Model reduction of nonlinear time-delay systems via ODE approximation and spectral submanifolds

TL;DR

This work addresses the challenge of analyzing nonlinear time-delay systems, which are inherently infinite-dimensional, by first converting DDEs into high-dimensional ODEs and then applying spectral submanifold (SSM) theory to derive two-dimensional reduced-order models. The resulting SSM-based ROMs capture key nonlinear phenomena, including isolas in forced response curves and bifurcations of periodic and quasi-periodic motions, with strong validation across multiple benchmark systems. The approach provides analytic backbones and efficient predictions for forced responses, limits cycles, and torus dynamics, enabling substantial computational speed-ups relative to full high-dimensional simulations. The methodology offers a practical framework for real-time analysis and design of nonlinear delay systems in engineering applications, subject to convergence and non-resonance constraints.

Abstract

Time-delay dynamical systems inherently embody infinite-dimensional dynamics, thereby amplifying their complexity. This aspect is especially notable in nonlinear dynamical systems, which frequently defy analytical solutions and necessitate approximations or numerical methods. These requirements present considerable challenges for the real-time simulation and analysis of their nonlinear dynamics. To address these challenges, we present a model reduction framework for nonlinear time-delay systems using spectral submanifolds (SSMs). We first approximate the time-delay systems as ordinary differential equations (ODEs) without delay and then compute the SSMs and their associated reduced-order models (ROMs) of the ODE approximations. These SSM-based ROMs successfully predict the nonlinear dynamical behaviors of the time-delay systems, including free and forced vibrations, and accurately identify critical features such as isolated branches in the forced response curves and bifurcations of periodic and quasi-periodic orbits. The efficiency and accuracy of the ROMs are demonstrated through examples of increasing complexity.

Figures (25)

• Figure 1: Backbone curves in polar (reduced) and physical coordinates under increasing orders of expansion for the slowest two-dimensional SSM of the system \ref{['eq:duffing']} with $\tau_d=1.0$.
• Figure 2: (Left panel) Time history plots of $x(t)$ and $\dot{x}(t)$ for the Duffing oscillator \ref{['eq:dde-duffing']}. We note that the initial conditions \ref{['eq: model1_dde_initial']} are off the SSM, which explains the observed discrepancies at the initial stage. Such discrepancies disappear as time increases because the SSM is attracting. (Right panel) Visualization of the SSM projected onto the coordinates $(\dot{x}(t), x(t), \dot{x}(t-\tau_d))$ (gray plane) along with the trajectories of the full system (blue dashed line) and the prediction via SSM-based ROM \ref{['eq:ROM1_stable']} (red solid line).
• Figure 3: Plot of the roots in the complex plane for $a(\rho)$ (cf. the first subequation of \ref{['eq:red-auto']}) for the ROM \ref{['eq:ROM1_unstable']}, with darker colors indicating an increasing truncation order up to $\mathcal{O}(15)$. The zeros from the highest approximation are highlighted in magenta. We observe that a non-trivial transverse zero persists for higher-order truncation, and it is clearly within the domain of analyticity.
• Figure 4: Time history curve of the free variable $x(t)$ (left panel) and the projection of the SSM (gray plane) onto the coordinates ($\dot{x}(t), x(t), \dot{x}(t-\tau_d)$) (right panel). The blue dashed line and the red solid line correspond to the trajectories of the full system and that obtained via the ROM \ref{['eq:ROM1_unstable']}. The left panel demonstrates that the limit cycle attracts nearby trajectories.
• Figure 5: The forced response curves subjected to the periodic excitation term 0.0009 cos $\Omega t$$(\epsilon = 0.0009)$ for the delayed Duffing system \ref{['eq:dde-duffing']}. The right panel is a zoomed plot of the left panel. The solid and dashed lines of the forced response curve of the periodic orbit represent stable and unstable periodic orbits, respectively, and the cyan circles indicate saddle-node bifurcation obtained from SSM-based prediction. Reference solutions are obtained from the collocation method applied to the ODE approximation of the DDEs. Results from the forward simulation of the DDEs are also provided in the right panel (dde23-stable).

Theorems & Definitions (2)

• Remark 1
• Remark 2