arXiv:2611.08692v1 Announce Type: new Abstract: Understanding the stability of large nonlinear dynamical systems is a fundamental problem in many fields, including biological, technological, and social networks. Traditional stability methods often rely on explicit eigenvalue calculations or the construction of Lyapunov functions, which become computationally difficult for large interconnected systems. In this work, we introduce a stability classifier that provides a computationally efficient way to determine the stability of complex systems without performing full spectral analysis. Our approach combines the Dynamical Jacobian Ensemble framework, which uses the network structure and the underlying dynamics to construct a Jacobian, with the Gershgorin disc theorem, which provides bounds on the eigenvalues of the system's Jacobian matrix. Using this combination, we derive an analytical classifier that predicts whether a dynamical system is asymptotically stable, unstable, or sensitively stable based on a small set of dynamical and network parameters. The analysis reveals that the stability of large dynamical systems is strongly influenced by highly connected hub nodes. This observation allows the stability condition to be simplified by focusing on the contribution of the hub-associated Gershgorin disc. Extensive numerical simulations on large ensembles of Jacobian matrices confirm that the proposed classifier accurately separates stable, unstable, and sensitively stable regimes across a wide range of dynamical parameters and network structures. The resulting framework provides a simple and scalable tool for predicting the stability of large dynamical networks while capturing the interplay between network topology and interaction dynamics.
Continue to arXiv Mathematics →You are being forwarded in 6s — stay on this page