Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2606.06728

Data-driven methods for computation of optimal linear response in high-dimensional dynamical systems

Gary Froyland·Dimitrios Giannakis·Nicholas Peters
View original source ↗Read source document ↗Export metadata ↓

Abstract

We develop a data-driven framework for estimating optimal linear response of nonlinear dynamical systems. Our approach is based on kernel-smoothed approximations of the transfer/Koopman operators of the system, built from possibly high-dimensional observations along trajectories. Combining these operator approximations with the theory developed in [Antown et al. (2018), J. Stat. Phys., 170(6), 1051-1087], we formulate a computationally tractable optimization problem for the optimal infinitesimal perturbation realising any desired manipulation of the spectrum. We also introduce a notion of optimal-response vector fields for visualising, and physically interpreting, the system's response to the optimal perturbation under arbitrary observations. Our focus is on finding perturbations that optimally increase the frequency or optimally suppress the decay of correlations of almost-cycles or almost-invariant sets associated with the eigenvalues of the kernel-smoothed transfer operator. We illustrate our approach with applications to low-dimensional periodic and chaotic systems, as well as a high-dimensional example involving the El Nino Southern Oscillation in a comprehensive Earth system model. In these examples our approach discovers nontrivial optimal perturbations of the system, which are post hoc natural and consistent with the desired dynamical objectives.

Explore related subjects

math.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Gary Froyland, Dimitrios Giannakis, Nicholas Peters. 2026-06-17. Data-driven methods for computation of optimal linear response in high-dimensional dynamical systems. https://arxiv.org/abs/2606.06728

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Pivoting technique for the circle homeomorphism group

We adapt Gou{ë}zel's pivoting technique to the circle homeomorphism group. As an application, we give different proofs of Gilabert Vio's probabilistic Tits alternative and Malicet's exponential synchronization.

math.DS↗
arXiv2026

Asymmetry of a class of Mellin transforms

We introduce the quantity $μ_η$, defined for every complex $s$ in the critical strip, as a transformation of the Mellin transform associated to the functions $η$. We establish a sufficient condition on $η$ under which $μ_η(s)$ and $μ_η(1-s)$ cannot both vanish outside the critical line. An application is given to the case in which $η$ is the fractional part function, and the zeros of $μ_η$ coincide with the zeros of the Riemann zeta function.

math.DS↗
arXiv2026

Infinite Set of Resonances in the Linear Damped Oscillator Subject to Harmonic Forcing with Non-standard Frequency Modulation

It is shown that harmonic signals incorporating a type of weak non-standard frequency modulation (wNSFM) have interesting spectral properties, namely, time-dependent bandwidths that become increasingly broader with increasing time. As such, they represent a class of signals with frequency-time coupling in their spectra. Specifically, the weakly damped oscillator exhibits always two transient resonance captures involving two distinct harmonics possessing relatively high amplitudes over finite time intervals, while the overall response decays as $~t^{-1/2}$ as $t\rightarrow\infty$. Considering the undamped oscillator, it possesses two types of resonances, referred to as simple and non-simple resonances. Simple resonances correspond to finite-amplitude steady-state responses caused by two sustained resonance captures, in the form of two distinct modulated quasi-periodic responses, which, however are "activated" at different time instances. The necessary and sufficient conditions for non-simple resonances are given in the form of a theorem which predicts the existence of resonant harmonics and specifies the special phase conditions that the resonant harmonics must satisfy for constructive interference; the resulting undamped non-simple resonance grows unboundedly as $~t^{-1/2}$ as $t\rightarrow\infty$, in contrast to the classical resonance growth of the linear resonator with unmodulated harmonic excitation whose response grows as $~t$ as $t\rightarrow\infty$. These resonant responses are persistent to changes in the parameters of the wNSFM. Our results reveal interesting infinite sets of resonances in linear SDOF resonators under frequency-modulated excitations.

math.DS↗