Constructive QP-Time-dependent KAM Algorithm for Lagrangian Tori

TL;DR

This work presents a constructive KAM algorithm for computing fiberwise Lagrangian tori in quasi-periodic time-dependent Hamiltonian systems using the parameterization method and a quasi-Newton refinement. It develops a complete numerical pipeline, including torus construction, continuation, and implementation considerations, and applies it to Tokamak magnetic-field-line models and fluid/plasma transport scenarios such as the single-wave model and a quasi-periodic pendulum. The results demonstrate the persistence and controllable continuation of invariant tori, with Sobolev-norm diagnostics providing insight into breakdown thresholds (e.g., near ε ≈ 0.00446 in the Tokamak example). The approach offers a direct vector-field formulation with reduced dimensionality and favorable computational properties for CAPs, with significant implications for confinement design and the study of quasi-periodic transport in plasmas and fluids.

Abstract

In this paper, we present an algorithm to computea fiberwise Lagrangian torus in quasi-periodic (QP) Hamiltonian systems, whose convergence is proved in the [CHP25]. We exhibit the algorithm with two models. The first is a Tokamak model [CVC+05, VL21], which proposes a control method to create barriers to the diffusion of magnetic field lines through a small modification in the magnetic perturbation. The second model [dCN00], known as the vorticity defect model, describes the nonlinear evolution of localized vorticity perturbations in a constant vorticity flow. This model was originally derived in the context of plasma physics and fluid dynamics.

Figures (14)

• Figure 1: The colors Cyan, Magenta and Teal correspond to $\varphi = 0.35, 0.3566248878338341665, 0.36$, respectively.
• Figure 2: Parametrized surface of the invariant torus, displaying the components $K_x(\theta,\varphi)$, $K_y(\theta,\varphi)$, and $\varphi$, represented with $2^{23}$ Fourier modes and an invariance error equal to $3.96005\times10^{-16}$. Where blue represents the lowest height and red the highest, based on the $K_y(\theta,\varphi)$ component.
• Figure 4: Norm of the Fourier coefficients on a logarithmic scale. The coefficients with the largest contributions, the 'significant' ones, align with the straight line shown in yelow in the image, which fits the equation $y= -0.58013x$.
• Figure 5: For each $k_\varphi$, we take the logarithm of the maximum of the $H^4$ Sobolev norm, equation \ref{['eq:H4-norm']}, of the Fourier coefficients for the torus with $\varepsilon=0.004$.
• Figure 6: We observed that near the parameter value $\varepsilon = 0.00445$, there is an exponential growth, indicating signs of the breakdown of the torus.

Theorems & Definitions (5)

• Remark 2.1
• Remark 2.2
• Remark 2.7
• Remark 2.8
• Remark 3.1