# American Institute of Mathematical Sciences

May  2021, 26(5): 2361-2370. doi: 10.3934/dcdsb.2020182

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

 1 Department of Mathematics, University of Mentouri Constantine 1, 25000, Algeria 2 Department of Sciences and Technology, Mathematics and their Interactions Laboratory, University of Mila, 43000, Algeria

* Corresponding author: Rabiaa Ouahabi

Received  October 2019 Revised  February 2020 Published  May 2021 Early access  June 2020

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.

Citation: Rabiaa Ouahabi, Nasr-Eddine Hamri. Design of new scheme adaptive generalized hybrid projective synchronization for two different chaotic systems with uncertain parameters. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2361-2370. doi: 10.3934/dcdsb.2020182
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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 $\alpha _{1} = 25, \alpha _{2} = 33, \alpha _{3} = 11, \alpha _{4} = 6$
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 $\beta _{1} = 36,$ $\beta _{2} = 25,$ $\beta _{3} = 3$
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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