Koopman Resolvent Frequency Response

• Koopman resolvent-based frequency response is a framework that generalizes classical LTI analysis to nonlinear and high-dimensional systems using Koopman operators.
• Data-driven methods like Dynamic Mode Decomposition and kernel projections enable precise state-space modeling, spectral decomposition, and identification of optimal input-output pathways.
• The formulation provides actionable insights into modal dynamics, energetic amplification, and control synthesis, with key applications in fluid mechanics and flow control.

A resolvent-based frequency-response formulation in the Koopman operator framework extends the notion of classical frequency response from finite-dimensional linear time-invariant (LTI) systems to general nonlinear and high-dimensional dynamical systems. In this approach, the resolvent of the Koopman generator plays a central role, enabling principled and often data-driven computation of transfer functions, modal content, and energetic amplification pathways, even in strongly nonlinear regimes. By establishing state-space representations, spectral decompositions, and robust computational procedures, Koopman resolvent-based frequency analysis provides a rigorous foundation for both the qualitative and quantitative study of input–output dynamics encountered in fluid mechanics, control, and dynamical systems theory.

1. Koopman State-Space Models and Discrete/Continuous Representations

The foundation of Koopman resolvent-based frequency response begins with an embedding of nonlinear dynamics into state-space models driven by an approximately finite-dimensional Koopman operator. For a system with measured fluid “excitation” $u \in \mathbb{R}^{m_u}$ and structural “output” $y \in \mathbb{R}^{m_y}$, the representation is

Discrete-time (Z-transform, Koopman-LTI):

$$x_{k+1} = A_d x_k + B_d u_k,\quad y_k = C x_k + D u_k.$$

Continuous-time (Laplace-transform, Koopman-LTI):

$$\dot x(t) = A x(t) + B u(t), \quad y(t) = C x(t) + D u(t),$$

where $x$ represents the latent Koopman state, $A$ ($A_d$ discrete) is the generator (or finite Koopman matrix approximation), and $B$, $C$, $D$ capture input, output, and feedthrough effects.

Data-driven estimation of $A$, $B$, $C$, $D$ proceeds by Dynamic Mode Decomposition (DMD) and its variants. For the unforced case, DMD identifies $A_d$ via the compact SVD of measured state increments. When input–output structure exists, Extended DMD or subspace regression is used to estimate all system matrices from paired input and response trajectories, converting the discrete-time fitted matrices to continuous time via logarithmic mapping and quadrature as required (Li et al., 2021).

2. Koopman Spectral Decomposition and the Resolvent Operator

Given a diagonalizable $A$, the Koopman generator admits the eigendecomposition

$$A = V \Lambda V^{-1},\quad \Lambda = \mathrm{diag}(\lambda_1,\ldots,\lambda_r),$$

where $\lambda_j$ are Koopman eigenvalues, and $V$ contains the corresponding eigenvectors.

The continuous- or discrete-time resolvent operator—which generalizes $(i\omega I - A)^{-1}$—is defined by

$$R(i\omega) = (i\omega I - A)^{-1} = V (i\omega I - \Lambda)^{-1} V^{-1}$$

in continuous time, or

$$R(z) = (z I - A_d)^{-1},\quad z = e^{i\omega\Delta t},$$

in discrete time.

For general nonlinear systems, the infinite-dimensional Koopman operator $L$ acting on observables yields $R(s;L) = (sI - L)^{-1}$, the resolvent acting in function space. This objects admits further spectral expansion,

$$R(s;L) f = \sum_{\alpha} \frac{\varphi_\alpha V_\alpha}{s - \lambda_\alpha} + \text{(continuous part)},$$

where $\lambda_\alpha$ are Koopman generator eigenvalues and $\varphi_\alpha V_\alpha$ are spectral projections (Susuki et al., 2020, Susuki et al., 6 Mar 2026).

3. Frequency Response and Transfer Function Formulation

The frequency response operator (transfer function) from input $u$ to output $y$ is

$$H(i\omega) = C (i\omega I - A)^{-1} B + D = C R(i\omega) B + D,$$

a direct analogue of the classical LTI frequency response. For general nonlinear systems forced at $u(t) = u_0 e^{i\omega t}$, the Laplace transform of the output,

$\hat y(s) = [R(s;L_{\text{forced}}) g](x_0, u_0)$

captures the effect of input on observable $g$, with $L_{\text{forced}}$ the skew-product Koopman generator incorporating the input as an autonomous extension. The fundamental $n$th-harmonic frequency response is identified as the residue at the pole $s = i n \omega$: $$H_n(\omega;g) = u_0^{-n} \operatorname{Res}_{s=i n\omega} \left\{[R(s;L_{\text{forced}}) g](x_0, u_0)\right\}$$ (Susuki et al., 6 Mar 2026). In the linear case, this recovers the standard $H(i\omega) = C(i\omega I - A)^{-1} B$.

Physical interpretation of $|H(i\omega)|$ yields amplitude gain, and $\arg H(i\omega)$ yields phase lag. These enable construction of Bode plots and input–output amplification diagrams, indicating the spectral sensitivity of structural or measurement observables to external periodic forcing.

4. Data-Driven Identification and Numerical Computation

Koopman-resolvent frequency response can be realized entirely from snapshot data, subject to sufficient temporal resolution and observable richness. In the finite-dimensional approximation, subspace regression or extended DMD yields $A_d, B_d, C, D$. For fully nonlinear or high-dimensional problems, kernel integral operator approaches and Galerkin projections are employed:

• Assemble time-ordered trajectory data and construct kernel-induced bases on the observable space (e.g., via leading eigenfunctions of a diffusion kernel).
• Use projected shift operators and quadrature to approximate the resolvent integral

$R(z) \approx \sum_{q=0}^Q w_q e^{-z t_q} U^q$

for appropriately projected Koopman operator $U$.

• Conduct polar/Schmidt decompositions to extract singular vectors (input/output modes) and gain curves, controlling accuracy through kernel bandwidth, integration quadrature, and rank truncation (Valva et al., 2023, Colbrook et al., 2021).

Residual dynamic mode decomposition (ResDMD) offers rigorous error bounds and convergence guarantees, extending to systems with continuous, discrete, or mixed spectra, and is compatible with high-dimensional settings (e.g., turbulent flows) (Colbrook et al., 2021).

5. Modal Decomposition, Energetics, and Input–Output Amplification

With the spectral decomposition of $A$ or $L$, modal analysis proceeds via two viewpoints:

Direct Koopman Mode Decomposition: Eigenvectors of $A$ yield spatial structures (modes) and temporal coefficients aligned with Koopman eigenvalues, reconstructing the state or observable trajectory as

$x(t) = \sum_{j=1}^r \phi_j a_j e^{\lambda_j t},$

with modal contributions quantified as $|a_j|^2 \|\phi_j\|^2$.

Resolvent-Based Decomposition: At a fixed frequency $\omega$, the transfer function $H(i\omega)$ is subjected to singular value decomposition,

$$H(i\omega) = \sum_j \sigma_j(\omega) \psi_j(\omega) \phi_j^*(\omega),$$

where $\sigma_j$ is the gain, and the corresponding $\phi_j$, $\psi_j$ are input (forcing) and output (response) modes—these maximize output energy for a unit-norm forcing at that frequency. This identifies the optimally amplified flow or state structures and direct mapping between system input and dominant response, which is particularly interpretable in fluid–structure interaction and flow control contexts (Li et al., 2021, Sharma et al., 2016, Padovan et al., 2019).

In the context of time-periodic base flows or invariant subspaces, the harmonic resolvent operator encodes cross-frequency (triadic) couplings and yields block-Toeplitz structures, directly encompassing mode–mode interactions observed in turbulent and oscillatory regimes (Padovan et al., 2019).

6. Physical Interpretation, Sufficient Conditions, and Applications

The existence of well-defined frequency response via the Koopman resolvent is guaranteed under several conditions:

• For LTI systems, the spectrum of $A$ is bounded away from the excitation frequency.
• For globally (asymptotically) stable periodic orbits with analytic nonlinearities, isolated simple poles at harmonic frequencies exist (Susuki et al., 6 Mar 2026).
• For ergodic attractors with pure point spectrum, the spectral projection yields isolated response peaks corresponding to the system's natural frequencies (Susuki et al., 2020).

In practical demonstrations, such as the subcritical prism wake at $\text{Re} = 22,000$, the Koopman-LTI approach yields a universal operator whose eigenvalues match across all measured degrees of freedom (flow and wall pressures), revealing direct one-to-one correspondences between coherent flow structures and structural response (Li et al., 2021). This methodology generalizes to any statistically stationary fluid–structure or dynamical system with accessible measurement time series.

The resolvent-based frequency response enables:

• Explicit phase and amplitude transfer characterization,
• Predictive low-order state-space models for nonlinear flow–structure systems,
• Efficient design and synthesis of targeted control and sensing strategies,
• Modal discrimination even in turbulent, high-dimensional, or multi-scale environments,
• Extraction of physically meaningful, optimally amplified input–output pathways.

Applications range from fluid–structure interaction, flow control, and mechanical systems to general nonlinear dynamical systems exhibiting complex nonlinear, periodic, or chaotic behavior.

7. Connections to Broader Operator-Theoretic and Modal Decomposition Frameworks

The Koopman resolvent-based frequency-response paradigm provides formal bridges between traditional linear systems theory, modern operator-theoretic modal analysis, and physical input–output approaches:

• The resolvent $(i\omega I - A)^{-1}$ in finite dimensions recovers the classical LTI transfer function and singular value decomposition identifies optimal input–output mode pairs, as in classical resolvent analysis (Sharma et al., 2016).
• Infinite-dimensional generalizations capture nonlinear, ergodic, or chaotic regimes via poles (point spectrum) or continuous spectral contributions (Susuki et al., 2020, Colbrook et al., 2021, Valva et al., 2023).
• Harmonic (block-Toeplitz) resolvent constructions capture temporally modulated base flows and explicit cross-frequency energy transfers (Padovan et al., 2019).
• Data-driven methods, including DMD, kernel-based Galerkin projection, and ResDMD, permit accurate, residual-controlled computation of Koopman spectral quantities, enabling robust operator-theoretic frequency response analysis from empirical data (Colbrook et al., 2021, Valva et al., 2023).

This unification facilitates informed model reduction, flow diagnostics, and synthesis of full-order or surrogate models for engineering and scientific applications. The method rigorously extends Bode plots, input–output gain analysis, and frequency-domain control tools to the field of nonlinear and data-rich systems through the resolvent theoretic lens.