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Synchronization in networks of chaotic systems with time-delay coupling
Toshiki Oguchi1, Henk Nijmeijer, Takashi Yamamoto
1Department of Mechanical Engineering, Graduate School of Science and Engineering, Tokyo Metropolitan University, 1-1, Minami-Osawa, Hachioji-shi, Tokyo 192-0397, Japan.
This study explores synchronizing nonlinear systems with time delays. We establish conditions for synchronization using network structure and Lyapunov-Krasovskii functionals, demonstrating convergence for coupled systems.
Area of Science:
- Control Theory
- Nonlinear Dynamics
- Network Systems
Background:
- Synchronization of coupled nonlinear systems is crucial in various fields.
- Time delays in coupled systems introduce significant challenges to achieving synchronization.
- Understanding network structures is key to analyzing system dynamics.
Purpose of the Study:
- To investigate the synchronization of N identical nonlinear systems with unidirectional or bidirectional coupling and time delay.
- To establish conditions for the convergence of coupled strictly semi-passive systems.
- To derive sufficient conditions for synchronization using linear matrix inequalities.
Main Methods:
- Application of the small-gain theorem to analyze system convergence.
- Analysis of network structures for synchronization error dynamics.
- Utilizing the Lyapunov-Krasovskii functional approach for deriving synchronization conditions.
- Formulation of conditions in terms of linear matrix inequalities.
Main Results:
- Demonstrated convergence of coupled strictly semi-passive systems to a bounded region.
- Derived a necessary condition for synchronization based on network structure.
- Established sufficient conditions for synchronization using linear matrix inequalities.
- Illustrated results on networks of Lorentz systems with coupling delay.
Conclusions:
- The study provides a comprehensive framework for analyzing and achieving synchronization in complex coupled systems with time delays.
- The derived conditions offer practical guidelines for designing and controlling synchronized networks.
- The Lyapunov-Krasovskii functional approach combined with network analysis proves effective for synchronization problems.
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