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