Volume 28, Number 3

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].

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 quasi-periodic 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 graph-transform 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 normal-form 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].

We discuss the holomorphic properties of the complex continuation of the classical Arnol'd – Liouville action-angle variables for real analytic 1 degree-of-freedom 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 action-angle 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.

In this paper, we provide a rigorous computer-assisted proof (CAP) of the conjecture that in the planar four-body 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.

This paper is a summary of results that prove the abundance of one-dimensional 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 two-dimensional 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 $n\geqslant 4$.

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 two-dimensional torus showing that our results do not necessarily hold when the cover is not infinite cyclic.

A brake orbit for the N-body 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 N-body problem at negative energies.

Affine transformations in Euclidean space generate a correspondence between integrable systems on cotangent bundles to a sphere, ellipsoid and hyperboloid embedded in $R^n$. Using this correspondence and the suitable coupling constant transformations, we can get real integrals of motion in the hyperboloid case starting with real integrals of motion in the sphere case. We discuss a few such integrable systems with invariants which are cubic, quartic and sextic polynomials in momenta.

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.

In this note we introduce the V-shaped action functional with delay in a symplectization, which is an intermediate action functional between the Rabinowitz action functional and the V-shaped action functional. It lives on the same space as the V-shaped 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 V-shaped action functional and the V-shaped 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 V-shaped action functional with delay and the ones of the Rabinowitz action functional.