Skip Navigation Links
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

Fax: +1 618 650 2555 Email:

Study on Dynamical System with Time-delay

Journal of Applied Nonlinear Dynamics 5(4) (2016) 441--456 | DOI:10.5890/JAND.2016.12.005

Amit Mondal$^{1}$; Nurul Islam$^{2}$

$^{1}$ Department of Mathematics, Jafarpur Kashinath High School, P.O.-Champahati, P.S.-Sonarpur, Dist-24 Pgs(S), Pin - 743330, West Bengal, India

$^{2}$ Department of Mathematics, Ramakrishna Mission Residential College(Autonomous), Narendrapur, Kolkata-700103, West Bengal, India

Download Full Text PDF



In this paper, stability analysis of the nonlinear time-delayed Sprott system is made by applying a small perturbation near the critical point. Next, we will discuss a scheme for delay synchronization. To achieve our claim, we will make the chaos synchronization between the coupled Sprott system with delay parameters. Here, we will discuss five distinct cases. Numerical simulation is done to verify our claim.


  1. [1]  Lorenz, E.N. (1963), Deterministic non-periodic flow, J.Atmos.Sci., 20, 130-141.
  2. [2]  Sprott, J.C. (1994), Some simple chaotic flow Phys.Rev.E, 50, R647-R650.
  3. [3]  Mondal, A., Islam, M. and Islam, N. (2015), Robust antisynchronization of chaos using sliding mode control strategy, Pramana-journal of physics, Indian Academy of Sciences, 84, 47-67.
  4. [4]  Vaidyanathan, S. (2013), Sliding mode controller design for the anti-synchronization of hyperchaotic LU systems, International Journal on Cybernatics and Informatics (IJCI), 2, 31-38.
  5. [5]  Mondal, A. and Islam, N. (2014), Generalized synchronization of nonlinear oscillators via OPCL coupling, International Journal on Cybernatics and Informatics (IJCI), 3, 21-33.
  6. [6]  Yongguang, Y. and Suachun, Z. (2003), Controlling uncertain Lu system using backstepping design, Chaos, Solitons and Fractals, 15, 897-902.
  7. [7]  Agiza, H.N. and Yassen M.T. (2001), Synchronization of Rossler and Chen chaotic dynamical systems using active control, Phys. Lett. A, 278, 191-197.
  8. [8]  Guo, R. (2008), A simple adaptive controller for chaos and hyperchaos synchronization, Phys. Lett. A, 372, 5593-5597.
  9. [9]  Yang, L.X., Chu, Y.D., Zhang, J.G., Li, X.F. and Chang, Y.X. (2009), Chaos synchronization in autonomous chaotic system via hybrid feedback control, Chaos, Solitons and Fractals , 41, 214-223.
  10. [10]  Ghosh, D., Chowdhury, A.R. and Saha, P. (2008), Multiple delay Rossler system -Bifurcation and chaos control, Chaos, Solitons and Fractals, 35, 472-485.
  11. [11]  Ghosh, D. (2009), Stabilty and projective synchronization in multiple delay Rossler system, International Journal of Nonlinear Science, 7, 207-214.
  12. [12]  Senthilkumar, D.V., Lakshmanan, M. and Kurths, J. (2006), Phase synchronization in time-delay systems, Phys. Rev. E., 74, 035205(R).
  13. [13]  Pecora, L.M. and Carroll, T.L. (1990) Synchronization in chaotic systems, Phys.Rev.Lett., 64, 821-825.
  14. [14]  Cao, L.Y. and Lai, Y.C. (1998), Anti-phase synchronization in chaotic systems, Phys. Rev. E., 58, 382-386.
  15. [15]  Zhan, M.,Wei, G.W. and Lai, C.H. (2002), Transition from intermittency to periodicity in lag synchronization in coupled Rossler oscillators, Phys. Rev. E, 65, 036202-05.
  16. [16]  Voss, H.U. (2001), Dynamic long-term anticipation of chaotic states, Phys. Rev. Lett., 87, 014102.
  17. [17]  Kocarev, L. and Parlitz, U. (1996), Generalized synchronization, predictability and equivalence of unidirectionally coupled dynamical systems, Phys. Rev. Lett., 76, 1816-1819.
  18. [18]  Kocarev, L.J., Halle, K.S., Eckert, K., Parlitz, U. and Chua, L.O. (1992), Experimental demonstration of secure communications via chaotic synchronization, Int. J. Bifur. Chaos, 2, 709-713.
  19. [19]  Lee, M.W., Larger, L. and Goedgebuer, J-P. (2003), Transmission system using chaotic delays between light waves, IEEE J Quantum Electron, 39, 931-935.
  20. [20]  Poria, S. and Tarai, A. (2007), Adaptive synchronization of two coupled chaotic neuronal systems, Rev. Bull. Calcutta Math. Soc., 15, 53-60.
  21. [21]  Roy, P.K., Hens, C., Grosu, I. and Dana, S.K. (2011), Engineering generalized synchronization in chaotic oscillators, Chaos, 21, 013106.
  22. [22]  Shahverdiev, E.M. and Shore, K.A. (2009), Impact of modulated multiple optical feedback time delays on laser diode chaos synchronization, Optics communication, 282, 3568-3572.
  23. [23]  Mackey, M.C. and Glass, L. (1977), Oscillations and chaos in a physiological control systems, Science, 197, 287-289.
  24. [24]  Mondal, A. and Islam, N. (2013), Stability and Control Analysis of the Sprott Model L, International Journal of Applied Mathematical Sciences (JAMS), 6, 63-68.
  25. [25]  Pai, B., Islam, N. and Mazumder, H.P. (2007), On the Stability and Control of Goodwin-Griffith System of Dynamical Equations Governing Tryptophan Operon, Proc. Indian National Science Academy, 73, 221-225.