School of Mathematics, Hefei University of Technology, Hefei, Anhui 230601, China.
Chaos. 2023 Jan;33(1):013117. doi: 10.1063/5.0131661.
This paper is mainly dedicated to defining an adequate notion of fractional Lyapunov exponent to the Hadamard-type fractional differential system (HTFDS). First, the continuous dependence of the solution to a nonautonomous HTFDS is discussed. Then, to characterize the specific chaotic dynamics of the HTFDS, a novel fractional Lyapunov exponent well correlated with both the Mittag-Leffler characteristic function and the fractional order is well established by the aid of the results of continuous dependence and variational principle to the HTFDS. Subsequently, the upper bound of fractional Lyapunov exponents for the general HTFDS is estimated on account of its variation system. Finally, an indispensable illustration is presented to verify our main results, which also infers that different kinds of fractional systems share different Lyapunov exponents indeed.
本文主要致力于为 Hadamard 型分数阶微分系统 (HTFDS) 定义一个适当的分数李雅普诺夫指数的概念。首先，讨论了非自治 HTFDS 的解的连续依赖性。然后，为了刻画 HTFDS 的特定混沌动力学，借助于连续依赖性和变分原理的结果，通过建立与 Mittag-Leffler 特征函数和分数阶都相关的新的分数李雅普诺夫指数，很好地描述了 HTFDS 的具体混沌动力学。随后，根据其变分系统估计了一般 HTFDS 的分数李雅普诺夫指数的上界。最后，通过一个不可或缺的实例来说明我们的主要结果，这也推断出不同类型的分数系统确实具有不同的李雅普诺夫指数。
