On the structure of complex spectra and eigenfunctions of transfer and Koopman operators

TL;DR

This work analyzes how small-noise perturbations shape the complex spectra of transfer and Koopman operators for dynamical systems with rotation speeds that vary over phase space. By proposing a canonical model on a discretized cylinder with multiple coexisting cycles, the authors derive a separable eigenfunction structure and characterize the zero-noise limit, including explicit quadratic and linear responses of eigenvalues and eigenfunctions. They establish convergence of eigenprojections to band-localized subspaces and reveal that cycle periods and locations are robust to noise, enabling practical cycle-detection algorithms from operator eigendata. The results generalize to banded configurations and higher-dimensional phase spaces, providing a principled framework for detecting approximately cyclic motion in complex systems. The findings have potential applications in climate dynamics, molecular dynamics, and data-driven spectral analysis where coexisting cycles play a critical role and noise is inevitable.

Abstract

Complex spectra of transfer and Koopman operators describe rotational motion in dynamical systems. A particularly relevant situation in applications is where the rotation speed depends on the position in the phase space. We consider a canonical model of such dynamics in the presence of small noise, and provide precise characterisations of the eigenspectrum and eigenfunctions of the corresponding transfer operators. Further, we study the limiting behaviour of the eigenspectrum and eigenfunctions in the zero-noise limit, including their quadratic and linear response. Our results clarify the structure of transfer and Koopman operator eigenspectra, and provide new interpretations for applications. Our theorems on support localisation of the eigenfunctions yield simple algorithms to detect the existence and phase-space location of approximately cyclic motion with distinct periods. Our response results demonstrate that information on the cycle periods and their locations determined by the operator eigendata is largely insensitive to the noise level. We believe the mechanisms creating the eigendata apply more broadly and enhance our understanding of approximate cycle detection in dynamical systems with operator methods.

Figures (5)

• Figure 1: Left: Magnitude of the leading complex eigenvector of the normalised transfer operator for the Lorenz flow. A peak in magnitude occurs in the vicinity of the lowest-period periodic orbit of the Lorenz equations with a period of $1.5586$ time units (shown in red, see Fig. 10a FP09 and FGZ93). Right: The argument of the complex eigenvector. In the periodic colormap, one sees a full rotation around each of the two wings as the argument proceeds from $-\pi$ to $\pi$; see FGLPS21.
• Figure 2: Streamlines of the cylinder rotation model with three bands of circular fibres with widths $L_1=11, L_2=7, L_3=15$ and rotation speeds $\beta_1=\pi/20, \beta_2=e/7, \beta_3=1/\sqrt2$, respectively.
• Figure 3: Left: The small coloured disks indicate the spectrum $\lambda_{1,\varepsilon}$ (for $\delta=\varepsilon=0.1$) of the $33\times 33$ matrix $D_{k,\alpha}W_\varepsilon = D_{k,\beta,L}W_\varepsilon$, associated to the model from Figure 1, where $\beta=(\pi/20, e/7, 1/\sqrt2)$ and $L=(11,7,15)$. The large black circle is the unit circle in the complex plane. Note that the 33 eigenvalues are distinct and that they appear in groups, with $L_i$ eigenvalues nearby $\exp(-2\pi i k \beta_i)$ for $i=1,2,3$. The relevant bounds \ref{['gbound']} from Proposition \ref{['prop:1']} are shown as dark blue circles. Right: Plot of $|f_{k,\varepsilon}^{(\ell)}(j)|$ vs lattice index $j$ for $\ell=1,2,4$, where the ordering of the eigenvalues are shown in the left panel. We plot the eigenvectors of the largest-magnitude eigenvalue from each of the three groups; the eigenvectors have been normalised to have unit norm.
• Figure 4: Details of leading eigenfunctions of $\mathcal{P}_\varepsilon$ on $M$, constructed from the eigenvectors in Figure \ref{['fig:pertmatrixfig']}. Upper row: Magnitudes of the extremal three eigenfunctions $F_{k,\varepsilon}^{(\ell)}(j, x) = f_{k,\varepsilon}^{(\ell)}(j) e^{2\pi i k x}$, $\ell=1,2,4$ and $k=1$ on $M=\{1,\ldots,33\}\times \mathbb{S}^1$ corresponding to the eigenvectors $f$ in Figure \ref{['fig:pertmatrixfig']}, using colourschemes to match the colouring in Figure \ref{['fig:pertmatrixfig']}. Note that the support of each eigenfunction is approximately restricted to the bands associated with one of the three rotation speeds illustrated in Figure \ref{['fig:model']}. Lower row: As for the upper row, but displaying the arguments of eigenfunctions instead of the magnitudes. Note that exactly one complex cycle occurs because $k=1$.
• Figure 5: Left: The small coloured disks indicate the response of the spectrum $\hat{\lambda}_{1}^{(\ell)}$, $\ell=1,\ldots,33$ of the $33\times 33$ matrix $\hat{P}_{1,\beta,L}$, associated to the model from Figure 1, where $\beta=(\pi/20, e/7, 1/\sqrt2)$ and $L=(11,7,15)$. Note that the responses of the 33 eigenvalues are grouped along three rays with $L_i$ responses on a ray with angle $2\pi\beta_i+\pi$ for $i=1,2,3$. The dashed lines and disk colouring are identical to those in Figure \ref{['fig:pertmatrixfig']}(left). Right: Plot of $|\hat{f}_{1}^{(\ell)}(j)|$ vs lattice index $j$ for $\ell=1,2,4$, where the ordering of the eigenvalues $\hat{\lambda}_1^{(\ell)}$ are shown in the left panels of this figure and Figure \ref{['fig:pertmatrixfig']}. The three response vectors have been normalised to have unit norm and correspond to the three eigenvectors displayed in Figure \ref{['fig:pertmatrixfig']}(right).

Theorems & Definitions (50)

• Remark 2.1
• Proposition 2.2
• proof
• Definition 2.3: $S$-banded
• Definition 2.4: $L$-admissible family of matrices
• Theorem 2.5
• proof
• Theorem 2.6
• Proposition 2.7
• proof