Nonlinear Dynamics

, Volume 86, Issue 3, pp 2003–2015 | Cite as

Synchronization of chaotic systems using particle swarm optimization and time-delay estimation

Original Paper


This paper presents an optimal chaos synchronization technique using particle swarm optimization (PSO) and time-delay estimation (TDE). Time-delay control (TDC), which uses the TDE to estimate and cancel the nonlinear terms in dynamics, is simple and robust, and has been recognized as a promising technique for chaos synchronization. The synchronization technique with TDC consists of three terms: a slave dynamics cancelation term using the TDE, a desired chaotic dynamics injection term, and a synchronization error dynamics describing term. In this paper, we propose the PSO algorithm for the gain of the last term, the slope of sliding surface of the synchronization error dynamics. An objective function is constructed using a sum of the absolute value of errors, and a set of particles is used to seek the optimal solution. We show that too small value of a gain results in weak robustness, and too large value of a gain worsens noise sensitivity. Our data suggest that an optimal gain set for minimizing synchronization errors of the TDC can be obtained through the proposed technique. The proposed technique can automatically find the optimized gain for the TDC in noisy environments although the estimation time-delay is altered significantly. The effectiveness of the proposed technique is verified through chaos synchronization simulations for secure communications.


Time-delay estimation Chaos synchronization Particle swarm optimization Secure communications Time-delay control 


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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Korea Electrotechnology Research Institute (KERI)ChangwonSouth Korea
  2. 2.Korea Institute of Robot and Convergence (KIRO)PohangSouth Korea

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