Chaos synchronization: a review

Abstract: This article provides a self-contained comprehensive review of the phenomenon of synchronization in dynamical systems, with a particular focus on chaotic systems in both continuous-time and discrete-time contexts. Synchronization, initially observed by Christiaan Huygens in 1665, has evolved from the study of periodic signals to encompass chaotic systems, where the sensitive dependence on initial conditions poses unique challenges. This review pointed to both theoretical foundations and contributions (concepts and methods) and practical insights, reinforcing the relevance of chaos synchronization in physics, biology, engineering, social sciences, economics and communication systems. The study investigates various coupling schemes, such as unidirectional and bidirectional coupling, and presents stability criteria under different configurations. In a very concise way, some ongoing research carried out by the authors is also indicated, using Lorenz, Rossler and hyperchaotic Rossler systems, and quadratic maps as case studies with parameter values leading to chaotic behavior. Special attention is given to the stability of synchronized states and the role of multi-stability and bifurcations, and its implications to loss of synchronization. We highlight the role of Lyapunov exponents, eigenvalues, and Lyapunov functions in guaranteeing local and global stability of the synchronized state. We aim to contribute to a broader understanding of chaos synchronization and its practical applications in diverse fields of knowledge. This text shed light on the control and stability of coupled chaotic systems, offering new perspectives on the synchronization of non-identical systems and the emergence of complex synchronization dynamics.