Sparse identification of nonlinear dynamical systems via non-convex penalty least squares.

This paper proposes a non-convex penalty regression method to identify governing equations of nonlinear dynamical systems from noisy state measurements. The idea to connect the non-convex penalty function instead of the l1 - norm with least squares is due to the fact that the l1 - norm excessively penalizes large coefficients and may incur estimation bias. The purpose of this work is to improve the accuracy and robustness in regression tasks. A threshold non-convex penalty sparse least squares optimization algorithm is developed, wherein the threshold parameter is selected using the L-curve criterion. With two examples of nonlinear dynamical systems, we illustrate the accuracy and robustness of the non-convex penalty least squares on noisy state measurements, indicating the validity of our method in a wide range of potential applications.

An improved cell mapping method based on dimension-extension for fractional systems.

Global analysis of fractional systems is a challenging topic due to the memory property. Without the Markov assumption, the cell mapping method cannot be directly applied to investigate the global dynamics of such systems. In this paper, an improved cell mapping method based on dimension-extension is developed to study the global dynamics of fractional systems. The evolution process is calculated by introducing additional auxiliary variables. Through this treatment, the nonlocal problem is localized in a higher dimension space. Thus, the one-step mappings are successfully described by Markov chains. Global dynamics of fractional systems can be obtained through the proposed method without memory losses. Simulations of the point mapping show great accuracy and efficiency of the method. Abundant global dynamics behaviors are found in the fractional smooth and discontinuous oscillator.