Volume 28, Numbers 45
Volume 28, Numbers 45, 2023
Special Issue: On the 80th Birthday of Professor A. Chenciner
Le Calvez P.
Abstract
We introduce the notion of abstract angle at a couple of points defined by two
radial foliations of the closed annulus. We will use for this purpose the digital line topology on
the set ${\mathbb Z}$ of relative integers, also called the Khalimsky topology. We use this notion to give
unified proofs of some classical results on area preserving positive twist maps of the annulus
by using the Lifting Theorem and the Intermediate Value Theorem. More precisely, we will
interpretate Birkhoff theory about annular invariant open sets in this formalism. Then we give
a proof of Mather’s theorem stating the existence of crossing orbits in a Birkhoff region of
instability. Finally we will give a proof of Poincaré – Birkhoff theorem in a particular case, that
includes the case where the map is a composition of positive twist maps.

Frauenfelder U.
Abstract
In this note we introduce the Vshaped action functional with delay in a symplectization,
which is an intermediate action functional between the Rabinowitz action functional and
the Vshaped action functional. It lives on the same space as the Vshaped action functional, but
its gradient flow equation is a delay equation as in the case of the Rabinowitz action functional.
We show that there is a smooth interpolation between the Vshaped action functional and the
Vshaped action functional with delay during which the critical points and its actions are fixed.
Moreover, we prove that there is a bijection between gradient flow lines of the Vshaped action
functional with delay and the ones of the Rabinowitz action functional.

Montgomery R.
Abstract
A brake orbit for the Nbody problem is a solution for which, at some instant, all
velocities of all bodies are zero. We reprove two “lost theorems” regarding brake orbits and
use them to establish some surprising properties of the completion of the Jacobi – Maupertuis
metric for the Nbody problem at negative energies.

Biasco L., Chierchia L.
Abstract
We discuss the holomorphic properties of the complex continuation of the classical
Arnol’d – Liouville actionangle variables for real analytic 1 degreeoffreedom Hamiltonian
systems depending on external parameters in suitable Generic Standard Form, with particular
regard to the behaviour near separatrices. In particular, we show that near separatrices the
actions, regarded as functions of the energy, have a special universal representation in terms
of affine functions of the logarithm with coefficients analytic functions. Then, we study the
analyticity radii of the actionangle variables in arbitrary neighborhoods of separatrices and
describe their behaviour in terms of a (suitably rescaled) distance from separatrices. Finally, we
investigate the convexity of the energy functions (defined as the inverse of the action functions)
near separatrices, and prove that, in particular cases (in the outer regions outside the main
separatrix, and in the case the potential is close to a cosine), the convexity is strictly defined,
while in general it can be shown that inside separatrices there are inflection points.

Fathi A., Pageault P.
Abstract
In this paper, we study the projected Aubry set of a lift of a Tonelli Lagrangian
$L$ defined on the tangent bundle of a compact manifold $M$ to an infinite cyclic covering of $M$.
Most of weak KAM and Aubry – Mather theory can be done in this setting. We give a necessary
and sufficient condition for the emptiness of the projected Aubry set of the lifted Lagrangian
involving both Mather minimizing measures and Mather classes of $L$. Finally, we give Mañè
examples on the twodimensional torus showing that our results do not necessarily hold when
the cover is not infinite cyclic.

Massetti J.
Abstract
In studying general perturbations of a dissipative twist map depending on two parameters, a frequency $\nu$ and a dissipation $\eta$, the existence of a Cantor set $\mathcal C$ of curves in the $(\nu,\eta)$ plane such that the corresponding equation possesses a Diophantine quasiperiodic invariant circle can be deduced, up to small values of the dissipation, as a direct consequence of a normal form theorem in the spirit of Rüssmann and the ``elimination of parameters'' technique. These circles are normally hyperbolic as soon as $\eta\not=0$, which implies that the equation still possesses a circle of this kind for values of the parameters belonging to a neighborhood $\mathcal V$ of this set of curves. Obviously, the dynamics on such invariant circles is no more controlled and may be generic, but the normal dynamics is controlled in the sense of their basins of attraction.
As expected, by the classical graphtransform method we are able to determine a first rough region where the normal hyperbolicity prevails and a circle persists, for a strong enough dissipation $\eta\sim O(\sqrt{\varepsilon}),$ $\varepsilon$ being the size of the perturbation. Then, through normalform techniques, we shall enlarge such regions and determine such a (conic) neighborhood $\mathcal V$, up to values of dissipation of the same order as the perturbation, by using the fact that the proximity of the set $\mathcal C$
allows, thanks to Rüssmann's translated curve theorem, an introduction of local coordinates of the type (dissipation, translation) similar to the ones introduced by Chenciner in [7].

Rodríguez J.
Abstract
This paper is a summary of results that prove the abundance of onedimensional
strange attractors near a Shil’nikov configuration, as well as the presence of these configurations
in generic unfoldings of singularities in $\mathbb{R}^{3}$ of minimal codimension. Finding these singularities
in families of vector fields is analytically possible and thus provides a tractable criterion
for the existence of chaotic dynamics. Alternative scenarios for the possible abundance of
twodimensional attractors in higher dimension are also presented. The role of Shil’nikov
configuration is now played by a certain type of generalised tangency which should occur for
families of vector fields $X_{\mu }$ unfolding generically some low codimension singularity in $\mathbb{R}^{n}$ with
$n\geqslant 4$.

Fontana McNally J., Miranda E., Oms C., PeraltaSalas D.
Abstract
In this short note, we prove that singular Reeb vector fields associated with generic $b$contact forms on three dimensional manifolds with compact embedded critical surfaces have either (at least) $2N$ or an infinite number of escape orbits, where $N$ denotes the number of connected components of the critical set. In case where the first Betti number of a connected component of the critical surface is positive, there exist infinitely many escape orbits. A similar result holds in the case of $b$Beltrami vector fields that are not $b$Reeb. The proof is based on a more detailed analysis of the main result in [19].

Sun S., Xie Z., You P.
Abstract
In this paper, we provide a rigorous computerassisted proof (CAP) of the conjecture
that in the planar fourbody problem there exists a unique convex central configuration for
any four fixed positive masses in a given order belonging to a closed domain in the mass space.
The proof employs the Krawczyk operator and the implicit function theorem (IFT). Notably,
we demonstrate that the implicit function theorem can be combined with interval analysis,
enabling us to estimate the size of the region where the implicit function exists and extend our
findings from one mass point to its neighborhood.

Moeckel R.
Abstract
For total collision solutions of the $n$body problem, Chazy showed that the overall size of the configuration converges to zero with asymptotic rate proportional to $T − t^{\frac{2}{3}}$ where $T$ is the collision time. He also showed that the shape of the configuration converges to the set of central configurations. If the limiting central configuration is nondegenerate, the rate of convergence of the shape is of order $O(T − t^p)$ for some $p > 0$. Here we show by example that in the planar fourbody problem there exist total collision solutions whose shape converges to a degenerate central configuration at a rate which is slower that any power of $T − t$.

Berti M., Maspero A., Murgante F.
Abstract
We present the almost global in time existence result in [13] of small amplitude space periodic solutions of the 1D gravitycapillary water waves equations with constant vorticity and we describe the ideas of proof. This is based on a novel Hamiltonian paradifferential Birkhoff normal form approach for quasilinear PDEs.

Chen K., Pan B.
Abstract
In this paper we provide estimates for mutual distances of periodic solutions for the Newtonian $N$body problem. Our estimates are based on masses, total variations of turning angles for relative positions, and predetermined upper bounds for action values. Explicit formulae will be proved by iterative arguments. We demonstrate some applications to actionminimizing solutions for three and fourbody problems.

Féjoz J., Guardia M.
Abstract
Chirikov’s celebrated criterion of resonance overlap has been widely used in celestial
mechanics and Hamiltonian dynamics to detect global instability, but is rarely rigorous. We
introduce two simple Hamiltonian systems, each depending on two parameters measuring,
respectively, the distance to resonance overlap and nonintegrability. Within some thin region
of the parameter plane, classical perturbation theory shows the existence of global instability
and symbolic dynamics, thus illustrating Chirikov’s criterion.

Gao Y., Neishtadt A. I., Okunev A.
Abstract
We consider a slowfast Hamiltonian system with one fast angle variable (a fast phase) whose frequency vanishes on some surface
in the space of slow variables (a resonant surface). Systems of such form appear in the study of dynamics of charged particles
in an inhomogeneous magnetic field
under the influence of highfrequency electrostatic waves. Trajectories of the system averaged over the fast phase cross the resonant surface.
The fast phase makes $\sim \frac 1\varepsilon$ turns before arrival at the resonant surface ($\varepsilon$
is a small parameter of the problem). An asymptotic formula for the value of the phase at the arrival at the resonance
was derived earlier in the context of study of charged particle dynamics on the basis of heuristic
considerations without any estimates of its accuracy. We provide a rigorous derivation of this formula and prove
that its accuracy is $O(\sqrt \varepsilon)$ (up to a logarithmic correction). This estimate for the accuracy is optimal.

GrycanGérard F., Marco J.
Abstract
Given a continuous fibered dynamical system, we first introduce the notion of polynomial torsion of a fiber,
which measures the ``infinitesimal variation'' of the dynamics between the fiber and the neighboring ones.
This gives rise to an (upper semicontinous) torsion function,
defined on the base of the system, which is a new
$C^0$ (fiber) conjugacy invariant. We prove that the polynomial entropy of the system is the supremum of
the torsion of its fibers, which yields a new insight into the creation of polynomial entropy in fibered systems.
We examine the relevance of these results in the context of integrable Hamiltonian
systems or diffeomorphisms, with the particular cases of $C^0$integrable twist maps on the annulus and geodesic flows.
Finally, we bound from below the polynomial entropy of $\ell$modal interval maps in terms of their lap number and answer a question by Gomes and Carneiro.

Knauf A., Montgomery R.
Abstract
For $n$ bodies moving in Euclidean $d$space under the influence of a homogeneous pair interaction we compactify every center of mass energy surface, obtaining a $\big(2d(n1)1\big)$dimensional manifold with corners in the sense of Melrose. After a time change, the flow on this manifold is globally defined and nontrivial on the boundary.

Alsedà L., Juher D., Los J., Mañosas F.
Abstract
We review some recent results on a class of maps, called Bowen – Serieslike maps,
obtained from a class of group presentations for surface groups. These maps are piecewise
homeomorphisms of the circle with finitely many discontinuities. The topological entropy of
each map in the class and its relationship with the growth function of the group presentation
is discussed, as well as the computation of these invariants.

Ureña A. J.
Abstract
The Lambert problem consists in connecting two given points in a given lapse of
time under the gravitational influence of a fixed center. While this problem is very classical, we
are concerned here with situations where friction forces act alongside the Newtonian attraction.
Under some boundedness assumptions on the friction, there exists exactly one rectilinear
solution if the two points lie on the same ray, and at least two solutions traveling in opposite
directions otherwise.

Fujiwara T., PérezChavela E.
Abstract
We study relative equilibria ($RE$) for the threebody problem on $\mathbb{S}^2$, under the influence of a general potential which only depends on $\cos\sigma_{ij}$ where $\sigma_{ij}$ are the mutual angles among the masses. Explicit conditions for masses $m_k$ and $\cos\sigma_{ij}$ to form relative equilibrium are shown. Using the above conditions, we study the equal masses case under the cotangent potential. We show the existence of scalene, isosceles, and equilateral Euler $RE$, and isosceles and equilateral Lagrange $RE$. We also show that the equilateral Euler $RE$ on a rotating meridian exists for general potential $\sum_{i\lt j} m_i m_j U(\cos \sigma_{ij})$ with any mass ratios.

Bustamante A., de la Llave R.
Abstract
We consider standardlike/Froeschl\'e dissipative maps
with a dissipation and nonlinear perturbation. That is,
\[
T_\varepsilon(p,q) = \left(
(1  \gamma \varepsilon^3) p + \mu + \varepsilon V'(q),
q + (1  \gamma \varepsilon^3) p + \mu + \varepsilon V'(q) \bmod 2 \pi \right)
\]
where $p \in {\mathbb R}^D$, $q \in {\mathbb T}^D$ are the dynamical
variables. We fix a frequency $\omega \in {\mathbb R}^D$ and study the existence of
quasiperiodic orbits. When there is dissipation, having
a quasiperiodic orbit of frequency $\omega$ requires
selecting the parameter $\mu$, called \textit{the drift}.
We first study the Lindstedt series (formal power series in $\varepsilon$) for quasiperiodic orbits with $D$ independent frequencies and the drift when $\gamma \ne 0$.
We show that, when $\omega$ is
irrational, the series exist to all orders, and when $\omega$ is Diophantine,
we show that the formal Lindstedt series are Gevrey.
The Gevrey nature of the Lindstedt series above was shown
in~\cite{BustamanteL22} using a more general method, but the present proof is
rather elementary.
We also study the case when $D = 2$, but the quasiperiodic orbits
have only one independent frequency (lowerdimensional tori).
Both when $\gamma = 0$ and when $\gamma \ne 0$, we show
that, under some mild nondegeneracy conditions on $V$, there
are (at least two) formal Lindstedt series defined to all orders
and that they are Gevrey.

Hu X., Ou Y., Tang X.
Abstract
It is well known that a planar central configuration of the $n$body problem gives rise to a solution where each particle moves in a Keplerian orbit with a common eccentricity $\mathfrak{e}\in[0,1)$. We call this solution an elliptic relative equilibrium (ERE for short). Since each particle of the ERE is always in the same plane, it is natural to regard it as a planar $n$body problem. But in practical applications, it is more meaningful to consider the ERE as a spatial $n$body problem (i.\,e., each particle belongs to $\mathbb{R}^3$). In this paper, as a spatial $n$body problem, we first decompose the linear system of ERE into two parts, the planar and the spatial part. Following the Meyer\,\,Schmidt coordinate~\cite{Meyer}, we give an expression for the spatial part and further obtain a rigorous analytical method to study the linear stability of the spatial part by the Maslovtype index theory. As an application, we obtain stability results for some classical ERE, including the elliptic Lagrangian solution, the Euler solution and the $1+n$gon solution.

Bernard P.
Abstract
The purpose of this paper is a pedagogical one. We provide a short and selfcontained
account of Siegel’s theorem, as improved by Bruno, which states that a holomorphic
map of the complex plane can be locally linearized near a fixed point under certain conditions
on the multiplier. The main proof is adapted from Bruno’s work.

De Maesschalck P., Dumortier F., Roussarie R.
Abstract
The paper deals with multilayer canard cycles, extending the results of [1]. As a practical tool we introduce the connection diagram of a canard cycle and we show how to determine it in an easy way. This connection diagram presents in a clear way all available information that is necessary to formulate the main system of equations used in the study of the bifurcating limit cycles. In a forthcoming paper we will show that both the type of the layers and the nature of the connections between the layers play an essential role in determining the number and the bifurcations of the limit cycles that can be created from a canard cycle.

Treschev D. V.
Abstract
We propose a new approach to the theory of normal forms for Hamiltonian systems
near a nonresonant elliptic singular point. We consider the space of all Hamiltonian functions
with such an equilibrium position at the origin and construct a differential equation in this space.
Solutions of this equation move Hamiltonian functions towards their normal forms. Shifts along
the flow of this equation correspond to canonical coordinate changes. So, we have a continuous
normalization procedure. The formal aspect of the theory presents no difficulties. As usual, the
analytic aspect and the problems of convergence of series are nontrivial.
