Evolutionary reconstruction of stability boundaries of dynamical systems in the parameter space

Abstract

The purpose of this paper is to formulate and implement a generalized approach to the evolutionary reconstruction of stability boundaries of dynamical systems in the parameter space, based on experimental observations or numerical simulation results. The problem of stability boundary recovery is formulated as an inverse optimization problem, in which observed time series are transformed into stability indicators such as variance, autocorrelation, or generalized spectral characteristics. To approximate the bifurcation surface Γ(t), a parametric model g(λ, θ) is employed, whose parameters are adaptively updated over time using recursive or filtering algorithms. The proposed method enables the reconstruction of the dynamic stability boundary without prior knowledge of the governing equations of the system. Through examples involving a linear stochastic system, the Van der Pol oscillator, and an SIS-type model, the approach demonstrates its capability to accurately identify critical parameters and transition zones even in the presence of noise. The proposed method is applicable for the analysis and monitoring of complex technical, biological, social, and economic systems in which stability can only be indirectly assessed through experimental observations.