This paper devotes to the adaptive globally synchronization within predefined-time of two time-delayed fractional-order chaotic systems. Firstly, through fractional calculus, two novel different fractional-order systems with time-delay are proposed, whose convergence is guaranteed and phase trajectory is given. Secondly, by exploiting the non-negative Lyapunov function and inequality theorem, a novel global predefined-time stability theorem is proposed, which can ensure the settling time tunable. And the upper bound of the settling time estimation is more accurate compared with the classical results. With the help of novel predefined-time stability theorem, two active controllers are designed, namely the fixed-time synchronization controller and predefined-time synchronization controller, to achieve the fixed-time synchronization and the predefined-time synchronization of two different time-delayed fractional-order chaotic systems respectively. Finally, several numerical simulations are presented in order to show the effectiveness of the proposed methods.

中文翻译：

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具有时滞的两个不同分数阶混沌系统的自适应预定义时间同步

本文致力于两个时滞分数阶混沌系统在预定时间内的自适应全局同步。首先，通过分数演算，提出了两个新颖的具有时滞的不同分数阶系统，它们的收敛性得到保证，并给出了相位轨迹。其次，利用非负李雅普诺夫函数和不等式定理，提出了一种新的全局预定义时间稳定性定理，可以保证稳定时间的可调性。与经典结果相比，建立时间估计的上限更加准确。借助新颖的预定义时间稳定性定理，设计了两个有源控制器，即固定时间同步控制器和预定义时间同步控制器，分别实现两个不同时滞分数阶混沌系统的固定时间同步和预定义时间同步。最后，提出了几种数值模拟，以证明所提出方法的有效性。