Corrado Falcolini | Searcharxiv

arXiv2026

Many coexisting attractors, a case study of the almost-conservative H\'enon map

For dynamical systems in the plane, there can be many periodic attractors coexisting in a bounded region. They become easier to find in systems with small dissipation, which we call ``almost-conservative''. We ask what happens when there are many periodic attractors. That is the vague question we start with. For a test study, we chose the H\'enon map with a tiny dissipation. We tuned the other parameter to yield a case with 50 attracting periodic orbits. They have a total of 4259 periodic points. We describe how these orbits can be organized into families. In addition to two low-period orbits, the remaining 48 orbits can be classified into three families, which we describe in detail.

math.DS↗

arXiv2025

On the analytic properties of the perturbing function in the PCR3Body Problem

We provide a new expansion of the Fourier coefficient of the Perturbing function of the PCR3Body problem in terms of Hansen Coefficients. This gives us a precise asymptotic formula for the coefficient in the region of application of KAM theory (i.e small value of eccentricity and semi-major axis see e.g. \cite{Celletti-Chierchia}). Moreover, in the above region, we study the presence of zeros of the Fourier coefficient for coprime modes $(m,k) \in \Z^2$ and the presence of common zeros between coefficients relative to modes $(m,k)$,$(2m,2k)$ and $(m,k)$,$(2m,2k)$,$(3m,3k)$. Thanks to the previous expansion, this numerical analysis is done up to order $60$ in the power of eccentricity and semimajor axis. This is a first step for a possible application of \cite{Singular KAM, BBCZ} to PCR3Body Problem that would imply a reduction in terms of measure in the phase space of the so called "non--torus" set from $O(1-\sqrt{\e})$ (implied by standard KAM theory) to $O(1-\e |\log\e|^c )$ for some $c>0$.

math.DS↗