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Asymptotic and finite-time cluster synchronization of neural networks via two different controllers

  • * Corresponding author: Fengli Ren

    * Corresponding author: Fengli Ren 
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  • In this paper, by using a pinning impulse controller and a hybrid controller respectively, the research difficulties of asymptotic synchronization and finite time cluster synchronization of time-varying delayed neural networks are studied. On the ground of Lyapunov stability theorem and Lyapunov-Razumikhin method, a novel sufficient criterion on asymptotic cluster synchronization of time-varying delayed neural networks is obtained. Utilizing Finite time stability theorem and hybrid control technology, a sufficient criterion on finite-time cluster synchronization is also obtained. In order to deal with time-varying delay and save control cost, pinning pulse control is introduced to promote the realization of asymptotic cluster synchronization. Following the idea of pinning control scheme, we design a progressive hybrid control to promote the realization of finite time cluster synchronization. Finally, an example is given to illustrate the theoretical results.

    Mathematics Subject Classification: Primary: 93C10, 93C27; Secondary: 05C82.


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  • Figure 1.  The error dynamic trajectory when $ i = 1, 2 $ without controller

    Figure 2.  The error dynamic trajectory when $ i = 3, 4, 5 $ without controller

    Figure 3.  The error dynamic trajectory when $ i = 1, 2 $ with the pinning impulsive controllers (6)

    Figure 4.  The error dynamic trajectory when $ i = 3, 4, 5 $ with the pinning impulsive controllers (6)

    Figure 5.  The error dynamic trajectory $ i = 1, 2 $ with the hybrid controllers (23)

    Figure 6.  The error dynamic trajectory when $ i = 3, 4, 5 $ with the hybrid controllers (23)

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