A new two-dimensional fractional discrete rational map: chaos and complexity

In this paper, a new two-dimensional fractional-order discrete rational map with γth-Caputo fractional difference operator is introduced. The study of the presence and stability of the fixed points shows that there are four types of these points; no fixed point, a line of fixed points, one fixed point and two fixed points. In addition, in the context of the commensurate and incommensurate instances, the nonlinear dynamics of the suggested fractional-order discrete map in different cases of fixed points are investigated through several numerical techniques including Lyapunov exponents, phase attractors and bifurcation diagrams. These dynamic behaviors suggest that the fractional-order discrete rational map has both hidden and self-excited attractors, which have rarely been described in the literature. Finally, to validate the presence of chaos, a complexity analysis is carried out using approximation entropy (ApEn) and the C ₀-measure.