Memory-Dependent Dynamics in a Caputo-Katugampola Fractional Neutral Delayed Neural Network: Fixed-Point Analysis, Ulam-Hyers Stability, and Computational Simulation

Abstract

We study a two-dimensional neutral delayed neural dynamical system governed by the Caputo-Katugampola derivative. Besides the fractional order $\alpha\in(0,1)$, the deformation parameter $\rho>0$ reshapes the internal clock through $q_\rho(t)=t^\rho/\rho$, which separates the strength of hereditary memory from the manner in which time itself is deformed. The present analysis extends the earlier, short-horizon sup-norm contraction argument in several directions. We first transform the model into an equivalent weakly singular Volterra equation and make explicit the exact clock transformation to a Caputo problem in $q_\rho$. Existence is obtained under continuity and boundedness assumptions through a Krasnosel'skii splitting, whereas uniqueness on any prescribed finite interval is proved in an exponentially weighted Bielecki norm whenever the neutral gain is smaller than one, thereby removing the restrictive growth of the original contraction constant as the final time increases. Ulam-Hyers robustness, continuous dependence on the history, and an explicit sensitivity estimate with respect to the discrete delays are then established. For computation, a product-integration method is derived directly in the transformed clock, together with consistency and convergence estimates. The method is validated against a scalar problem with an exact Mittag-Leffler solution and against an independent high-resolution fractional Adams predictor-corrector calculation. Parameter sweeps over $\alpha$, $\rho$, the delays, the neutral coefficients, and the connection gains quantify their distinct effects on the transient response. \myedit{These} results clarify the mathematical and computational differences among the Caputo, Caputo-Fabrizio, and Caputo-Katugampola neural models and offer a reproducible framework for delayed systems with deformed power-law memory.