Numerical and Analytical Study of KR-Regulated Attractor Dynamics in Nonlinear Parabolic and Fractional Evolution Equations

This manuscript presents a unified analytical and numerical study of KR-regulated nonlinear parabolic and fractional evolution equations. The proposed excitation–regulation framework decomposes nonlinear reaction dynamics into structurally independent growth and damping components governed by distinct parameters. Within a Sobolev space setting, the study establishes well-posedness, dissipativity, and existence of parameter-dependent global attractors. The attractor radius is shown to scale explicitly with the excitation–regulation ratio μ=K/R\mu = K/Rμ=K/R. A stability-preserving finite-difference discretization is developed for long-time simulations in both classical and fractional settings. Numerical investigations validate convergence, perturbation robustness, fractional memory effects, and parameter-driven attractor scaling. The results demonstrate equilibrium-type attractor behavior with near-zero stability exponent and fractal dimension, confirming regulated stability rather than chaotic dynamics. The work integrates attractor theory, fractional evolution analysis, and numerical stability methods, aligning with the scope of applied numerical analysis for nonlinear evolution equations.