On the fractional Lyapunov exponent for Hadamard-type fractional differential system

Abstract

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.