On stability analysis via Lyapunov exponents calculated based on radial basis function networks

Abstract

The concept of Lyapunov exponents is a powerful tool for analysing the stability of nonlinear dynamic systems, especially when the mathematical models of the systems are available. For real world systems, such models are often unknown. Estimating Lyapunov exponents using a time series has the advantage in that no mathematical model is required. However the time-series-based methods are believed to be reliable only for estimating positive exponents. Furthermore, when nonlinear mapping is applied for deriving the neighbourhood-to-neighbourhood matrices, the loads of mathematical deduction and programming increase significantly, which makes it unfeasible to nonlinear systems with high dimensions. In contrary, the model-based methods are constructive and reliable for calculating both positive and non-positive exponents. The use of the system Jacobians is the key to the advantage of the model-based methods. In this article, a novel approach is proposed, where the system Jacobians are derived based on system approximation using the radial basis function network. The proposed method inherits the advantage of the model-based methods, yet no mathematical model is required. Two case studies are presented to demonstrate the efficacy of the proposed method. We believe that the work can contribute to the stability analysis of nonlinear systems of which the dynamics are either difficult to model due to complexities or unknown.