IOP PUBLISHING PHYSICA SCRIPTA Phys. Scr. 81 (2010) 045007 (13pp) doi:10.1088/0031-8949/81/04/045007 Spatiotemporal chaos in coupled logistic maps Andre Varella Guedes and Marcelo Amorim Savi Universidade Federal do Rio de Janeiro, COPPE - Department of Mechanical Engineering, PO Box 68.503, 21.941.972, Rio de Janeiro, RJ, Brazil E-mail: savi@mecanica.ufrj.br Received 9 September 2009 Accepted for publication 18 February 2010 Published 19 March 2010 Online at stacks.iop.org/PhysScr/81/045007 Abstract The objective of this work is to investigate the spatiotemporal dynamics of coupled logistic maps. These maps are prototypes of high-dimensional dynamical systems and have been used to describe the evolution and pattern formation in different systems. Here, the logistic map lattice is coupled by a power law and, therefore, each map is influenced by other maps in its neighborhood. The Kolmogorov–Sinai entropy density is employed to quantify the complexity of system behavior, permitting a general qualitative understanding of different aspects of system dynamics. Three kinds of boundary conditions are treated and the influence of initial conditions is also of concern. Non-homogeneous maps are investigated, showing interesting aspects of spatiotemporal dynamics. The idea is to analyze the spatial interaction between two qualitative different types of behavior from a grid that is split into two parts. Numerical simulations show what types of conditions present a greater tendency to develop chaotic, periodic and synchronized responses. It should be highlighted that non-homogeneous grids have situations where a chaotic pattern can emerge from two periodic responses and also situations where a periodic pattern can emerge from chaos. PACS numbers: 05.45.-a, 05.45.Jn (Some figures in this article are in colour only in the electronic version.) 1. Introduction Natural systems have nonlinear characteristics that are responsible for a great variety of possibilities. Chaos is one of these possibilities, and natural systems can adopt chaotic regimes as desirable behavior due to the intrinsic richness related to the existence of an infinite number of unstable periodic orbits. In the past, most research was dedicated to investigating the temporal evolution of low-dimensional systems. Recently, the spatiotemporal evolution of dynamical systems has increasing importance (Savi 2007, Viana et al 2005, Vasconcelos et al 2004, Lai and Grebogi 1999, Shibata 1998a, 1998b, Awrejcewicz 1991, Umberger et al 1989). The spatiotemporal characteristics of a dynamical system are important in the analysis of complex behavior. This paper presents an investigation of the spatiotemporal dynamics of coupled maps. The system is composed of a logistic map lattice connected by a communication protocol. This coupling is described by a power law that can represent either local- or global-type couplings. Therefore, each map is influenced by other maps in its neighborhood, and boundary conditions are important in defining the coupling characteristics. This map lattice represents a mathematical idealization of physical systems that are discrete in time and space. It is used to describe the evolution and pattern formation in chemical reactions, turbulence, neural networks and population dynamics. Because of that, the investigation of this system became important in nonlinear dynamics analysis (Wysham and Hastings 2008, Lloyd 1995, Holden and Zhang 1992). The literature presents numerous investigations concerning coupled logistic maps. Willeboordse (2003) argued that the key motivation is the search for universal properties and behavior that apply to all dynamical systems. Therefore, coupled maps can be understood as prototypes of high-dimensional dynamical systems. Shen et al (2008) studied the synchronization and pattern dynamics of coupled logistic maps on a type of complex network, 0031-8949/10/045007+13$30.00 Printed in the UK & the USA 1 © 2010 The Royal Swedish Academy of Sciences