Nonlinear Dynamics

, Volume 71, Issue 1–2, pp 279–290 | Cite as

Complete and generalized synchronization of chaos and hyperchaos in a coupled first-order time-delayed system

  • Tanmoy Banerjee
  • Debabrata Biswas
  • B. C. Sarkar
Original Paper


This paper explores the synchronization scenario of coupled chaotic and hyperchaotic time delay systems that are coupled through linear, dissipative and unidirectional coupling. For the present study, we choose a prototype first-order nonlinear time delay system, which is recently reported in Banerjee et al. (Nonlinlinear. Dyn., doi: 10.1007/s11071-012-0490-3, 2012); the system shows well-characterized chaotic and hyperchaotic oscillations even for a small time delay, and also, experimental implementation of the system is easy. We show that, keeping all the system design parameters the same for the two systems, if the time delays associated with the two systems are equal, then complete synchronization occurs beyond a threshold coupling strength. On the contrary, above a certain coupling strength, generalized synchronization between two identical coupled systems occurs for the unequal time delays. We derive an estimate of the coupling strength and sufficient stability conditions for all the synchronization processes using Krasovskii–Lyapunov theory. We simulate the coupled system numerically to support the analytical results. Also, we implement the coupled system in an electronic circuit to verify all the synchronization phenomena. It is shown that the experimental results agree well with our analytical and numerical results.


Delay dynamical system Chaos synchronization Hyperchaos Time delay electronic circuit 



One of the authors (D.B.) thankfully acknowledges the financial support provided by the University of Burdwan.


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Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  • Tanmoy Banerjee
    • 1
  • Debabrata Biswas
    • 1
  • B. C. Sarkar
    • 1
  1. 1.Department of PhysicsThe University of BurdwanBurdwanIndia

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