Modeling of nonlinear dynamical systems based on deterministic learning and structural stability

Recently, a deterministic learning (DL) theory was proposed for accurate identification of system dynamics for nonlinear dynamical systems. In this paper, we further investigate the problem of modeling or identification of the partial derivative of dynamics for dynamical systems. Firstly, based on the locally accurate identification of the unknown system dynamics via deterministic learning, the modeling of its partial derivative of dynamics along the periodic or periodic-like trajectory is obtained by using the mathematical concept of directional derivative. Then, with accurately identified system dynamics and the partial derivative of dynamics, a C-1-norm modeling approach is proposed from the perspective of structural stability, which can be used for quantitatively measuring the topological similarities between different dynamical systems. This provides more incentives for further applications in the classification of dynamical systems and patterns, as well as the prediction of bifurcation and chaos. Simulation studies are included to demonstrate the effectiveness of this modeling approach.