Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · math/0602468

A new uniqueness criterion for the number of periodic orbits of Abel equations

M. J. Alvarez·A. Gasull·H. Giacomini
View original source ↗Read source document ↗Export metadata ↓

Abstract

A solution of the Abel equation $\dot{x}=A(t)x^3+B(t)x^2$ such that $x(0)=x(1)$ is called a periodic orbit of the equation. Our main result proves that if there exist two real numbers $a$ and $b$ such that the function $aA(t)+bB(t)$ is not identically zero, and does not change sign in $[0,1]$ then the Abel differential equation has at most one non-zero periodic orbit. Furthermore, when this periodic orbit exists, it is hyperbolic. This result extends the known criteria about the Abel equation that only refer to the cases where either $A(t)\not\equiv0$ or $B(t)\not\equiv0$ does not change sign. We apply this new criterion to study the number of periodic solutions of two simple cases of Abel equations: the one where the functions $A(t)$ and $B(t)$ are 1-periodic trigonometric polynomials of degree one and the case where these two functions are polynomials with three monomials. Finally, we give an upper bound for the number of isolated periodic orbits of the general Abel equation $\dot{x}=A(t)x^3+B(t)x^2+C(t)x,$ when $A(t), B(t)$ and $C(t)$ satisfy adequate conditions.

Explore related subjects

math.DS

Keep this discovery

BibTeXRIS

M. J. Alvarez, A. Gasull, H. Giacomini. 2006-02-21. A new uniqueness criterion for the number of periodic orbits of Abel equations. https://arxiv.org/abs/math/0602468

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

KEEP EXPLORING

Related papers

arXiv2026

Admissible Fourier Lengths, KAM Reducibility, and Spectral Applications

We develop a perturbative KAM reducibility theory for one-frequency $\mathrm{SL}(2,\mathbb{R})$ cocycles based on an admissible Fourier length $\ell$. The regularity relevant to the iteration is measured by positive adapted Fourier width rather than ordinary smoothness in the Euclidean length $|n|$. The same length governs Fourier decay, truncation and resonance scales, and the arithmetic condition controlling the small divisors. This framework contains the classical analytic and Gevrey settings, while non-monotone choices of $\ell$ allow classical nowhere differentiable Weierstrass-type perturbations and continuous perturbations outside every positive H\"older class. As spectral applications, we obtain purely absolutely continuous spectrum for every phase and $1/2$-H\"older continuity of the integrated density of states for the associated quasiperiodic Schr\"odinger operators. The Aubry dual has pure point spectrum for Lebesgue almost every dual phase, with eigenfunctions exponentially localized in the metric induced by $\ell$. We also construct nowhere differentiable quasiperiodic potentials with purely absolutely continuous Cantor spectrum.

math.DS↗
arXiv2026

Dynamics inside the attracting basins of some skew products

Polynomial skew products in $\mathbb{C}^2$ are maps of the form $F(z,w)=(P(z),Q(z,w))$, where $P$ and $Q$ are polynomials. Their local dynamics have been widely investigated. In this paper, we study the global dynamics inside Fatou components of some skew products. We consider all the inverse images in a Fatou component of a given point and use the Kobayashi metric to measure the distance between points. In the cases we consider, there are always arbitrarily large Kobayashi balls in the complement of these inverse sets.

math.DS↗
arXiv2026

Ergodicity of dynamical systems without uniqueness of orbits

Recently, there has been considerable interest in the study of non-deterministic dynamical systems. To analyze the chaotic behavior of such systems from a measure-theoretic viewpoint, it is desirable to consider ergodicity. However, the classical definition of ergodicity involves invariant sets, whose definition is not unique for non-deterministic dynamical systems. Thus, we are led to the question of which invariance yields an interesting definition of ergodicity. Here, we propose a definition based on the strong backward invariance and show that analogs of classical results hold. We also consider implications of the Birkhoff ergodic theorem for systems without uniqueness of orbits.

math.DS↗