On synchronization of a forced delay dynamical system via the Galerkin approximation Dibakar Ghosh, Papri Saha, A. Roy Chowdhury * High Energy Physics Division, Department of Physics, Jadavpur University, Kolkata 700 032, India Received 12 August 2005; received in revised form 24 August 2005; accepted 24 August 2005 Available online 4 October 2005 Abstract A forced scalar delay dynamical system is analyzed from the perspective of bifurcation and synchronization. In general first order differential equations do not exhibit chaos, but introduction of a delay feedback makes the system infinite dimensional and shows chaoticity. In order to study the dynamics of such a system, Galerkin projection technique is used to obtain a finite dimensional set of ordinary differential equations from the delay differential equation. We compare the results of simulation with those obtained from direct numerical simulation of the delay equation to ascertain the accuracy of the truncation process in the Galerkin approximation. We have considered two cases, one with five and the other with eight shape functions. Next we study two types of synchronization by considering coupling of the above derived equations with a forced dynamical system without delay. Our analysis shows that it is possible to have synchronization between two such systems. It has been shown that the chaotic system with delay feedback can drive the system without delay to achieve synchronization and the opposite case is also equally valid. This is confirmed by the evaluation of the conditional Lyapu- nov exponents of the systems. Ó 2005 Elsevier B.V. All rights reserved. PACS: 05.45; 45.40.f; 02.90.+p; 03.20.+i; 03.40 Keywords: Delay system; Galerkin approximation; Chaos; Synchronization 1. Introduction Delay induced instabilities as described by delay differential equations, play an important role in modelling natural phenomena. Such models are used in many different scientific disciplines like electronics, laser physics, ecology, engineering, economics and cognitive sciences [1–5]. As the dynamical systems given by DDEÕs have an infinite dimensional state space, the attractors of the solutions are also high dimensional. The numerical simulation method for solving such system of equations is mainly based on interpolation approximations 1007-5704/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2005.08.006 * Corresponding author. Tel.: +91 3324 163708; fax: +91 3324 137121. E-mail addresses: drghosh_chaos@yahoo.com (D. Ghosh), papri_saha@yahoo.com (P. Saha), asesh_r@yahoo.com (A.R. Chowdhury). Communications in Nonlinear Science and Numerical Simulation 12 (2007) 928–941 www.elsevier.com/locate/cnsns