Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2605.19493

Breaking of invariant curves: from the Fermi-Ulam map to the breathing circle billiard

José Lamas·Stefano Marò
View original source ↗Read source document ↗Export metadata ↓

Abstract

We consider the breathing circle billiard, in which a point particle moves freely inside a disk. The radius varies periodically in time, with elastic reflections at the moving boundary. In this system the angular momentum is preserved, and fixing its value $c$ reduces the dynamics to a two-dimensional exact symplectic map on a cylinder. In the high-energy regime this map is a twist map generated by a diagonally periodic generating function $h_c$. We study the small angular momentum regime as a perturbation of the limiting case $c=0$, which corresponds to the Fermi-Ulam dynamics along a diameter. Using this perturbative structure and a quantitative version of Mather's converse-KAM criterion, we exclude invariant Lipschitz graphs for suitable rotation numbers. Combined with Aubry-Mather theory and Forni's theorem, this yields positive topological entropy for sufficiently small $c\geq0$. Our result improves previous similar results obtained via the standard Mather's converse-KAM criterion by giving a sharper quantitative threshold for the destruction of invariant curves.

Explore related subjects

math.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

José Lamas, Stefano Marò. 2026-05-19. Breaking of invariant curves: from the Fermi-Ulam map to the breathing circle billiard. https://arxiv.org/abs/2605.19493

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↗