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Synchronization of chaotic networks with time-delayed couplings: an analytic study
A Englert1, S Heiligenthal, W Kinzel
1Institute for Theoretical Physics, University of Würzburg, D-97074 Würzburg, Germany.
Summary
Nonlinear networks with time-delayed couplings can achieve complete chaos synchronization. This study derives analytic results for synchronization stability in networks with single and multiple delay times.
Area of Science:
- Nonlinear dynamics
- Complex systems
- Network synchronization
Background:
- Networks of nonlinear units with time-delayed couplings can synchronize.
- Synchronization can occur even with very large delay times, without time shifts.
Purpose of the Study:
- To derive analytic results for the stability of chaotic synchronization in networks with time-delayed couplings.
- To investigate the relationship between synchronization and network properties for single and multiple delay times.
Main Methods:
- Analysis of coupled Bernoulli maps to derive analytic results for synchronization stability.
- Application of polynomial theory for networks with multiple delay times.
- Comparison with iterated tent maps and Lang-Kobayashi equations.
Main Results:
- Chaos synchronization stability is linked to the spectral gap of the coupling matrix for single delay times.
- Analytic results for multiple delay times are obtained using polynomial theory.
- Demonstrated synchronization in networks mimicking semiconductor laser behavior.
Conclusions:
- Complete chaos synchronization is achievable in large-scale, time-delayed nonlinear networks.
- The spectral gap and polynomial theory provide tools to analyze synchronization stability.
- The findings have implications for understanding and controlling synchronized behavior in complex systems.
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