A Novel Variable-Order Fractional Chaotic Map and Its Dynamics

In recent years, fractional-order chaotic maps have been paid more attention in publications because of the memory effect. This paper presents a novel variable-order fractional sine map (VFSM) based on the discrete fractional calculus. Specially, the order is defined as an iterative function that incorporates the current state of the system. By analyzing phase diagrams, time sequences, bifurcations, Lyapunov exponents and fuzzy entropy complexity, the dynamics of the proposed map are investigated comparing with the constant-order fractional sine map. The results reveal that the variable order has a good effect on improving the chaotic performance, and it enlarges the range of available parameter values as well as reduces non-chaotic windows. Multiple coexisting attractors also enrich the dynamics of VFSM and prove its sensitivity to initial values. Moreover, the sequence generated by the proposed map passes the statistical test for pseudorandom number and shows strong robustness to parameter estimation, which proves the potential applications in the field of information security.