Globalizing the Carleman linear embedding method for nonlinear dynamics

TL;DR

Globalizing Carleman embedding addresses the convergence limitations of the classical approach by partitioning the state space into multiple convergence-charts, enabling accurate representation of nonlinear dynamics with multiple fixed points and chaotic attractors. The paper introduces ACE (adaptive piecewise Carleman embedding), GCE (grid-anchored static embedding), and compares them to the standard SCE baseline, with convergence governed by a per-chart radius $\xi$ and adaptive criteria based on density of points and a tolerance $\epsilon_{tol}$. Across 1D, 2D limit cycles, LV-type systems, and 3D chaotic attractors such as Rössler, Lorenz, and Chen, the globalized methods yield bounded errors where SCE fails, with PCE offering strong baseline accuracy and ACE providing adaptive refinement at higher computational cost. The results illuminate tradeoffs between speed and accuracy, highlight the potential for quantum-inspired or hybrid quantum-classical implementations, and show how a robust linear embedding can advance prediction and control of complex nonlinear dynamics using a modular atlas-based approach.

Abstract

The Carleman embedding method is a widely used technique for linearizing a system of nonlinear differential equations, but fails to converge in regions where there are multiple fixed points. We propose and test three different versions of a global piecewise Carleman embedding technique, based on partitioning space into multiple regions where the center and size of the embedding region are chosen to control convergence. The first method switches between local linearization regions of fixed size once the trajectory reaches the boundary of the current linearization chart. During the transition, the embedding is reconstructed within the newly created chart, centered at the transition point. The second method also adapts the chart size dynamically, enhancing accuracy in regions where multiple fixed points are located. The third method partitions the state space using a static grid with precomputed linearization charts of fixed size, making it more suitable for applications that require high speed. All techniques are numerically tested on multiple integrable and chaotic nonlinear dynamical systems demonstrating their applicability for problems that are completely intractable for the standard Carleman embedding method. Simulations of chaotic dynamical systems such as various types of strange attractors demonstrate the power of the adaptive methods, if a sufficiently low tolerance is imposed. Still, the non-adaptive version of the method, with fixed centers and sizes of the linearization charts, can be faster in simulating dynamical systems while providing similar accuracy and may be more appropriate as the basis of algorithms for future quantum computers.

Figures (12)

• Figure 1: Schematic depiction of the dynamic Piecewise Carleman Embedding (PCE) for a system with $n = 2$ degrees of freedom. A new linearization chart must be chosen each time the trajectory leaves the original linearization chart.
• Figure 2: The logical flow diagram for the ACE algorithm that adapts the new chart radius $\xi$ based on the need to satisfy an imposed error threshold $\epsilon_{\rm tol}$.
• Figure 3: Schematic depiction of the static piecewise Grid-anchored Carleman Embedding (GCE) for the case with $n = 2$ variables. The original nonlinear variables $X^{(i)}$ are shown in blue. The red dot at $\left(X_{\rm GC}^{(0)}, X_{\rm GC}^{(1)}\right)$ marks the center of the linearization grid. The position of each tile-shaped element is determined by the integers $l^{(i)}$. The distance between the centers of two neighboring elements is $2\xi^{(i)}$ in the $i$-th direction. The orange cross indicates the location of the initial condition, $\left(X_{0}^{(0)}, X_{0}^{(1)}\right)$.
• Figure 4: Results from simulations of the problem \ref{['eq:1d']} with $X_{c,1} = -0.6$, $X_{c,2} = -0.1$, $X_{c,3} = 0.4$ (a); $X_{c,1} = -2.2$, $X_{c,2} = 0.2$, $X_{c,3} = 1.6$ (b); $X_{c,1} = 0.8$, $X_{c,2} = 0.9$, $X_{c,3} = 1.6$ (c); $X_{c,1} = 0.1$, $X_{c,2} = 0.9$, and $X_{c,3} = 1.6$ (d). The CL simulations are shown by gray lines. (a) and (b): Comparison with the SCE (black dashed lineas) and PCE (red dashed lines) simulations for various initial conditions. (c) and (d): Comparison with the ACE and PCE simulations. Plots (a2), (b2), (c2), and (d2) shown numerical errors in the PCE and ACE simulations. All PCE and ACE simulations have $P = 6$. All ACE simulations have $\xi_{\rm init} = \xi_{\rm max}$ and $\Delta\xi = 0.02$.
• Figure 5: Results of simulating the Van der Pol limit cycle \ref{['eq:vdp-NORM']} with the initial conditions $(X,Y) = (0.2, 0.0)$. (a1) SCE (blue dashed line) and PCE (red dashed line) simulations and the comparison with the CL simulation (gray line). (a2) Error of the SCE and PCE results. (b1) Evolution of the convergence radius $\xi$ in the ACE simulations under various values of the tolerance and with a fixed step size of $\Delta\xi = 0.02$. (b2) Maximum error of the ACE results vs. time.