Dynamical analysis of the improper fractional-order 2D-SCLMM and its DSP implementation

In this paper, the definition of fractional calculus introduces a newly constructed discrete chaotic map. Interestingly, when the order is extended to the improper fractional order, the map still shows a chaotic state with appropriate parameters. The related dynamical behavior is analyzed by a phase diagram, bifurcation model, maximum Lyapunov exponent diagram, and approximate entropy algorithm method. The research results show that both the chaotic range of the fractional-order and the improper fractional-order are larger than the chaotic range of the integer-order, and they have richer dynamical behaviors. Besides, we found that multi-stability phenomena also exist in discrete chaotic systems, and the multiple stability of fractional order is more complicated than that of integer order, and the types of coexisting attractors are more abundant. Finally, the digital circuit and pseudo-random sequence generator of the discrete system are designed. This research guides the application and teaching of discrete fractional-order systems.