Range Failure, Resolvent Growth, and Resonance in Partially Dissipative Systems
Abstract
We introduce several notions of resonance for systems exhibiting weak or partial dissipation: u̇=Au+f(t), with A the generator of a strongly stable semigroup S(t) on a Hilbert space H. We refer to resonance as the phenomenon where a time-periodic f may yield an unbounded u, classically occurring when A has imaginary eigenvalues. In infinite dimensions, however, unbounded growth may depend on topological choices, and several ``resonances" may occur. Classically, existence of T-periodic u under T-periodic f is equivalent to a range condition: ℛ(I−S(T))=H. Consequently, its failure permits unbounded solution growth and we elucidate that connection through properties of S(t). We describe a hierarchy of range failures, and provide an associated resonance taxonomy. The growth rate of ~||(A−λI)−1||H on ~iR dictates a regularity gap between u and f, and, for rapid growth in λ, that gap can be infinite. In this case, we demonstrate implications for periodic solvability, and a mechanism for constructing smooth resonant forces. The theory developed here is motivated by (and demonstrated for) a hyperbolic-parabolic system, where the latter component provides the only system dissipation. Such dynamics are a simplification of fluid-structure phenomena, which demonstrate strong stability but do not provide unconditional periodic well-posedness. Though point-spectral resonance is ruled out, we note that ℛ(I−S(T))≠ H. Our main result here shows that, for a carefully constructed geometry, a classical heat-wave system possesses periods T which yield infinite derivative loss and, via supporting results, yield resonance with smooth forcing.
Related
- A discrete fractional Grönwall inequality for dissipative systems via resolvent kernel estimate
- Long-time dynamics of partially dissipative hyperbolic systems with non-autonomous coefficients
- Dissipative Systems Analysis and Control. Theory and Applications: Erratum/Addendum. Springer Nature Switzerland AG, Communications and Control Engineering, Third uncorrected Edition, 2020
- A complete classification for n-species Lotka-Volterra stably dissipative systems based on a generalized AHU algorithm
- Adiabatic Invariant Actions for Partially Integrable Systems
BibTeX
@misc{muha2026range,
title = {{Range Failure, Resolvent Growth, and Resonance in Partially Dissipative Systems}},
author = {Boris Muha and Sebastian Schwarzacher and Justin T. Webster},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.17712},
eprint = {2609.17712},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17712},
}
RIS
TY - UNPB
TI - Range Failure, Resolvent Growth, and Resonance in Partially Dissipative Systems
AU - Boris Muha
AU - Sebastian Schwarzacher
AU - Justin T. Webster
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2609.17712
UR - https://arxiv.org/abs/2609.17712
AB - We introduce several notions of resonance for systems exhibiting weak or partial dissipation: u̇=Au+f(t), with A the generator of a strongly stable semigroup S(t) on a Hilbert space H. We refer to resonance as the phenomenon where a time-periodic f may yield an unbounded u, classically occurring when A has imaginary eigenvalues. In infinite dimensions, however, unbounded growth may depend on topological choices, and several ``resonances" may occur. Classically, existence of T-periodic u under T-periodic f is equivalent to a range condition: ℛ(I−S(T))=H. Consequently, its failure permits unbounded solution growth and we elucidate that connection through properties of S(t). We describe a hierarchy of range failures, and provide an associated resonance taxonomy. The growth rate of ~||(A−λI)^−1||_H on ~iR dictates a regularity gap between u and f, and, for rapid growth in λ, that gap can be infinite. In this case, we demonstrate implications for periodic solvability, and a mechanism for constructing smooth resonant forces. The theory developed here is motivated by (and demonstrated for) a hyperbolic-parabolic system, where the latter component provides the only system dissipation. Such dynamics are a simplification of fluid-structure phenomena, which demonstrate strong stability but do not provide unconditional periodic well-posedness. Though point-spectral resonance is ruled out, we note that ℛ(I−S(T))≠ H. Our main result here shows that, for a carefully constructed geometry, a classical heat-wave system possesses periods T which yield infinite derivative loss and, via supporting results, yield resonance with smooth forcing.
ER -