Complexity of the Caputo Fractional Chialvo Neuron Map of Nilpotent Matrices is Robust with Respect to Fractionality

The Caputo fractional Chialvo neuron map is extended to the algebra of nilpotent matrices. It is demonstrated that the nilpotent structure of the initial conditions is preserved at every iteration, and that the resulting dynamics exhibit qualitatively new nonlinear effects (transient divergence and blow-up) that are entirely invisible to the scalar map alone. A computationally efficient iterative algorithm is used for the simulation of the fractional recurrence, reducing the computational cost by approximately two orders of magnitude ([Formula: see text]) and rendering full-resolution analysis of the parameter plane feasible. High-resolution computation of the parameter plane reveals band-like boundary structures separating distinct dynamical regimes. The complexity of these boundaries is quantified using the Wada index. Remarkably, while the location of these structures shifts monotonically across the parameter plane as the degree of fractionality varies, the complexity of the regime boundaries remains robust across the entire range of fractionality investigated. This finding demonstrates that the rich boundary structure of the Chialvo neuron model is an intrinsic property of its nonlinear dynamics, insensitive to the strength of fractional memory introduced by the Caputo operator.