66I6P516

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 66 | Issue 6 | Year 2020 | Article Id. IJMTT-V66I6P516 | DOI : https://doi.org/10.14445/22315373/IJMTT-V66I6P516

Dynamic Behavior for a System of Four Coupled Nonlinear Oscillators with Delays

Gail Sellers

Abstract

In this paper, a system of two coupled damped Duffing resonators driven by two van der Pol oscillators with delays is studied. Some sufficient conditions to ensure the periodic and partial periodic oscillations for the system are established. Computer simulation is given to demonstrate our result.

Keywords

nonlinear system, time delay, periodic and partial periodic oscillation

References

[1] R.B. Karabalin, M.C. Cross and M.L. Roukes, “Nonlinear dynamics and chaos in two coupled nanomechanical resonators,” Physical Review B, vol. 79, 165309, 2009.
[2] A. Y. Leung, Z. Guo and H. Yang, “Residue harmonic balance analysis for the damped Duffing resonator driven by a van der Pol oscillator,” Internat. J. Mech. Sci. vol. 63, pp. 59-65, 2012.
[3] R. Rand and J. Wong, “Dynamics of four coupled phase-only oscillators,” Commun. Nonlinear Sci. Numer. Simulat. vol. 13, pp. 501-507, 2008.
[4] A. Sharma, “Time delay induced partial death patterns with conjugate coupling in relay oscillators,” Physics Letters A, vol. 383, pp. 1865-1870, 2019.
[5] A.A. Kashchenko, “Multistability in a system of two coupled oscillators with delayed feedback,” J. Diff. Equa. vol. 266, pp. 562-579, 2019.
[6] S. Gao, H.H. Guo and T.R. Chen, “The existence of periodic solutions for discrete-time coupled systems on networks with time-varying delay,” Physica A: Stat. Mech. Appl. vol. 526, 120876, 2019.
[7] X. Xu, D.Y. Yu and Z.H. Wang, “Inter-layer synchronization of periodic solutions in two coupled rings with time delay,” Physica D: Nonlinear Phenomena, vol. 396, pp. 1-11, 2019.
[8] K. Miyamoto, K. Ryono and T. Oguchi, “Delay-independent synchronization in networks of timedelay coupled systems with uncertainties,” IFAC-PapersOnLine, vol. 51, pp. 211-216.
[9] D. Dileep, F. Borgioli, L. Hetel, J.P. Richard and W. Michiels, “A scalable design method for stabilising decentralised controllers for networks of delay-coupled systems,” IFAC-PapersOnLine, vol. 51, pp. 68-73.
[10] J.M. Grzybowski, E.E.Macau and T. Yoneyama, “The Lyapunov-Krasovskii theorem and a sufficient criterion for local stability of isochronal synchronization in networks of delay-coupled oscillators,” Physica D: Nonlinear Phenomena, vol. 346, pp. 28-36, 2017.
[11] J.M. Zhang and X.S. Gu, “Stability and bifurcation analysis in the delay-coupled van der Pol oscillators,” Appl. Math. Model. vol. 34, pp. 2291-2299, 2010.
[12] C. Zhang, B. Zheng and L.Wang, “Multiple Hopf bifurcations of three coupled van der Pol oscillators with delay,” Appl. Math. Comput. vol. 217, pp. 7155-7166, 2011.
[13] M. V. Tchakui and P. Woafo, “Dynamics of three unidirectionally coupled autonomous Duffing oscillators and application to inchworm piezoelectric motors: Effects of the coupling coefficient and delay,” Chaos, vol. 26, 113108, 2016.
[14] I.V. Sysoev, “Reconstruction of ensembles of generalized Van der Pol oscillators from vector time series,” Physica D: Nonlinear Phenomena, vol. 364-385, pp. 1-11, 2018.
[15] P. Kumar, S. Narayanan and S. Gupta, “Investigations on the bifurcation of a noisy Duffing-Van der Pol oscillator,” Probab. Engin. Mech. vol. 45, pp. 70-86, 2016.
[16] T.A. Giresse and K.T. Crepin, “Chaos generalized synchronization of coupled Mathieu-Van der Pol and coupled Duffing-Van der Pol systems using fractional order-derivative,” Chaos, Solitons, Fractals, vol. 98, pp. 88-100, 2017.
[17] A.M. Santos, S.R. Lopes and R.L. Viana, “Rhythm synchronization and chaotic modulation of coupled Van der Pol oscillators in a model for the heartbeat,” Physica A, vol. 338, pp. 335-355, 2004.
[18] H.Y. Hu, “Global Dynamics of a Duffing system with delayed velocity feedback,” Proc. IUTAM Sympos. pp.335-344, 2003.
[19] C.J. Xu and Y.S. Wu, “Bifurcation control for a Duffing oscillator with delayed velocity feedback,” Intern. J. Auto. Comput. vol. 13, pp. 596-606, 2016.
[20] H. Zang, T.H. Zhang and Y.D. Zhang, “Stability and bifurcation analysis of delay coupled Van der Pol-Duffing oscillators,” Nonlinear Dynamics, vol. 75, pp. 35-47, 2014.
[21] C.H. Feng and C.M. Akujuobi, “Partial periodic oscillation: an interesting phenomenon for a system of three coupled unbalanced damped Duffing oscillators with delays,” Electronic J. Qualitative Theory Diff. Equa. vol. 86, pp. 1-16, 2018.
[22] N. Chafee, “A bifurcation problem for a functional differential equation of finitely retarded type,” J. Math. Anal. Appl. vol. 35, pp. 312-348, 1971.
[23] C.H. Feng and R. Plamondon, “An oscillatory criterion for a time delayed neural ring network model,” Neural Networks, vol. 29-30, pp. 70-79, 2012.
[24] K. Gopalsamy, Stability and oscillations in delay differential equations of populations dynamics, Kluwer Academic Publishers, 1992.

Citation :

Gail Sellers, "Dynamic Behavior for a System of Four Coupled Nonlinear Oscillators with Delays," International Journal of Mathematics Trends and Technology (IJMTT), vol. 66, no. 6, pp. 168-183, 2020. Crossref, https://doi.org/10.14445/22315373/IJMTT-V66I6P516