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Design of new scheme adaptive generalized hybrid projective synchronization for two different chaotic systems with uncertain parameters

  • * Corresponding author: Rabiaa Ouahabi

    * Corresponding author: Rabiaa Ouahabi 
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  • This paper proposes a new scheme generalized hybrid projective synchronization for two different chaotic systems using adaptive control, where the master and slave systems do not necessarily have the same number of uncertain parameters. In this method the master system is synchronized by the sum of hybrid state variables for the slave system. Based on Lyapunov stability theory, an adaptive controller for the synchronization of two different chaotic systems is proposed, This method is also applicable if the master and slave systems are identical. As example the generalized hybrid projective synchronization between Vaidyanathan and Zeraoulia chaotic systems are discussed. Numerical simulation are provided to demonstrate the effectiveness of the proposed method.

    Mathematics Subject Classification: Primary: 34H10, 37C75; Secondary: 93D21, 93D05.


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  • Figure 1.  Estimated unknown parameters $ \overset{\sim }{\alpha }_{1}\left( t\right) , $ $ \overset{\sim }{\alpha }_{2}\left( t\right) , $ $ \overset{\sim }{\alpha }_{3}\left( t\right) $ and $ \overset{\sim }{\alpha }_{4}\left( t\right) $ of the master Vaidyanathan system (19), we observe that the estimation values of unknown parameters converge to their real values $ \alpha _{1} = 25, \alpha _{2} = 33, \alpha _{3} = 11, \alpha _{4} = 6 $

    Figure 2.  Estimated unknown parameters $ \overset{\sim }{\beta } _{1}\left( t\right) , $ $ \overset{\sim }{\beta }_{2}\left( t\right) , $ $ \overset{\sim }{\beta }_{3}\left( t\right) $ of the slave Zeraoulia system (20), we observe that the estimation values of unknown parameters converge to their real values $ \beta _{1} = 36, $ $ \beta _{2} = 25, $ $ \beta _{3} = 3 $

    Figure 3.  Synchronization errors $ e_1, e_2, e_3 $ between Vaidynathan and Zeraoulia systems (19) and (20), we observe that the errors converge to zero when the time increases

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