Data-Driven Modeling of Global Bifurcations and Chaos in a Mechanical System under Delayed and Quantized Control

TL;DR

The authors address the challenge of modeling nonlinear mechanical systems under time-delayed and discretized control by employing Spectral Submanifolds (SSMs) to construct parametric reduced-order models directly from data. They demonstrate that SSMs can capture both local and global bifurcations in a Furuta pendulum controlled with delayed PD feedback, and extend the approach to non-smooth dynamics arising from spatial discretization, reproducing a microchaotic attractor. A 4D parametric SSM captures a sequence of bifurcations (including a heteroclinic connection to an unstable 3-torus and a subcritical Hopf) near a codimension-two point, while a 6D SSM with Radial Basis Function (RBF) mapping models experimental microchaos with high fidelity in Lyapunov statistics and PDFs. The results show that data-driven SSM reduction is a robust framework for identifying, predicting, and analyzing complex delay- and discretization-induced dynamics in PID-stabilized mechanical systems, with potential applicability to a broad class of robotics and control problems where nonlinearity, delay, and non-smoothness interact.

Abstract

We illustrate how the recent theory of Spectral Submanifolds (SSM) can capture global bifurcations and complex dynamics in mechanical systems even under delay and spatial discretization. Specifically, we build a parameter-dependent SSM-reduced model that predicts global heteroclinic and local bifurcations in a Furuta pendulum under control with delay, and verify these predictions numerically. Under additional spatial discretization of the digital controller, we also obtain an SSM-reduced model that correctly reproduces a numerically and experimentally observed microchaotic attractor in the system.

Figures (26)

• Figure 1: Left: experimental rig used for controlling the Furuta pendulum. Right: schematic of the experimental setup (adopted from Vizi2024).
• Figure 2: Stability map of the Furuta pendulum in the plane of the control gains. The theoretical stability boundaries are presented for three different sampling delays with the parameters presented in Table \ref{['tab:furutaParams2']}. The experimentally observed stable and unstable points are adapted from Vizi2024. For numerical simulations, the control gains at the red cross are used, which is slightly outside the stable region, close to the double-Hopf bifurcation point.
• Figure 5: Left: Phase portrait of the parametric SSM model of the Furuta pendulum at a sampling time of $31.60\,\mathrm{ms}$. Three full-system trajectories are initialized at the colored markers (IC1: blue, IC2: brown, IC3: yellow). Right: the same three full-system trajectories shown along time and in a 3D projection of the delay-embedding space.
• Figure 6: Amplitude spectrum obtained from the fast Fourier transform of the initial transient of full model trajectory of the Furuta pendulum initialized at IC2, confirming three distinct peaks associated with the unstable 3-torus predicted by the parametric SSM-reduced model.
• Figure 7: Left: Comparison of probability density functions of reduced coordinates obtained from numerical simulations (black) and the polynomial-based SSM model (red). Right: Chaotic attractor reconstructed from numerical simulations (black) and from the polynomial-based SSM model (red).