Volume 28, Numbers 4-5

Volume 28, Numbers 4-5, 2023
Special Issue: On the 80th Birthday of Professor A. Chenciner

Citation: To Alain Chenciner. On his 80th Birthday, Regular and Chaotic Dynamics, 2023, vol. 28, nos. 4-5, pp. 333-342
Le Calvez P.
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.
Keywords: positive twist map, Birkhoff theory, region of instability, periodic orbit
Citation: Le Calvez P., Twist Maps of the Annulus: An Abstract Point of View, Regular and Chaotic Dynamics, 2023, vol. 28, nos. 4-5, pp. 343-363
Frauenfelder U.
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.
Keywords: Symplectic homology, Rabinowitz – Floer homology, delay equation
Citation: Frauenfelder U., V-Shaped Action Functional with Delay, Regular and Chaotic Dynamics, 2023, vol. 28, nos. 4-5, pp. 364-373
Montgomery R.
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.
Keywords: N-body problem, Hill region, Jacobi – Maupertuis metric, brake orbit, zero velocity surface
Citation: Montgomery R., Brake Orbits Fill the N-body Hill Region, Regular and Chaotic Dynamics, 2023, vol. 28, nos. 4-5, pp. 374-394
Biasco L.,  Chierchia L.
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.
Keywords: Hamiltonian systems, action-angle variables, Arnol’d – Liouville integrable systems, complex extensions of symplectic variables, KAM theory
Citation: Biasco L.,  Chierchia L., Complex Arnol'd – Liouville Maps, Regular and Chaotic Dynamics, 2023, vol. 28, nos. 4-5, pp. 395-424
Fathi A.,  Pageault P.
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.
Keywords: Aubry – Mather theory, weak KAM theory, infinite cyclic coverings
Citation: Fathi A.,  Pageault P., Aubry Set on Infinite Cyclic Coverings, Regular and Chaotic Dynamics, 2023, vol. 28, nos. 4-5, pp. 425-446
Massetti  J.
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].
Keywords: nonconservative twist maps, invariant circles, elimination of parameters, normal forms
Citation: Massetti  J., Attractive Invariant Circles à la Chenciner, Regular and Chaotic Dynamics, 2023, vol. 28, nos. 4-5, pp. 447-467
Rodríguez J.
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 $n\geqslant 4$.
Keywords: Shil’nikov orbits, strange attractors, unfolding of a singularity, expanding baker maps, two-dimensional strange attractors
Citation: Rodríguez J., Emergence of Strange Attractors from Singularities, Regular and Chaotic Dynamics, 2023, vol. 28, nos. 4-5, pp. 468-497
Fontana McNally J.,  Miranda E.,  Oms C.,  Peralta-Salas D.
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].
Keywords: contact geometry, Beltrami vector fields, escape orbits, celestial mechanics
Citation: Fontana McNally J.,  Miranda E.,  Oms C.,  Peralta-Salas D., From $2N$ to Infinitely Many Escape Orbits, Regular and Chaotic Dynamics, 2023, vol. 28, nos. 4-5, pp. 498-511
Sun S.,  Xie Z.,  You P.
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.
Keywords: central configuration, convex central configuration, uniqueness, $N$-body problem, Krawczyk operator, implicit function theorem
Citation: Sun S.,  Xie Z.,  You P., On the Uniqueness of Convex Central Configurations in the Planar $4$-Body Problem, Regular and Chaotic Dynamics, 2023, vol. 28, nos. 4-5, pp. 512-532
Moeckel R.
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 four-body problem there exist total collision solutions whose shape converges to a degenerate central configuration at a rate which is slower that any power of $|T − t|$.
Keywords: celestial mechanics, $n$-body problem, total collision
Citation: Moeckel R., Total Collision with Slow Convergence to a Degenerate Central Configuration, Regular and Chaotic Dynamics, 2023, vol. 28, nos. 4-5, pp. 533-542
Berti M.,  Maspero A.,  Murgante F.
We present the almost global in time existence result in [13] of small amplitude space periodic solutions of the 1D gravity-capillary 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 quasi-linear PDEs.
Keywords: water waves equations, vorticity, Hamiltonian Birkhoff normal form, paradifferential calculus
Citation: Berti M.,  Maspero A.,  Murgante F., Hamiltonian Paradifferential Birkhoff Normal Form for Water Waves, Regular and Chaotic Dynamics, 2023, vol. 28, nos. 4-5, pp. 543-560
Chen K.,  Pan B.
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 four-body problems.
Keywords: $N$-body problem, variational method, mutual distance
Citation: Chen K.,  Pan B., Distance Estimates for Action-Minimizing Solutions of the $N$-Body Problem, Regular and Chaotic Dynamics, 2023, vol. 28, nos. 4-5, pp. 561-577
Féjoz J.,  Guardia M.
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.
Keywords: Hamiltonian system, instability, resonance overlap, nonintegrability, Chirikov’s criterion, symbolic dynamics
Citation: Féjoz J.,  Guardia M., A Remark on the Onset of Resonance Overlap, Regular and Chaotic Dynamics, 2023, vol. 28, nos. 4-5, pp. 578-584
Gao Y.,  Neishtadt A. I.,  Okunev A.
We consider a slow-fast 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 high-frequency 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.
Keywords: slow-fast systems, averaging method, resonance
Citation: Gao Y.,  Neishtadt A. I.,  Okunev A., On Phase at a Resonance in Slow-Fast Hamiltonian Systems, Regular and Chaotic Dynamics, 2023, vol. 28, nos. 4-5, pp. 585-612
Grycan-Gérard F.,  Marco J.
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.
Keywords: zero entropy, polynomial entropy, integrability, fibered system
Citation: Grycan-Gérard F.,  Marco J., Polynomial Entropy and Polynomial Torsion for Fibered Systems, Regular and Chaotic Dynamics, 2023, vol. 28, nos. 4-5, pp. 613-627
Knauf A.,  Montgomery R.
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 $\big(2d(n-1)-1\big)$-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.
Keywords: regularization, scattering, collision
Citation: Knauf A.,  Montgomery R., Compactification of the Energy Surfaces for $n$ Bodies, Regular and Chaotic Dynamics, 2023, vol. 28, nos. 4-5, pp. 628-658
Alsedà L.,  Juher D.,  Los J.,  Mañosas F.
We review some recent results on a class of maps, called Bowen – Series-like 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.
Keywords: Surface groups, Bowen – Series maps, topological entropy, volume entropy
Citation: Alsedà L.,  Juher D.,  Los J.,  Mañosas F., On Families of Bowen – Series-Like Maps for Surface Groups, Regular and Chaotic Dynamics, 2023, vol. 28, nos. 4-5, pp. 659-667
Ureña A. J.
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.
Keywords: Kepler problem, Dirichlet boundary conditions, friction
Citation: Ureña A. J., On the Lambert Problem with Drag, Regular and Chaotic Dynamics, 2023, vol. 28, nos. 4-5, pp. 668-689
Fujiwara T.,  Pérez-Chavela E.
We study relative equilibria ($RE$) for the three-body 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.
Keywords: relative equilibria, Euler and Lagrange configurations
Citation: Fujiwara T.,  Pérez-Chavela E., Three-Body Relative Equilibria on $\mathbb{S}^2$, Regular and Chaotic Dynamics, 2023, vol. 28, nos. 4-5, pp. 690-706
Bustamante A.,  de la Llave R.
We consider standard-like/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 quasi-periodic orbits. When there is dissipation, having a quasi-periodic 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 quasi-periodic 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 quasi-periodic orbits have only one independent frequency (lower-dimensional 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.
Keywords: Lindstedt series, Gevrey series, asymptotic expansions, resonances, whiskered tori
Citation: Bustamante A.,  de la Llave R., A Simple Proof of Gevrey Estimates for Expansions of Quasi-Periodic Orbits: Dissipative Models and Lower-Dimensional Tori, Regular and Chaotic Dynamics, 2023, vol. 28, nos. 4-5, pp. 707-730
Hu X.,  Ou Y.,  Tang X.
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 Maslov-type 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.
Keywords: linear stability, elliptic relative equilibrium, Maslov-type index, spatial $n$-body problem
Citation: Hu X.,  Ou Y.,  Tang X., Linear Stability of an Elliptic Relative Equilibrium in the Spatial $n$-Body Problem via Index Theory, Regular and Chaotic Dynamics, 2023, vol. 28, nos. 4-5, pp. 731-755
Bernard P.
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.
Keywords: linearization, normal forms
Citation: Bernard P., The Siegel – Bruno Linearization Theorem, Regular and Chaotic Dynamics, 2023, vol. 28, nos. 4-5, pp. 756-762
De Maesschalck  P.,  Dumortier F.,  Roussarie R.
The paper deals with multi-layer 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.
Keywords: multi-layer canard cycles, orientation of $SF$ and $FS$ passages, side-comparison at a canard connection, connection diagram, equations for limit cycles
Citation: De Maesschalck  P.,  Dumortier F.,  Roussarie R., Side-Comparison for Transition Maps in Multi-Layer Canard Problems, Regular and Chaotic Dynamics, 2023, vol. 28, nos. 4-5, pp. 763-780
Treschev D. V.
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.
Keywords: normal forms, Hamiltonian systems, small divisors
Citation: Treschev D. V., Normalization Flow, Regular and Chaotic Dynamics, 2023, vol. 28, nos. 4-5, pp. 781-804

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