Data-driven ODE modeling of the high-frequency complex dynamics via a low-frequency dynamics model

Abstract

In our previous paper [N. Tsutsumi, K. Nakai and Y. Saiki, Chaos 32, 091101 (2022)], we proposed a method for constructing a system of differential equations of chaotic behavior from only observable deterministic time series, which we call the radial function-based regression (RfR) method. However, when the targeted variable's behavior is rather complex, the direct application of the RfR method does not function well. In this study, we propose a novel method of modeling such dynamics, including the high-frequency intermittent behavior of a fluid flow, by considering another variable (base variable) showing relatively simple, less intermittent behavior. We construct an autonomous joint model composed of two parts: the first is an autonomous system of a base variable, and the other concerns the targeted variable being affected by a term involving the base variable to demonstrate complex dynamics. The constructed joint model succeeded in not only inferring a short trajectory but also reconstructing chaotic sets and statistical properties obtained from a long trajectory such as the density distributions of the actual dynamics.

Figures (6)

• Figure 1: Outline of the joint RfR method for constructing a joint model composed of the base and fiber models. The proposed method constructs a system of ODEs that infer the time series of two variables $x(t)$ and $y(t)$ from only their observable time series data, where $x(t)=g_1(\bm{p})$ and $y(t)=g_2(\bm{p})$ for some functions $g_1$ and $g_2$, and ${\bm{p}}$ is determined by some unknown dynamical system $\frac{d\bm{p}}{dt}=f(\bm{p}).$ Using the time series of $x(t)$ that shows relatively simple dynamics, we construct a "base model" using a system of ODEs of a variable ${\bm X}$ composed of $x$ and its time delay variables employing the RfR method tsutsumi22tsutsumi23. The relatively complex behavior of the $y$ variable is described as fiber dynamics using a system of skew-product-type ODEs of a variable ${\bm Y}$ composed of $y$, its time delay variables, and the variable ${\bm X}$ of the "base model." The constructed model for the fiber dynamics is called the "fiber model". Here the first component of the variables ${\bm X}$ and ${\bm Y}$ are $X_1$ and $Y_1$, respectively.
• Figure 2: Short time series for the Rössler dynamics. The estimated time series of the targeted variable $z$ ($Y_1$ of the fiber model) (red) is shown together with that of the actual time series (blue). The time series including noise used for the estimation is also plotted (yellow). We added Gaussian noise, with a standard deviation of 1% of the raw data, to the time series of the $z$ variable.
• Figure 3: Projection of a chaotic set for the Rössler dynamics. The projection of a long trajectory generated from the original system onto the $(x,z)$ plane (actual) is shown in the left panel and that of the model (model) in the right panel. The two trajectories exhibit a similar shape.
• Figure 4: Short time series of two different variables for the macroscopic behavior of the fluid dynamics. The relatively moderate estimated time series of the base variable $E_1$ ($X_1$ of the base model) (red) is shown in the upper panel. The relatively complex estimated time series of the targeted variable $E_{10}$ ($Y_1$ of the fiber model) (red) is shown in the lower panel. The actual trajectories are plotted together with the estimations, which are shown by the blue lines.
• Figure 5: Density distribution of a targeted variable $Y_1$ for the macroscopic behavior of the fluid dynamics. The density distribution created from a long model trajectory ($T=50,000$) of $Y_1$ (estimation of $E_{10}$) (red) is shown together with the actual trajectory (blue). The distributions exhibit similar shapes.