Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2601.21073

Dynamics of Newton's method for odd and even elliptic functions

Adrián Esparza-Amador·Mónica Moreno Rocha
View original source ↗Read source document ↗Export metadata ↓

Abstract

We investigate Newton's method applied to any odd or any even elliptic function with an arbitrary period lattice. For any function of this type whose set of poles coincides with its period lattice, we show that the Julia set of its Newton map is connected, as long as no Herman rings exist. Moreover, we provide sufficient conditions on the Newton's method of any odd or even elliptic function to exhibit wandering domains coexisting with attracting basins. This phenomenon was first reported by Florido and Fagella for the Newton's method applied to a family of entire functions; however, our approach does not employ the lifting technique. We also provide a detailed study of a one-parameter family of elliptic functions given by $\wp_Λ+b$ with $Λ$ any triangular period lattice and $b\in\mathbb{C}$. We show that their associated Newton maps do not exhibit Herman rings or Baker domains, and any other Fatou component, including wandering, is bounded.

Explore related subjects

math.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Adrián Esparza-Amador, Mónica Moreno Rocha. 2026-01-28. Dynamics of Newton's method for odd and even elliptic functions. https://arxiv.org/abs/2601.21073

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↗