Dynamic analysis of a new two-dimensional map in three forms: integer-order, fractional-order and improper fractional-order

In this paper, a new 2-dimensional chaotic map with a simple algebraic form is proposed. And the numerical solution of the corresponding fractional-order map is derived. It is novel that the new map still exhibits chaotic behaviors when the new map is expanded to fractional-order and improper fractional-order. The dynamical characteristics of these three forms are detected through the bifurcation diagram, the maximum Lyapunov exponent spectrum, Kolmogorov entropy and attractor portraits. More interestingly, the new map has multiple coexisting attractors, but the multistability of the fractional-order and improper fractional-order is more complicated than the integer-order form. In addition, the Permutation entropy (PE) complexity algorithm and 2-dimensional maximum Lyapunov exponent diagram are used to explore the dynamic changes of three forms when the amplitude changes simultaneously. The analysis shows that under the appropriate order, the chaotic range of the fractional-order and improper fractional-order is larger than that of the integer order. This research provides guidance on the application and teaching of discrete fractional-order systems