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Carleman-Fourier Linearization of Complex Dynamical Systems: Convergence and Explicit Error Bounds

Panpan Chen, Nader Motee, Qiyu Sun

Abstract

This paper presents a Carleman-Fourier linearization method for nonlinear dynamical systems with periodic vector fields involving multiple fundamental frequencies. By employing Fourier basis functions, the nonlinear dynamical system is transformed into a linear model on an infinite-dimensional space. The proposed approach yields accurate approximations over extended regions around equilibria and for longer time horizons, compared to traditional Carleman linearization with monomials. Additionally, we develop a finite-section approximation for the resulting infinite-dimensional system and provide explicit error bounds that demonstrate exponential convergence to the original system's solution as the truncation length increases. For specific classes of dynamical systems, exponential convergence is achieved across the entire time horizon. The practical significance of these results lies in guiding the selection of suitable truncation lengths for applications such as model predictive control, safety verification through reachability analysis, and efficient quantum computing algorithms. The theoretical findings are validated through illustrative simulations.

Carleman-Fourier Linearization of Complex Dynamical Systems: Convergence and Explicit Error Bounds

Abstract

This paper presents a Carleman-Fourier linearization method for nonlinear dynamical systems with periodic vector fields involving multiple fundamental frequencies. By employing Fourier basis functions, the nonlinear dynamical system is transformed into a linear model on an infinite-dimensional space. The proposed approach yields accurate approximations over extended regions around equilibria and for longer time horizons, compared to traditional Carleman linearization with monomials. Additionally, we develop a finite-section approximation for the resulting infinite-dimensional system and provide explicit error bounds that demonstrate exponential convergence to the original system's solution as the truncation length increases. For specific classes of dynamical systems, exponential convergence is achieved across the entire time horizon. The practical significance of these results lies in guiding the selection of suitable truncation lengths for applications such as model predictive control, safety verification through reachability analysis, and efficient quantum computing algorithms. The theoretical findings are validated through illustrative simulations.

Paper Structure

This paper contains 19 sections, 12 theorems, 162 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Carleman Linearization using Monomials
  3. Carleman-Fourier Linearization and Convergence of the Finite-Section Approximations
  4. Carleman-Fourier linearization using Fourier basis functions
  5. Exponential convergence of finite-section approximation in a time range
  6. Exponential convergence of finite-section approximation in the entire time range
  7. Carleman-Fourier Linearization of Complex Dynamical Systems with Multiple Fundamental Frequencies
  8. Carleman-Fourier linearization for dynamical systems with multiple fundamental frequencies
  9. Convergence of finite-section approximation in a time range
  10. Exponential convergence of finite-section approximation over the entire range
  11. Numerical demonstrations
  12. Comparing Carleman-Fourier Linearization with Carleman Linearization
  13. Carleman-Fourier linearization of the first-order Kuramoto model
  14. Proofs
  15. Proof of Theorem \ref{['maintheoremanalytic.thm1']}
  16. ...and 4 more sections

Key Result

Theorem 2.1

Suppose that ${\bf x}(t)$ is the solution of the dynamical system dynamicsystem with the vector field ${\bf g}(t,{\bf x})$ satisfying zeroequil.eq and Assumption assump-1 for some $D_0>0$ and $R>1$, and ${\bf y}_{1, N}(t), N\ge 1$, is the first block of the solution of the finite-section approximati hold for all $0\le t\le T_C^*$ and $N\ge 1$, where $T_C^*$ is given in Tstar0.def.

Figures (5)

  • Figure 1: Plotted are the vector fields $a(1-e^{ix})$ of the complex dynamical system \ref{['simpleexample2.eq1']} with $a=1$ (left), $a=i$ (middle) and $a=-i$ (right), where $-\pi\le \Re x\le \pi$ and $-\pi/2\le \Im x\le \pi/2$. Trajectories on the left figure have parameters $a=1$ and initial $x_0= i\ln (1-e^{ai\pi/2})\approx 0.7854 + 0.3466i$ (in black), $-1/2$ (in cyan) and $-3/2$ (in red). Presented in the middle are trajectories with $a=i$ and $x_0= i\ln (1-e^{ai\pi/2}) \approx - 0.2330i$ (in black) and $-1/2$ (in red), while on the right are trajectories with $a=-i$ and $x_0= i\ln (1-e^{ai\pi/2}) \approx -3.1416 + 1.3378i$ (in black) and $3/2$ (in blue). Trajectories shown in the figures may blow up in a finite time (in black), have limit cycle (in cyan), converge (in blue) and diverge (in red).
  • Figure 2: Plotted on the left is the function $\min\{ h(\varphi, t), 10\}, -\pi/2\le \varphi\le \pi/2, 0\le t\le 5$, where $h$ is given in \ref{['hh.def00']}. Presented in the middle is the actual time range $\min(T^*(\varphi), 3), -\pi/2\le \varphi\le 0$, in \ref{['actualtimerage.orderone']}, where $\Im x_0=0$ (in green) and $\Im x_0=2$ (in blue), and the time range $T_{CF}^*$ in Theorem \ref{['maintheoremanalytic.thm1']} when $\Im x_0=0$ (in red) and when $\Im x_0=2$ (in magenta). Shown on the right is the requirement on the initial $x_0$ for the exponential convergence on $[0, \infty)$, which is $\frac{1}{2} \ln \sup_{t\ge 0} h(\varphi, t), 0<\varphi\le \pi/2$, in \ref{['timerange.orderoneeq1']} (in green) and the theoretical lower bound $-\ln \sin \varphi, 0<\varphi\le \pi/2$ in Theorem \ref{['maintheoremanalytic.thm2']} (in red).
  • Figure 3: Plotted on the top are the finite-section approximation errors $\max(\min(E_{CF}(x_0, T^*, N), 2), -5)$ of the Carleman-Fourier linearization, defined in \ref{['fouriererror.def']}, where $-\pi/2\le\phi\le \pi/2$ as the $x$-axis and $-2\le \Im x_0\le 2$ as the $y$-axis, and level curve $E_{CF}(x_0, T^*, N)=0$ (in black) for $N=10$ and $T^*=2$ (left), $1/2$ (middle) and $1/4$ (right) respectively. Shown in the middle are $\max(\min(E_{CF}(x_0, T^*, N), 2), -5)$ with $-2\le \Re x_0 \le 2$ as the $x$-axis and $-2\le \Im x_0\le 2$ as the $y$-axis, for $N=10, T^*=1/2$ and $\phi=-\pi/2$ (left), $0$ (middle) and $\pi/2$ (right) respectively. Plotted at the bottom are the finite-section approximation errors $\max(\min(E_{C}(x_0, T^*, N), 2), -5)$ of Carleman linearization, defined in \ref{['OrderOneCarlemanErrod.def']}, where $-2\le \Re x_0, \Im x_0\le 2, T^*=1/2, N=10$ and $a=-i$ (left), $0$ (middle) and $i$ (right).
  • Figure 4: Plotted on the top row are the vector fields of the dynamical system \ref{['kuramotodimension3.def']} for $-4\pi/3\le \theta_1(0), \theta_2(0)\le 4\pi/3$ and the shadowed regions on which the vector field has relatively small magnitude, where $\tilde{K}=-1$ and $(\omega_1, \omega_2)= (0, 0)$ (top left), $(0, 1)$ (top middle), $(0.5, 0.5)$ (top right) respectively. Plotted in the middle and bottom rows are the approximation error $E_{CF}(\theta_1(0), \theta_2(0), \omega_1, \omega_2, N, T)$ in \ref{['hatECFNt.def']} and $\tilde{E}_{CF}(\theta_1(0), \theta_2(0), \omega_1, \omega_2, N, T)$ in \ref{['tildeECNt.def']}, where $-4\pi/3\le \theta_1(0), \theta_2(0)\le 4\pi/3, N=10, T=0.5, K=-1, \theta_3(0)=-\theta_1(0)-\theta_2(0), \omega_3=-\omega_1-\omega_2$, and $(\omega_1, \omega_2)=(0, 0)$ (left), $(0, 1)$ (middle) and $(0.5, 0.5)$ (right) respectively.
  • Figure 5: Plotted are the approximation error $E_C(\theta_1(0), \theta_2(0), \omega_1, \omega_2, N, T), -4\pi/3\le \theta_1(0), \theta_2(0)\le 4\pi/3$, in \ref{['hatECNt.def']} of the finite-section method to the Carleman linearization of the dynamical system \ref{['kuramotodimension3.Taylordef']}, where $\tilde{K}=-1, N=10, T=0.5$ and $(\omega_1, \omega_2)=(0, 0)$(left), $(0, 1)$(middle) and $(0.5, 0.5)$(right).

Theorems & Definitions (16)

  • Theorem 2.1
  • Theorem 3.1
  • Corollary 3.2
  • Theorem 3.3
  • Corollary 3.4
  • Theorem 4.1
  • Corollary 4.2
  • Theorem 4.3
  • Lemma 6.1
  • proof
  • ...and 6 more