Data-driven reconstruction of chaotic dynamical equations: the Hénon-Heiles type system

TL;DR

This work analyzes a family of Hénon–Heiles–type systems with potentials $V_N(r,\theta)$, examining bounded dynamics and the emergence of chaos as a function of the dihedral symmetry index $N$ and energy $E$. By combining time-series analysis, Poincaré sections, symmetry lines, and Lyapunov exponents, it maps regular and chaotic regimes and identifies periodic orbits seeded by symmetry. The study then benchmarks the Sparse Identification of Nonlinear Dynamical Systems (SINDy) algorithm on this Hamiltonian, showing robustness to chaos when time steps are sufficiently small and revealing that SINDy can recover the governing equations and even yield approximate analytical descriptions of periodic trajectories. The results highlight both the potential and limitations of data-driven equation discovery in nonlinear, reversible Hamiltonian systems and point to future directions, including quantum extensions and the use of SINDy to uncover system symmetries and invariants.

Abstract

In this study, the classical two-dimensional potential $V_N=\frac{1}{2}\,m\,ω^2\,r^2 + \frac{1}{N}\,r^N\,\sin(N\,θ)$, $N \in {\mathbb Z}^+$, is considered. At $N=1,2$, the system is superintegrable and integrable, respectively, whereas for $N>2$ it exhibits a richer chaotic dynamics. For instance, at $N=3$ it coincides with the Hénon-Heiles system. The periodic, quasi-periodic and chaotic motions are systematically characterized employing time series, Poincaré sections, symmetry lines and the largest Lyapunov exponent as a function of the energy $E$ and the parameter $N$. Concrete results for the lowest cases $N=3,4$ are presented in complete detail. This model is used as a benchmark system to estimate the accuracy of the Sparse Identification of Nonlinear Dynamical Systems (SINDy) method, a data-driven algorithm which reconstructs the underlying governing dynamical equations. We pay special attention at the transition from regular motion to chaos and how this influences the precision of the algorithm. In particular, it is shown that SINDy is a robust and stable tool possessing the ability to generate non-trivial approximate analytical expressions for periodic trajectories as well.

Figures (13)

• Figure 1: Level curves, on the plane $(x,y)$, of the potential $V_N$ (\ref{['VN']}) as a function of $N$. The corresponding values of the critical energy $E_{\rm crit}=\frac{N-2}{2\,N}$ are displayed as well. The value of the combination $m\,\omega^2=1$ is used in the calculations.
• Figure 2: Top panel shows level curves on the plane $(x,y)$ for the case $N=3$ and considering the potential $V_3$ (\ref{['pots']}) with $m\,\omega^2=1$. The corresponding critical energy $E_{\rm crit}=\frac{1}{6}\approx 0.1666$ is indicated as well as certain trajectories: (a) periodic trajectory (green line) with initial conditions $(x_0=0,\,y_0=-0.21341508,\,p_{x_0}=0.17656123,\,p_{y_0}=0)$ and energy $E=\frac{1}{4}E_{c}$, (b) quasi-periodic orbit (dark line) with initial conditions $(x_0=0,\,y_0=0.15,\,p_{x_0}=0.479322,\,p_{y_0}=0)$ and energy $E=\frac{3}{4}E_{c}$, and (c) a chaotic trajectory (red line) with initial conditions $(x_0=0,\,y_0=0.1,\,p_{x_0}=0.520545,\,p_{y_0}=0.23)$ and energy $E=\frac{99}{100}E_{c}$. Bottom panel shows level curves on the plane $(x,y)$ for the case $N=4$ with potential $V_4$ (\ref{['pots']}) and parameters $m\,\omega^2=1$. The critical energy $E_{\rm crit}=\frac{1}{4}=0.25$ is indicated as well as certain trajectories: (d) periodic trajectory (green line) with initial conditions $(x_0=0,\,y_0=0,\,p_{x_0}=0.32666587,\,p_{y_0}=0.13523834)$ and energy $E=\frac{1}{4}E_{c}$, (e) quasi-periodic orbit (dark line) with initial conditions $(x_0=0,\,y_0=-0.4,\,p_{x_0}=0.234521,\,p_{y_0}=-0.4)$ and energy $E=\frac{3}{4}E_{c}$, and (f) a chaotic trajectory (red line) with initial conditions $(x_0=0,\,y_0=-0.3,\,p_{x_0}=0.640312,\,p_{y_0}=0)$ and energy $E=\frac{99}{100}E_{c}$.
• Figure 3: Case $N=3$: time series of the dynamical variables $y(t)$ and $p_{y}(t)$ of ${\cal H}_{{}_{N=3}}$ (\ref{['HNc']}) for three representative trajectories of the periodic (green dashed), quasi-periodic (black solid) and chaotic (red dotted) motion at different values of the energy. Here $E=\frac{1}{4}E_c$ in (a), $E=\frac{3}{4}E_c$ in (b), whereas $E=\frac{99}{100}E_c$ for (c). The corresponding initial conditions $(x_{0},y_{0},p_{x_{0}},p_{y_{0}})$ are those used for the top panel in Fig. \ref{['XY_N3']}.
• Figure 4: Case $N=4$: time series $y(t)$ and $p_{y}(t)$ of ${\cal H}_{{}_{N=4}}$ (\ref{['HNc']}) for three representative trajectories of the periodic (green dashed), quasi-periodic (black solid) and chaotic (red dotted) motion at different values of the energy. Here $E=\frac{1}{4}E_c$ in (a), $E=\frac{3}{4}E_c$ in (b), whereas $E=\frac{99}{100}E_c$ for (c). The corresponding initial conditions $(x_{0},y_{0},p_{x_{0}},p_{y_{0}})$ are those used for the bottom panel of Fig. \ref{['XY_N3']}.
• Figure 5: Cases $N=3$ (top) and $N=4$ (bottom): numerical trajectories $p_y=p_y(y)$, corresponding to those displayed in Fig. \ref{['XY_N3']}, for representative orbits of the periodic (green dashed), quasi-periodic (black solid) and chaotic (red dotted) motion. Here $E=\frac{1}{4}E_c$ in (a) and (d), $E=\frac{3}{4}E_c$ in (b) and (e), while $E=\frac{99}{100}E_c$ for (c) and (f).