Geometric trajectory stabilization and parabolic modal rate matching in impulsive delayed Cohen-Grossberg neural networks

Abstract

This study investigated how exponential stability in impulsive delayed Cohen-Grossberg neural networks is reflected in the geometry of the error trajectory and in an auxiliary reaction-diffusion description. The analysis distinguished rigorous consequences of exponential stability from regular-point and finite-horizon geometric diagnostics. It was shown that exponential convergence of the error norm does not, in general, imply decay of the classical curvature. Instead, a speed-weighted curvature, representing the normal component of the error acceleration, satisfies an exponential upper bound under the stated higher-order regularity assumptions. An auxiliary reaction-diffusion model with multiplicative contractive impulses was then analysed by semigroup and Duhamel estimates, and a periodic Fourier-mode formulation was used to characterize transient rate matching between modal and forcing decay rates. Spectral graph estimates further provided a rigorous exponential envelope for the network disagreement energy. Numerical experiments supported these distinctions: The weighted-curvature diagnostic exhibited a fitted decay rate of 1.0478 with R2 = 0.9914, while 456 of 500 Monte Carlo realizations satisfied the convergence criterion. The near-matching modal response had a peak ratio of 1.2553 relative to the selected detuned case, and the topology experiment detected statistically significant differences in both classical-curvature behavior and finite-horizon accumulated curvature. Overall, the results showed that norm stability rigorously controls weighted geometric and graph-energy quantities, whereas classical curvature and topology-dependent geometric effects should be interpreted as empirical finite-horizon diagnostics.