Fractional-Order Discrete Memristive Chialvo Neuron Map: Chaos Boundary Control and Reservoir Computing Application

A three-dimensional fractional-order discrete memristive Chialvo neuron map is proposed and investigated using the Grunwald–Letnikov difference scheme. The system couples the classical Chialvo neuron with a flux-controlled memristor under incommensurate fractional orders. Through twin-trajectory Lyapunov exponent computation, bifurcation analysis, and the 0–1 test for chaos, it is shown that the fractional order and memristor coupling jointly shift the chaos boundary in parameter space. Reducing the fractional order below unity progressively stabilizes the dynamics, requiring a larger recovery-decay parameter to sustain chaos. This shift provides a continuous two-parameter control mechanism for positioning the system at the edge of chaos. The edge-of-chaos property is exploited by deploying the map as the nonlinear node of a time-multiplexed single-node reservoir computer. At a matched operating point, the fractional-order reservoir gives a small improvement over its integer-order counterpart in linear memory capacity, while on the NARMA-10 benchmark, the difference between the two lies within the spread produced by different input masks. The incommensurate orders give access to a family of memory–nonlinearity operating points in the information processing capacity spectrum.