Comparative symmetry analysis and qualitative properties of impulsive multi-delay systems across a family of power Caputo fractional kernels

In this study, we perform a comparative symmetry analysis and investigate the qualitative properties of a nonlinear impulsive fractional differential system with multiple delays and nonlocal boundary conditions. By utilizing the generalized power Caputo fractional derivative, we present a unified theoretical framework that encompasses several operators—including the Atangana-Baleanu, Caputo-Fabrizio, and weighted Hattaf derivatives—as special cases. This generality guarantees that our findings are relevant to a range of fractional kernels, emphasizing the inherent symmetry characteristics of these operators. We establish adequate criteria for the existence and uniqueness of solutions through fixed-point theory. We also show that the system is Ulam-Hyers stable, an important property for maintaining its strength in the face of change. A convergent numerical scheme confirms the theoretical results, and a sensitivity analysis demonstrates the effect of kernel symmetry on stability margins. The networked control system application shows how useful the framework is in real life. The results show that the framework can include complex genetic phenomena and spontaneous interactions that are often ignored in traditional models.