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arxiv: 2603.05771 · v2 · submitted 2026-03-06 · 📡 eess.SY · cs.SY· math.DS· math.OC

On Koopman Resolvents and Frequency Response of Nonlinear Systems

Yoshihiko Susuki , Natsuki Katayama , Alexandre Mauroy , Igor Mezi\'c This is my paper

Pith reviewed 2026-05-15 16:00 UTC · model grok-4.3

classification 📡 eess.SY cs.SYmath.DSmath.OC
keywords Koopman operatorresolventfrequency responsenonlinear systemsBode plotLaplace transformdynamical systems
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The pith

The frequency response of a nonlinear system is the Laplace transform of its output, derived using the resolvent of the Koopman operator.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper shows how to define the frequency response of a nonlinear dynamical system by taking the Laplace transform of its time-domain output under sinusoidal forcing. The derivation uses the resolvent operator associated with the Koopman operator, which provides a linear representation of the nonlinear evolution. This generalizes the classical definition used for linear time-invariant systems and supports the construction of Bode plots that capture gain and phase shift as functions of driving frequency. Sufficient conditions are given to guarantee that such a response exists for three families of nonlinear dynamics. Readers would care because it offers a route to frequency-domain analysis tools for systems that do not obey superposition.

Core claim

The frequency response is defined as the complex-valued function obtained from the Laplace transform of the plant output, and this function is shown to exist under the resolvent theory of the Koopman operator for three classes of dynamics.

What carries the argument

The Koopman resolvent, the resolvent operator of the Koopman operator, which linearizes the nonlinear dynamics and directly supplies the frequency response via the output Laplace transform.

Load-bearing premise

The nonlinear system belongs to one of the three classes for which existence of the frequency response is guaranteed.

What would settle it

A concrete nonlinear system outside the three classes where the Laplace transform of the output changes with input amplitude or fails to converge to a unique complex value at each frequency.

Figures

Figures reproduced from arXiv: 2603.05771 by Alexandre Mauroy, Igor Mezi\'c, Natsuki Katayama, Yoshihiko Susuki.

Figure 1
Figure 1. Figure 1: Bode plots for frequency responses of the 2d nonlinear example: [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
read the original abstract

This paper proposes a novel formulation of frequency response for nonlinear systems in the Koopman operator framework. This framework is a promising direction for the analysis and synthesis of systems with nonlinear dynamics based on (linear) Koopman operators. We show that the frequency response of a nonlinear plant is derived through the Laplace transform of the output of the plant, which is a generalization of the classical approach to LTI plants and is guided by the resolvent theory of Koopman operators. The response is a complex-valued function of the driving angular frequency, allowing one to draw the so-called Bode plots, which display the gain and phase characteristics. Sufficient conditions for the existence of the frequency response are presented for three classes of dynamics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript proposes a Koopman operator framework for defining the frequency response of nonlinear systems. The response is obtained via the Laplace transform of the output under harmonic excitation, generalizing the classical LTI derivation and guided by Koopman resolvent theory; this yields a complex-valued function of driving frequency from which Bode plots can be constructed. Sufficient existence conditions are stated for three classes of dynamics.

Significance. If the derivation is valid and the three classes encompass a useful range of engineering nonlinearities, the work would supply a principled, operator-theoretic route to frequency-domain analysis of nonlinear plants, potentially aiding identification and control design where classical Bode methods fail. The explicit link to resolvent theory is a technical strength.

major comments (1)
  1. [§3] §3 (or equivalent section defining the classes): the three classes of dynamics for which sufficient conditions are given are not characterized with respect to standard nonlinearities (e.g., polynomial, saturation, or trigonometric). Without this, it is impossible to judge whether the claimed generalization applies to typical plants or only to narrowly structured systems where the output Laplace transform remains free of harmonic contamination; this directly affects the scope of the central claim.
minor comments (2)
  1. [Introduction] Notation for the resolvent operator and the frequency-response function should be introduced with a single consistent symbol set early in the paper to avoid later ambiguity.
  2. [Abstract] The abstract would benefit from a one-sentence indication of the breadth of the three classes.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback, particularly for highlighting the potential utility of the Koopman resolvent framework. We address the single major comment below.

read point-by-point responses

  1. Referee: §3 (or equivalent section defining the classes): the three classes of dynamics for which sufficient conditions are given are not characterized with respect to standard nonlinearities (e.g., polynomial, saturation, or trigonometric). Without this, it is impossible to judge whether the claimed generalization applies to typical plants or only to narrowly structured systems where the output Laplace transform remains free of harmonic contamination; this directly affects the scope of the central claim.

    Authors: We agree that explicit characterization of the three classes with respect to standard nonlinearities is needed to clarify the scope. In the revised manuscript we will add a dedicated subsection to §3 that maps common nonlinearities (polynomials, saturation, trigonometric) to each class, states which satisfy the sufficient conditions, and notes any implications for harmonic content in the output Laplace transform. This will allow readers to assess applicability to typical plants. revision: yes

Circularity Check

0 steps flagged

No significant circularity; frequency response derived directly as Laplace transform of output under Koopman resolvent guidance

full rationale

The derivation chain begins with the Laplace transform of the nonlinear plant output to obtain a complex-valued frequency response function, presented as a direct generalization of the classical LTI Bode-plot construction. This is guided by (but not reduced to) the resolvent theory of Koopman operators. Sufficient existence conditions are stated for three classes of dynamics without any fitted parameters, self-referential definitions, or load-bearing self-citations that collapse the result to its inputs. The central claim remains independent of the target frequency-response expression and does not rename known empirical patterns or smuggle ansatzes via prior work. The approach is self-contained against external benchmarks such as the standard LTI Laplace-transform definition.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, invented entities, or ad-hoc axioms are stated beyond reliance on standard Koopman operator existence and Laplace transform properties.

axioms (2)
  • domain assumption The nonlinear system admits a Koopman operator representation
    Required for the resolvent framework to apply to the plant dynamics.
  • standard math The Laplace transform of the output exists for the driving frequencies of interest
    Central to defining the frequency response as a complex-valued function.

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