Multistability and offset boosting in a fractional-order discrete memristive neuron model with application to image encryption

This paper presents a three-dimensional discrete fractional-order memristor-coupled neuron (DFOMCN) map derived from coupling a 2D neuron model through a discrete memristor, exhibiting unique offset-boosting dynamics. Unlike integer-order chaotic systems, the offset-boosting behavior in this fractional-order map achieved through parameter offset demonstrates explicit dependence on initial conditions, attributable to the memory effects of fractional-order operators. Rigorous dynamical analysis is conducted using Lyapunov exponent spectra, bifurcation diagrams, and phase portraits, revealing significantly richer hyperchaotic behavior compared to its integer-order counterpart. The system’s multistability and conditional symmetry are rigorously investigated, leading to initial-induced heterogeneous multistability and initial-boosted homogeneous multistability. Furthermore, permutation entropy (PE) complexity analysis confirms exceptional unpredictability and pseudo-randomness in generated sequences. The proposed DFOMCN map is successfully applied to a novel color image encryption scheme, demonstrating outstanding security performance with high resistance to statistical, differential, and brute-force attacks.