April 16, 2015 | Louis M. Pecora; Thomas L. Carroll
This article reviews the history and early work on synchronization of chaotic systems, focusing on the discovery and development of the concept. It begins with the authors' own discovery of the phenomenon, but traces the history back to the earliest known papers. Chaotic systems are known to resist synchronization due to their positive Lyapunov exponents, but synchronization is possible when two systems exchange information in the right way. The concept of synchronization evolved from a purely mathematical view to a more geometric one using synchronization manifolds. The authors show how building synchronizing systems leads to more complex systems with chaotic components that can be tuned to output various chaotic signals. The article also discusses synchronization in networks of oscillators, which remains an active area of research.
The authors describe their initial experiments with chaotic systems, including a 2D logistic map system and the Lorenz and Rössler systems. They show how synchronization can be achieved by coupling systems in specific ways, and how this can be extended to larger networks. They also discuss the stability of synchronized systems, using conditional Lyapunov exponents to analyze the behavior of different systems. The authors also explore synchronization in networks of oscillators, showing how synchronization can be achieved in different configurations and how the stability of the synchronized state depends on the coupling strength and the system's dynamics.
The article also discusses the challenges of synchronizing chaotic systems in the presence of noise, and how techniques for overcoming this sensitivity may exist in systems of coupled neurons. The authors also explore the concept of generalized synchronization, where the response is a different system from the drive, but still synchronizes with it. They show how this can be achieved using an auxiliary system method, and how this has been used in various studies.
The article concludes with a discussion of the master stability function, a tool that allows for the analysis of synchronization in arbitrary networks of oscillators. The function separates the dynamics of the system from the network structure, allowing for the stability analysis of any network configuration with the same dynamics. The authors also discuss the importance of synchronization in networks, and how it has been used in various applications, including communication and signal processing. The article highlights the ongoing research in this area and the potential for further developments in the understanding and application of synchronization in chaotic systems.This article reviews the history and early work on synchronization of chaotic systems, focusing on the discovery and development of the concept. It begins with the authors' own discovery of the phenomenon, but traces the history back to the earliest known papers. Chaotic systems are known to resist synchronization due to their positive Lyapunov exponents, but synchronization is possible when two systems exchange information in the right way. The concept of synchronization evolved from a purely mathematical view to a more geometric one using synchronization manifolds. The authors show how building synchronizing systems leads to more complex systems with chaotic components that can be tuned to output various chaotic signals. The article also discusses synchronization in networks of oscillators, which remains an active area of research.
The authors describe their initial experiments with chaotic systems, including a 2D logistic map system and the Lorenz and Rössler systems. They show how synchronization can be achieved by coupling systems in specific ways, and how this can be extended to larger networks. They also discuss the stability of synchronized systems, using conditional Lyapunov exponents to analyze the behavior of different systems. The authors also explore synchronization in networks of oscillators, showing how synchronization can be achieved in different configurations and how the stability of the synchronized state depends on the coupling strength and the system's dynamics.
The article also discusses the challenges of synchronizing chaotic systems in the presence of noise, and how techniques for overcoming this sensitivity may exist in systems of coupled neurons. The authors also explore the concept of generalized synchronization, where the response is a different system from the drive, but still synchronizes with it. They show how this can be achieved using an auxiliary system method, and how this has been used in various studies.
The article concludes with a discussion of the master stability function, a tool that allows for the analysis of synchronization in arbitrary networks of oscillators. The function separates the dynamics of the system from the network structure, allowing for the stability analysis of any network configuration with the same dynamics. The authors also discuss the importance of synchronization in networks, and how it has been used in various applications, including communication and signal processing. The article highlights the ongoing research in this area and the potential for further developments in the understanding and application of synchronization in chaotic systems.