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Carleman-Fourier linearization of nonlinear real dynamical systems with quasi-periodic fields

Nader Motee, Qiyu Sun

TL;DR

The paper introduces Carleman-Fourier linearization for nonlinear real dynamical systems with (quasi-)periodic vector fields, leveraging Fourier representations and extended-state augmentation to obtain a block-upper-triangular, sparse linear embedding. It proves that finite-section approximations of the primary block converge exponentially to the original dynamics, with bounds independent of equilibrium and initial conditions for short time windows. The methodology is extended from a single-frequency setting to multi-frequency quasi-periodic systems and validated numerically on the Kuramoto model, where it outperforms classical Carleman linearization, especially away from zero initial states. These results offer a principled, scalable framework for reachability analysis and learning in complex nonlinear systems with periodic structure, with potential relevance to engineering, physics, biology, and quantum computing.

Abstract

This paper presents Carleman-Fourier linearization for analyzing nonlinear real dynamical systems with periodic vector fields. Using Fourier basis functions, this novel framework transforms such dynamical systems into equivalent infinite-dimensional linear dynamical systems. In this paper, we establish the exponential convergence of the primary block in the finite-section approximation of this linearized system to the state vector of the original nonlinear system. To showcase the efficacy of our approach, we apply it to the Kuramoto model, a prominent model for coupled oscillators. The results demonstrate promising accuracy in approximating the original system's behavior.

Carleman-Fourier linearization of nonlinear real dynamical systems with quasi-periodic fields

TL;DR

The paper introduces Carleman-Fourier linearization for nonlinear real dynamical systems with (quasi-)periodic vector fields, leveraging Fourier representations and extended-state augmentation to obtain a block-upper-triangular, sparse linear embedding. It proves that finite-section approximations of the primary block converge exponentially to the original dynamics, with bounds independent of equilibrium and initial conditions for short time windows. The methodology is extended from a single-frequency setting to multi-frequency quasi-periodic systems and validated numerically on the Kuramoto model, where it outperforms classical Carleman linearization, especially away from zero initial states. These results offer a principled, scalable framework for reachability analysis and learning in complex nonlinear systems with periodic structure, with potential relevance to engineering, physics, biology, and quantum computing.

Abstract

This paper presents Carleman-Fourier linearization for analyzing nonlinear real dynamical systems with periodic vector fields. Using Fourier basis functions, this novel framework transforms such dynamical systems into equivalent infinite-dimensional linear dynamical systems. In this paper, we establish the exponential convergence of the primary block in the finite-section approximation of this linearized system to the state vector of the original nonlinear system. To showcase the efficacy of our approach, we apply it to the Kuramoto model, a prominent model for coupled oscillators. The results demonstrate promising accuracy in approximating the original system's behavior.

Paper Structure

This paper contains 8 sections, 3 theorems, 69 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Classical Carleman Linearization
  3. Carleman-Fourier linearization and its finite-section approximation
  4. Carleman-Fourier linearization of systems with multiple fundamental frequencies
  5. Applications: Kuramoto model
  6. Applications: Kuramoto model
  7. Proof of Theorem \ref{['main.thm']}
  8. Conclusions and Discussions

Key Result

Theorem 3.1

Consider a periodic function $g$ as defined in Fourierexpansion.def and fulfilling the condition detailed in fouriercoefficient.decay. Let $x(t)$ represent the state of the dynamical system given by dynamicsystem.def, and ${\bf y}_{1, N}(t)$ as specified in Carleman.eq8. Let us write ${\bf y}_{1, N} where constants $D>0$ and $r\in (0, 1)$ are from fouriercoefficient.decay. Then, for $N\ge 1$, the

Figures (1)

  • Figure 1: Plotted from top to bottom are the approximation error $E_{\rm C}(\theta_1(0), \omega_1, N, t)$ in \ref{['ECNt.def']} for the finite-section approach to the classical Carleman linearization, the approximation error $E_{\rm CF}(\theta_1(0), \omega_1, N, t)$ in \ref{['ECFNt.def']} for the finite-section approach to the Carleman-Fourier linearization, and the approximation error $E(\theta_1(0), \omega_1, N, t)$ in \ref{['ENt.def']} for the finite-section approach to the conventional linearization respectively, where $-\pi/2\le \theta_1(0)\le \pi/2, 0\le t\le 1/2$. Here $\tilde{K}=1, N=10$ and $\omega_1=0$ (all left) and $\omega_1=1$ (all right).

Theorems & Definitions (4)

  • Theorem 3.1
  • Corollary 3.2
  • Remark 3.3
  • Theorem 4.1