Dynamics of a higher dimensional fractional-order chaotic map

Abstract

Fractional calculus has been a hot topic in nonlinear science, because it can describe physical phenomena more accurately. In recent years, the theory of fractional difference has become one of the foundations for fractional-order chaotic maps. Based on this concept, a higher dimensional fractional-order chaotic map is investigated. Different from other proposed fractional-order chaotic maps, it has multiple dimensions, and can be set as incommensurate fractional order. The bifurcation diagrams and attractors of this system with changing fractional orders are plotted by numerical simulations, and the maximum Lyapunov exponent (MLE), the permutation entropy complexity, and the probability density distribution are also calculated. Results show it has rich dynamical behaviors. Many of the periodic windows in the integer-order map become chaos when they are in the fractional-order form. And the complexity of the fractional-order form is higher than that of its integer-order counterpart and other nine typical chaotic maps. In addition, the distribution of fractional-order system is also better than that of the integer-order one. These results lay the theoretical foundation for its practical applications and relevant to the study of other fractional-order chaotic maps.