Lyapunov exponents of a class of piecewise continuous systems of fractional order

In this paper, we prove that a class of autonomous piecewise continuous systems of fractional order has well-defined Lyapunov exponents. For this purpose, based on some known results from differential inclusions of integer and fractional order and differential equations with discontinuous right-hand side, the associated discontinuous initial value problem is approximated with a continuous one of fractional order. Then, the Lyapunov exponents are numerically determined using, for example, the known Wolf's algorithm. Three examples of piecewise continuous chaotic systems of fractional order are simulated and analyzed: Sprott's system, Chen's system and Shimizu-Morioka's system.