Nonlinear Dyn (2013) 71:279–290 DOI 10.1007/s11071-012-0660-3 ORIGINAL PAPER Complete and generalized synchronization of chaos and hyperchaos in a coupled first-order time-delayed system Tanmoy Banerjee · Debabrata Biswas · B.C. Sarkar Received: 25 September 2012 / Accepted: 24 October 2012 / Published online: 8 November 2012 © Springer Science+Business Media Dordrecht 2012 Abstract This paper explores the synchronization scenario of coupled chaotic and hyperchaotic time de- lay systems that are coupled through linear, dissipa- tive 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 synchroniza- tion 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 nu- merically to support the analytical results. Also, we implement the coupled system in an electronic cir- cuit to verify all the synchronization phenomena. It T. Banerjee ( ) · D. Biswas · B.C. Sarkar Department of Physics, The University of Burdwan, Burdwan 713104, West Bengal, India e-mail: tanbanrs@yahoo.co.in T. Banerjee e-mail: tanban.buphys@gmail.com is shown that the experimental results agree well with our analytical and numerical results. Keywords Delay dynamical system · Chaos synchronization · Hyperchaos · Time delay electronic circuit 1 Introduction For the last two decades synchronization of chaos has been an active area of research in various fields, in- cluding physics, biology, mathematics, engineering, etc. In a seminal paper Pecora and Carrol [2] have first shown that two chaotic trajectories having dif- ferent initial conditions can be synchronized. Since then researchers around the world have been actively engaged to explore different possible synchronization scenario of the chaotic systems; the following variety of synchronization schemes have been observed and identified: Complete synchronization [2], Generalized synchronization [3–5], Phase synchronization [6], Lag synchronization [7, 8], Anticipatory synchronization [9], Impulsive synchronization [10, 11], etc. Two ex- cellent review papers [12, 13] describe the state of the present status of synchronization of chaos. In the initial years, emphasis has been on the syn- chronization of low dimensional chaotic systems, i.e., chaotic systems having a single positive Lyapunov ex- ponent (LE). Later on, synchronization of higher di- mensional chaos (i.e. hyperchaos) [14], and chaotic