A class of fractional-order discrete map with multi-stability and its digital circuit realization

Abstract

In this paper, a class of nonlinear functions and Gaussian function are modulated to construct a new high-dimensional discrete map. Based on Caputo fractional-order difference definition, the fractional form of the map is given, and its dynamical behaviors are explored. The three discrete maps with different nonlinear functions are compared and analyzed by bifurcation diagrams and Lyapunov exponents, especially the dynamical phenomena that evolve with the order. In addition, the maps have multiple rich stability, including homogeneous and heterogeneous coexistence attractors and hyperchaos coexistence attractors. The spectral entropy (SE) algorithm is used to measure the complexity of one-dimensional and two-dimensional maps. Performance tests show that the fractional-order map has more complex dynamics than the original map. Finally, the new maps were successfully implemented on the digital platform, which shows the simplicity and feasibility of the map implementation. The experimental results provide a reference for the research on the multi-stability of fractional discrete maps.