Compressive-sensing model reconstruction of nonlinear systems with multiple attractors

Abstract

In this study, facing the challenges on model reconstruction for multi-attractor nonlinear systems, the data generation and sparse regression processes in sparse identification method are generalized, referring to compressive sensing, to obtain the accurate description under the least volume of test data set. In the data generation process, we introduce compressive sensing process by an arbitrary initial condition and arbitrary perturbations applied on time domain to transcend the local attractors. In the sparse regression process, the optimum sparse parameter is obtained by bi-optimization criterion according to accuracy and interpretability. Then, accuracy criterion is proposed, and when it discovers the unperceived dynamic behaviors, more dynamic data could be recorded and added into the modeling reconstruction data set by perturbations. It requires less dynamic signals by dynamical compressive sensing with perturbations compared to the previous method with uniform point fetching on the state space. Several numerical cases and two experiments of nonlinear systems with different kinds of multi-attractors are proposed to illustrate the effectiveness of the reconstruction method. In experiment, for the dynamic systems with multi-steady states phenomenon, the most obvious problem is the calibration of the equilibrium at the symmetrical configuration, which cannot be obtained for local dynamic behaviors due to its instability. In applications, this generalized modeling reconstruction method can continuously and compressively sense the dynamic behaviors and the stability of multiple attractors to figure out the accurate governing equations under a small quantity of data. In summary, different from the previous sparse regression algorithm for nonlinear systems with multiple attractors under huge amount of data set assembled offline filling a large enough state space, the proposed model reconstruction process can intelligently sense the dynamic behaviors and give the accurate prediction.