Int J Theor Phys (2008) 47: 1005–1015 DOI 10.1007/s10773-007-9527-7 Global Synchronization & Anti-Synchronization in N-Coupled Map Lattices M.A. Jafarizadeh · S. Behnia · E. Faizi · S. Ahadpour Received: 1 June 2007 / Accepted: 15 August 2007 / Published online: 5 September 2007 © Springer Science+Business Media, LLC 2007 Abstract By considering a symmetric N-dimensional map which possesses invariant mea- sure in its diagonal and anti-diagonal invariant sub-manifolds, we have been able to propose an N-coupled map which possesses invariant measure in synchronized or anti-synchronized states. Then chaotic synchronization and anti-synchronization are investigated in the intro- duced model. We have calculated Kolmogrov–Sinai entropy and Lyapunov exponent as an- other tool to study the stability of N-coupled map in synchronized and anti-synchronization states. Keywords Synchronization · Anti-synchronization · Invariant measure · Kolmogrov–Sinai entropy · Lyapunov exponent 1 Introduction How to simulate of the natural phenomena has been a major field of research in nowadays world meanwhile coupled map lattices has been a favored paradigm for studying fundamen- tal questions in spatially extended dynamical systems [1–5]. Synchronization of two identi- cal chaotic dynamical systems has received much attention in the mathematics and physics literature in the last few years [6]. In this field, the key concept of complete synchroniza- tion refers to a state where the trajectories of dynamical systems approach each other [7–9]. By consideration a symmetric N-dimensional map possessing invariant measure in its diago- nal and anti-diagonal invariant sub-manifolds, we have been able to introduce the N-coupled M.A. Jafarizadeh · E. Faizi Department of Theoretical Physics and Astrophysics, Tabriz University, Tabriz, Iran S. Behnia ( ) Department of Physics, IAU, Ourmia, Iran e-mail: s.behnia@iaurmia.ac.ir S. Ahadpour Department of Physics, Mohaghegh Ardabili University, Ardabil, Iran