Beyond Carleman Linearization of Nonlinear Dynamical System: Insights from a Case Study

TL;DR

The paper advances the analysis of nonlinear dynamical systems by developing a general linearization framework based on Carleman linearization and introducing Carleman-Fourier linearization to handle periodic, trigonometric governing fields. It shows that Carleman-Fourier, via an extended state representation, yields a block-upper-triangular infinite-dimensional system that can be analyzed with finite-section approximations, providing exponential convergence over a finite time window. A detailed case study with $g(x)=a(1-e^{ix})$ demonstrates explicit solution forms and compares the two methods, revealing that Carleman-Fourier outperforms Carleman for large imaginary initial conditions while Carleman is superior near the origin. Collectively, the results offer a structured, scalable approach to embed periodic nonlinear dynamics into linear frameworks, with potential impact on analysis, control, and design of nonlinear systems.

Abstract

Nonlinear dynamical systems are widely encountered in various scientific and engineering fields. Despite significant advances in theoretical understanding, developing complete and integrated frameworks for analyzing and designing these systems remains challenging, which underscores the importance of efficient linearization methods. In this paper, we introduce a general linearization framework with emphasis on Carleman linearization and Carleman-Fourier linearization. A detailed case study on finite-section approximation to the lifted infinite-dimensional dynamical system is provided for the dynamical system with its governing function being a trigonometric polynomial of degree one.

Figures (5)

• Figure 1: The figures above display the finite-section approximation errors, given by $\max(\min(E_{C}(x_0, T^*, N), 2), -5)$ in \ref{['ETC.def00']}, over the domain $-2 \le \Re x_0, \Im x_0 \le 2$, with a fixed time range $T^*= 1/2$. The columns correspond to value $a = -i$ (left), $1$ (middle), and $i$ (right), while the rows correspond to $N = 1$ (top), $5$ (middle), and $10$ (bottom).
• Figure 2: The finite-section approximation errors, given by $\max(\min(E_{C}(x_0, T^*, N), 2), -5)$ in \ref{['ETC.def00']}, over the domain $-2 \le \Re x_0, \Im x_0 \le 2$, with a fixed time range $T^*= 1/2$ and truncation order $N=10$. The columns correspond to value $a = -i$ (left), $1$ (middle), and $i$ (right), while the rows correspond to $b=2/3$ (top) and $b=4/3$ (bottom).
• Figure 3: 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 on the right 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).
• Figure 4: Plotted on the top are the finite-section approximation errors $\max(\min(E_{CF}(x_0, T^*, N), 2), -5)$ with $\-\pi/2\le\phi\le \pi/2$ 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 on the bottom are $\max(\min(E_{CF}(x_0, T^*, N), 2), -5)$ with $-2\le \Re x_0 \pi/2$ 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.
• Figure 5: 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 has parameter $a=1$ (left) and initial $x_0= i\ln (1-e^{ai\pi/2})\approx 0.7854 a + 0.3466i$ (in black), $-1/2a$ (in cyan) and $-3/2a$ (in red). Presented in the middle is 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).

Theorems & Definitions (6)

• Theorem 2.1
• Theorem 4.1
• Remark 4.2
• Lemma B.1
• proof
• proof : Proof of Theorem \ref{['Carleman.thm']}