Fractional neutral integro-differential dynamical systems with periodic BVPs

Fractional neutral integro-differential dynamical systems with periodic Boundary Value Problems The primary objective of this study is to comprehensively investigate the outcomes concerning the existence and Ulam stability of a fractional dynamic system, speci cally one involving a neutral partial integro-differential equation with periodic boundary conditions on time scales, using the Caputo fractional nabla derivative. The study applies standard fixed point methods(Krasnoselskii fixed point theorem) to derive its results, with a focus on controllability and Ulam stability. Additionally, the practical relevance of the theoretical fi ndings is showcased through an illustrative example, which includes a graphical representation through MATLAB Software. Future work will focus on advancing numerical methods for fractional systems with delaysor nonlocal conditions, exploring new control strategies, and applying these systemsto communication networks and biomedical elds. Key areas include enhancing stabilityanalysis and leveraging machine learning for optimized control.Keywords: Neutral equations, Caputo-Nabla derivative, Fixed point, Time Scales, Fractionaldynamic equation.2020 Mathematics Subject Classification : 39A05, 37C25, 34K40, 34N05.