Kybernetika 53 no. 5, 765-779, 2017

Synchronization of time-delayed systems with discontinuous coupling

Hong-jun Shi, Lian-ying Miao and Yong-zheng SunDOI: 10.14736/kyb-2017-5-0765

Abstract:

This paper concerns the synchronization of time-delayed systems with periodic on-off coupling. Based on the stability theory and the comparison theorem of time-delayed differential equations, sufficient conditions for complete synchronization of systems with constant delay and time-varying delay are established. Compared with the results based on the Krasovskii-Lyapunov method, the sufficient conditions established in this paper are less restrictive. The theoretical results show that two time-delayed systems can achieve complete synchronization when the average coupling strength is sufficiently large. Numeric evidence shows that the synchronization speed depends on the coupling strength, on-off rate and time delay.

Keywords:

time-delayed system, complete synchronization, discontinuous coupling

Classification:

34F05, 34H10

References:

  1. M. Aghababa, S. Khanmohammadi and G. Alizadeh: Finite-time synchronization of two different chaotic systems with unknown parameters via sliding mode technique. Appl. Math. Model. 35 (2011), 3080-3091.   DOI:10.1016/j.apm.2010.12.020
  2. M. Akhmet: Self-synchronization of the integrate-and-fire pacemaker model with continuous couplings. Nonlinear Anal. Hybrid Syst. 6 (2012), 730-740.   DOI:10.1016/j.nahs.2011.07.003
  3. S. Boccaletti, J. Kurths, G. Osipov, D. Valladares and C. Zhou: The synchronization of chaotic systems. Phys.Rep. 366 (2002), 1-101.   DOI:10.1016/s0370-1573(02)00137-0
  4. X. Chen and J. Lu: Adaptive synchronization of different chaotic systems with fully unknown parameters. Phys. Lett. A 364 (2007), 123-128.   DOI:10.1016/j.physleta.2006.11.092
  5. D. Chen, R. Zhang, X. Ma and S. Liu: Chaotic synchronization and anti-synchronization for a novel class of multiple chaotic systems via a sliding mode control scheme. Nonlinear Dyn. 69 (2012), 35-55.   DOI:10.1007/s11071-011-0244-7
  6. R. Erban, J. Haskovec and Y. Sun: A Cucker-Smale model with noise and delay. SIAM J. Appl. Math. 76 (2016), 1535-1557.   DOI:10.1137/15m1030467
  7. D. Ghosh: Projective synchronization in multiple modulated time-delayed systems with adaptive scaling factor. Nonlinear Dyn. 62 (2010), 751-759.   DOI:10.1007/s11071-010-9759-6
  8. A. Hmamed: Further results on the delay-independent asymptotic stability of Linear systems. Int. J. Syst. Sci. 22 (1991), 1127-1132.   DOI:10.1080/00207729108910686
  9. J. Hu: On robust consensus of multi-agent systems with communication delays. Kybernetika 45 (2009), 768-784.   CrossRef
  10. V. Lakshmikantham and S. Leela: Differential and Integral Inequalities. Academic Press, New York 1969.   CrossRef
  11. Y. Li, X. Wu, J. Lu and J. Lü: Synchronizability of duplex networks. IEEE Trans. Circuits Syst. II 63 (2016), 206-210.   DOI:10.1109/tcsii.2015.2468924
  12. W. Lin: Adaptive chaos control and synchronization in only locally Lipschitz systems. Phys. Lett. A 372 (2008), 3195-3200.   DOI:10.1016/j.physleta.2008.01.038
  13. J. Lin and J. Yan: Adaptive synchronization for two identical generalized Lorenz chaotic systems via a single controller. Nonlinear Anal.: Real World Appl. 10 (2009), 1151-1159.   DOI:10.1016/j.nonrwa.2007.12.005
  14. J. Lu, J. Cao and D. Ho: Adaptive stabilization and synchronization for chaotic Lur'e systems with time-varying delay. IEEE Trans. Circuits Syst. I 55 (2008), 1347-1356.   DOI:10.1109/tcsi.2008.916462
  15. D. Ning, X. Wu, J. Lu and J. Lü: Driving-based generalized synchronization in two-layer networks via pinning control. Chaos 25 (2016), 113104.   DOI:10.1063/1.4935069
  16. N. Noroozi, M. Roopaei and M. Jahromi: Adaptive fuzzy sliding mode control scheme for uncertain systems. Commun. Nonlinear Sci. Numer. Simul. 14 (2009), 3978-3992.   DOI:10.1016/j.cnsns.2009.02.015
  17. L. Pan, W. Zhou, J. Fang and D. Li: A novel active pinning control for synchronization and anti-synchronization of new uncertain unified chaotic systems. Nonlinear Dyn. 62 (2010), 417-425.   DOI:10.1007/s11071-010-9728-0
  18. L. Pecora and T. Carroll: Synchronization in chaotic systems. Phys. Rev. Lett. 64 (1990), 821-824.   DOI:10.1103/physrevlett.64.821
  19. A. Pototsky and N. Janson: Synchronization of a large number of continuous one-dimensional stochastic elements with time-delayed mean-field coupling. Physica D 238 (2009), 175-183.   DOI:10.1016/j.physd.2008.09.010
  20. M. Pourmahmood, S. Khanmohammadi and G. Alizadeh: Synchronization of two different uncertain chaotic systems with unknown parameters using a robust adaptive sliding mode controller. Commun. Nonlinear Sci. Numer. Simul. 16 (2011), 2853-2868.   DOI:10.1016/j.cnsns.2010.09.038
  21. M. Roopaei and M. Jahromi: Synchronization of two different chaotic systems using novel adaptive fuzzy sliding mode control. Chaos 18 (2008), 033133.   DOI:10.1063/1.2980046
  22. M. Roopaei, B. Sahraei and T. Lin: Adaptive sliding mode control in a novel class of chaotic systems. Commun. Nonlinear Sci. Numer. Simul. 15 (2010), 4158-4170.   DOI:10.1016/j.cnsns.2010.02.017
  23. H. Shi, Y. Sun, L. Miao and Z. Duan: Outer synchronization of uncertain complex delayed networks with noise coupling. Nonlinear Dyn. 85 (2016), 2437-2448.   DOI:10.1007/s11071-016-2836-8
  24. H. Shi, Y. Sun and D. Zhao: Synchronization of chaotic systems with on-off periodic coupling. Phys. Scr. 88 (2013), 045003.   DOI:10.1088/0031-8949/88/04/045003
  25. H. Shi, Y. Sun and D. Zhao: Synchronization of two different chaotic systems with discontinuous coupling. Nonlinear Dyn. 75 (2014), 817-827.   DOI:10.1007/s11071-013-1106-2
  26. X. Shi and Z. Wang: The alternating between complete synchronization and hybrid synchronization of hyperchaotic Lorenz system with time delay. Nonlinear Dyn. 69 (2012),1177-1190.   DOI:10.1007/s11071-012-0339-9
  27. W. Sun, C. Huang, J. Lü and X. Li: Velocity synchronization of multi-agent systems with mismatched parameters via sampled position data. Chaos 26 (2016), 023106.   DOI:10.1063/1.4941373
  28. Y. Sun, W. Li and D. Zhao: Outer synchronization between two complex dynamical networks with discontinuous coupling. Chaos 22 (2012), 043125.   DOI:10.1063/1.4941373
  29. Y. Sun, W. Li and D. Zhao: Finite-time stochastic outer synchronization between two complex dynamical networks with different topologies. Chaos 23 (2012), 023152.   DOI:10.1063/1.4731265
  30. S. Tan, J. Lü and Z. Lin: Emerging behavioral consensus of evolutionary dynamics on complex networks. SIAM J. Control Optim. 54 (2016), 3258-3272.   DOI:10.1137/151004276
  31. S. Tan, Y. Wang and J. Lü: Analysis and control of networked game dynamics via a microscopic deterministic approach. IEEE Trans. Automat. Control 61 (2016), 4118-4124.   DOI:10.1109/tac.2016.2545106
  32. J. Wu, Z. Ma, Y. Sun and F. Liu: Finite-time synchronization of chaotic systems with noise perturbation. Kybernetika 51 (2015), 137-149.   DOI:10.14736/kyb-2015-1-0137
  33. J. Yan, M. Hung, T. Chiang and Y. Yang: Robust synchronization of chaotic systems via adaptive sliding mode control. Phys. Lett. A 356 (2006), 220-225.   DOI:10.1016/j.physleta.2006.03.047
  34. W. Yu, J. Lü, X. Yu and G. Chen: Distributed adaptive control for synchronization in directed complex network. SIAM J. Control Optim 53 (2015), 2980-3005.   DOI:10.1137/140970781
  35. H. Zhang, W. Huang, Z. Wang and T. Chai: Adaptive synchronization between two different chaotic systems with unknown parameters. Phys. Lett.A 350 (2006), 363-366.   DOI:10.1016/j.physleta.2005.10.033
  36. G. Zhang, Z. Liu and J. Zhang: Adaptive synchronization of a class of continuous chaotic systems with uncertain parameters. Phys. Lett. A 372 (2008), 447-450.   DOI:10.1016/j.physleta.2007.07.080
  37. J. Zhou, C. Juan, J. Lu and J. Lü: On applicability of auxiliary system approach to detect generalized synchronization in complex networks. IEEE Trans. Automat. Control 99 (2016), 1-6.   CrossRef