Design and analysis of discrete fractional-order chaotic map with offset-boosting behavior

Fractional calculus, as a more accurate tool for depicting the dynamics of complex systems, has been introduced into discrete chaotic maps. To further describe the offset-boosting behavior in discrete systems, a discrete fractional-order chaotic map (DFOCM) based on the Caputo difference operator is constructed. The mapping order of this fractional-order model controls the stability of the fixed point, thereby affecting the dynamic behavior of the map. The dynamics of DFOCM is studied using numerical simulation methods such as bifurcation diagrams and maximum Lyapunov exponents, revealing the presence of multistability. By comparing with integer-order map, it is found that DFOCM exhibit a larger chaotic region. Based on this, the difference between fractional order and integer order offset-boosting behavior is theoretically derived. Specifically, the offset-boosting behavior of fractional-order maps concerning mapping parameters is related to the initial state, which was further demonstrated through numerical simulations. SE complexity proves that the chaotic sequences generated by DFOCM have high unpredictability and pseudo-randomness. Finally, the proposed DFOCM is implemented on the DSP hardware platform, and the physical feasibility of numerical simulation is verified.