Data-driven modeling fast-slow coupling delayed dynamics using noisy data and sparse enhancement

In this work, we focus on fast-slow delayed dynamical systems, which are usually described by coupled delayed differential equations with two-time scales. The purpose of this article is to propose a data-driven discovery method to identify the fast-slow delayed differential equations from noisy measurements. The proposed method blends delay dictionary-based sparse identification of nonlinear dynamics (SINDy) framework and a fixed interval sampling strategy. Specifically, we leverage the SINDy methodology to describe the fast-slow delayed differential equations by a few active terms of a large dictionary containing the delay terms and many candidate nonlinear basis functions. To identify the active terms and the corresponding coefficients, a sequential thresholded re-weighted ℓ
2
least squares algorithm is used to solve the resulting sparse regression problem to enhance the sparsity. Once a sparse model with fixed delay is selected, the proposed method further estimates the delayed time by calculating the reconstruction error from the sparse model until the one that makes the reconstruction error minimum. We verify the effectiveness of the proposed method with several classical fast-slow coupling delayed systems. The experimental results show that the proposed method can well automate the fast-slow delayed differential equations from noisy data and the delayed terms also can be detected accurately.