ANALYTICAL BOUNDS ON THE TOPOLOGICAL STABILITY OF DYNAMICAL SYSTEMS UNDER NOISE AND STRUCTURAL ANOMALIES

Abstract

The paper develops a generalized analytical model for determining the bounds of topological stability of dynamical systems under noise and structural anomalies. Topological stability is interpreted as the invariance of the qualitative type of dynamics– preservation of the Conley index, Morse decomposition, and the homological class of the attractor–under stochastic and discrete structural perturbations that modify the geometry and connectivity of system components. Unlike classical approaches based on local metric stability, the proposed methodology integrates the operator formalism of evolutionary flows, the spectral analysis of the semigroup generator, and topological–statistical invariants. The core of the method is the representation of stochastic and structural effects as parametric perturbations of the system, which enables tracing the continuity of the generator spectrum and deriving the analytical conditions for the preservation of topological invariants. An integral functional of topological stiffness is introduced as a quantitative measure of the mean structural change of the attractor; its behavior determines the threshold condi- tions for the loss of stability and delineates the invariance domain of the system. Analytical relations are derived that define the region of analytical stability–the set of noise intensities, correlation times, and depths and ranks of structural defects–within which the probability of changing the topological invariant does not exceed a prescribed level. Numerical testing confirms the adequacy of the model: analytical bounds are consistent with empirical results obtained through topological data analysis using persistence diagrams. It is shown that in the low-perturbation regime the empirical boundary is nearly identical to the analytical one, while deviations in the transition zone remain within the expected stochastic uncertainty. The results demonstrate the effectiveness of the proposed approach for quantitative prediction of the onset of topological instability and for assessing the stability margin in reaction–diffusion, networked, neurodynamic, and controlled technical systems. The methodology is universal, reproducible, and can be integrated into adaptive control modules of complex systems operating under uncertainty and noisy environments.