Sparse Recovery and Dictionary Learning to Identify the Nonlinear Dynamical Systems: One Step Toward Finding Bifurcation Points in Real Systems

Abstract

Modeling real dynamical systems is an important challenge in many areas of science. Extracting governing equations of systems from their time-series is a possible solution for such a challenge. In this paper, we use the sparse recovery and dictionary learning to extract governing equations of a system with parametric basis functions. In this algorithm, the assumption of sparsity in the functions of dynamical equations is used. The proposed algorithm is applied to different types of discrete and continuous nonlinear dynamical systems to show the generalization ability of this method. On the other hand, transition from one dynamical regime to another is an important concept in studying real world complex systems like biological and climate systems. Lyapunov exponent is an early warning index. It can predict bifurcation points in dynamical systems. Computation of Lyapunov exponent is a major challenge in its application in real systems, since it needs long time data to be accurate. In this paper, we use the predicted governing equation to generate long time-series, which is needed for Lyapunov exponent calculation. So the proposed method can help us to predict bifurcation points by accurate calculation of Lyapunov exponents.

Investigation of Early Warning Indexes in a Three-Dimensional Chaotic System with Zero Eigenvalues

“…The time-series of the three state variables and two-dimensional projections in the phase-spaces of the proposed chaotic attractor given in system (1) are presented in Figure 1. It can be seen that the chaotic attractor is symmetric around the line y = 0. model, as shown in [29]. Recently, the Lyapunov exponent was proposed as an indicator of tipping points [30].…”

“…There are two types of features that can predict bifurcation points: Metric-based and model-based indicators [28]. Metric-based indicators use various features of time series to predict tipping points [25], while model-based indicators estimate some models from the time series and then predict tipping points using the model, as shown 2 of 8 in [29]. Recently, the Lyapunov exponent was proposed as an indicator of tipping points [30].…”

Predicting Tipping Points in Chaotic Maps with Period-Doubling Bifurcations

“…So, a new version of the well-known indicators was proposed to solve those issues [56]. In [57,58], the Lyapunov exponent was studied as an indicator of bifurcation points. However, some points in the calculation of Lyapunov exponents should be considered [59].…”

Entropy Analysis and Image Encryption Application Based on a New Chaotic System Crossing a Cylinder

“…Many methods have been proposed for data encryption [ 44 , 45 , 46 ]. Chaotic systems have many applications in various areas, such as biology and communication [ 47 , 48 , 49 ]. In [ 50 ], chaotic dynamics were investigated in the cryptocurrency market.…”