Physics Letters A 374 (2010) 2636–2639 Contents lists available at ScienceDirect Physics Letters A www.elsevier.com/locate/pla Delay-coupled discrete maps: Synchronization, bistability, and quasiperiodicity Manish Dev Shrimali a,∗ , Rangoli Sharan b,1 , Awadhesh Prasad c , Ram Ramaswamy d,2 a The LNM Institute of Information Technology, Jaipur 302 031, India b Department of Electrical Engineering, Indian Institute of Technology, Kharagpur 721 302, India c Department of Physics and Astrophysics, Delhi University, Delhi 110 007, India d Graduate School of Arts and Sciences, University of Tokyo, Komaba, Meguro-ku, Tokyo 153 8902, Japan article info abstract Article history: Received 10 February 2010 Received in revised form 8 April 2010 Accepted 19 April 2010 Available online 23 April 2010 Communicated by A.R. Bishop Keywords: Coupled maps Delay coupling Synchronization The synchronization transition is studied in delay-coupled logistic maps. For low coupling, in-phase and out-of-phase synchronous dynamics coexist, and with increasing coupling there is a regime of quasiperi- odicity before eventual attraction to a fixed point at a critical value of coupling that depends on the nonlinearity. The presence of a region of asynchrony separating two synchronized regimes—termed anomalous behaviour—has been observed earlier in continuous systems and is shown here to occur in delay mappings as well. There are regions of in-phase, anti-phase, and out-of-phase dynamics of periodic as well as chaotic attractors.  2010 Elsevier B.V. All rights reserved. 1. Introduction Amplitude death [1] and synchronization [2] are among the most extensively studied of the various dynamical phenomena that can arise when two nonlinear systems are coupled. In the former case, two oscillatory systems, when coupled drive each other to fixed points, resulting in a loss of oscillation (and therefore am- plitude death). In the latter case, the two systems continue to oscillate with unique responses to one another [2]. When dealing with coupled systems, most studies have taken the interaction to be instantaneous. It has often been pointed out that in order to properly treat many physical or biological systems—in which the signals that mediate the interaction have finite transmission time—the interaction could be time-delayed. Recent studies have probed the manner in which systems synchro- nize when there is time-delay in the coupling [3–5], and have also examined the nature of amplitude death [4–6] and synchronization [3,7,8] when there is delay. In this Letter, we examine the dynamics of discrete mappings [9] coupled with delays. Earlier studies of either synchronization or amplitude death which have largely focused on time-continuous dynamical systems, namely flows. An advantage in studying map- pings is that some aspects of the analysis become simpler, particu- * Corresponding author. Tel.: +91 141 2689011; fax: +91 141 2689014. E-mail address: m.shrimali@gmail.com (M.D. Shrimali). 1 Present address: Control and Dynamical Systems, California Institute of Technol- ogy, Pasadena, CA 91125, USA. 2 Permanent address: School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110 067, India. larly with reference to multistability [10]. Although delay increases the dimensionality of the problem, unlike the case of flows, the system remains finite-dimensional in discrete delay mappings. We find that the dynamics can be periodic, quasiperiodic or chaotic as the coupling strength is varied, and above a critical coupling (which depends on the nonlinearity parameter) the dynamics goes to a fixed point attractor: this, effectively, is the analogue of am- plitude death in flows. In the transition from periodic motion to a fixed point (which can also be considered a synchronized state), there is an intervening asynchronous regime where the motion is quasiperiodic; this behaviour has been termed anomalous in the sense that the transition is not uniformly in the synchronized regime [11,7]. Below we describe the model system of delay coupled maps studied here. Our results on different synchronization transitions are presented in Section 3, and this is followed by a summary and conclusion in Section 4. 2. The model system We consider bidirectionally coupled maps, with variables de- noted by x and y respectively, x n+1 = (1 − β) f (x n ) + β g ( y n ), y n+1 = (1 − β) f ( y n ) + β g (x n+1 ). (1) In the present work we take f (x) to be the logistic function, αx(1 − x). The coupling is asymmetric in the delay, and the cou- pling function g(.) is taken to be linear. Such maps (in the absence of delay) have been studied extensively in the past [12], especially 0375-9601/$ – see front matter  2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2010.04.048