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# Volume 18, Number 6, 2013 On the 70th birthday of professor A. Chenciner

 Alain Chenciner. On his 70th birthday
 Ekeland I.,  Long Y.,  Zhou Q. A New Class of Problems in the Calculus of Variations Abstract This paper investigates an infinite-horizon problem in the one-dimensional calculus of variations, arising from the Ramsey model of endogeneous economic growth. Following Chichilnisky, we introduce an additional term, which models concern for the well-being of future generations. We show that there are no optimal solutions, but that there are equilibrium strateges, i.e. Nash equilibria of the leader-follower game between successive generations. To solve the problem, we approximate the Chichilnisky criterion by a biexponential criterion, we characterize its equilibria by a pair of coupled differential equations of HJB type, and we go to the limit. We find all the equilibrium strategies for the Chichilnisky criterion. The mathematical analysis is difficult because one has to solve an implicit differential equation in the sense of Thom. Our analysis extends earlier work by Ekeland and Lazrak. Keywords: minimization problem, sustainable economy, time-inconsistency, existence Citation: Ekeland I.,  Long Y.,  Zhou Q., A New Class of Problems in the Calculus of Variations, Regular and Chaotic Dynamics, 2013, vol. 18, no. 6, pp. 553-584 DOI:10.1134/S1560354713060014
 Ortega R. Stable Periodic Solutions in the Forced Pendulum Equation Abstract Consider the pendulum equation with an external periodic force and an appropriate condition on the length parameter. It is proved that there exists at least one stable periodic solution for almost every external force with zero average. The stability is understood in the Lyapunov sense. Keywords: Lyapunov stability, forced pendulum, prevalence, periodic solution, regular value, discriminant Citation: Ortega R., Stable Periodic Solutions in the Forced Pendulum Equation, Regular and Chaotic Dynamics, 2013, vol. 18, no. 6, pp. 585-599 DOI:10.1134/S1560354713060026
 Montgomery R. MICZ-Kepler: Dynamics on the Cone over $SO(n)$ Abstract We show that the $n$-dimensional MICZ-Kepler system arises from symplectic reduction of the "Kepler problem" on the cone over the rotation group $SO(n)$. As a corollary we derive an elementary formula for the general solution of the MICZ-Kepler problem. The heart of the computation is the observation that the additional MICZ-Kepler potential, $|\phi|^2/r^2$, agrees with the rotational part of the cone’s kinetic energy. Keywords: Kepler problem, MICZ-K system, co-adjoint orbit, Sternberg phase space, symplectic reduction, superintegrable systems Citation: Montgomery R., MICZ-Kepler: Dynamics on the Cone over $SO(n)$, Regular and Chaotic Dynamics, 2013, vol. 18, no. 6, pp. 600-607 DOI:10.1134/S1560354713060038
 Zung N.,  Minh T. Commuting Foliations Abstract The aim of this paper is to extend the notion of commutativity of vector fields to the category of singular foliations using Nambu structures, i.e., integrable multi-vector fields. We will classify the relationship between singular foliations and Nambu structures and show some basic results about commuting Nambu structures. Keywords: commuting foliations, integrable differential forms, Nambu structures Citation: Zung N.,  Minh T., Commuting Foliations, Regular and Chaotic Dynamics, 2013, vol. 18, no. 6, pp. 608-622 DOI:10.1134/S156035471306004X
 Marco J. Polynomial Entropies and Integrable Hamiltonian Systems Abstract We introduce two numerical conjugacy invariants of dynamical systems — the polynomial entropy and the weak polynomial entropy — which are well-suited for the study of "completely integrable" Hamiltonian systems. These invariants describe the polynomial growth rate of the number of balls (for the usual "dynamical" distances) of covers of the ambient space. We give explicit examples of computation of these polynomial entropies for generic Hamiltonian systems on surfaces. Keywords: dynamical complexity, entropy, integrability, Morse Hamiltonians Citation: Marco J., Polynomial Entropies and Integrable Hamiltonian Systems, Regular and Chaotic Dynamics, 2013, vol. 18, no. 6, pp. 623-655 DOI:10.1134/S1560354713060051
 Maderna E. Minimizing Configurations and Hamilton–Jacobi Equations of Homogeneous $N$-body Problems Abstract For $N$-body problems with homogeneous potentials we define a special class of central configurations related with the reduction of homotheties in the study of homogeneous weak KAM solutions. For potentials in $1/r^\alpha$ with $\alpha \in (0,2)$ we prove the existence of homogeneous weak KAM solutions. We show that such solutions are related to viscosity solutions of another Hamilton–Jacobi equation in the sphere of normal configurations. As an application we prove for the Newtonian three-body problem that there are no smooth homogeneous solutions to the critical Hamilton–Jacobi equation. Keywords: $N$-body problem, central configuration, Hamilton–Jacobi Citation: Maderna E., Minimizing Configurations and Hamilton–Jacobi Equations of Homogeneous $N$-body Problems, Regular and Chaotic Dynamics, 2013, vol. 18, no. 6, pp. 656-673 DOI:10.1134/S1560354713060063
 Bernard P. Semi-concave Singularities and the Hamilton–Jacobi Equation Abstract We study the Cauchy problem for the Hamilton–Jacobi equation with a semiconcave initial condition.We prove an inequality between two types of weak solutions emanating from such an initial condition (the variational and the viscosity solution).We also give conditions for an explicit semi-concave function to be a viscosity solution. These conditions generalize the entropy inequality characterizing piecewise smooth solutions of scalar conservation laws in dimension one. Keywords: Hamilton–Jacobi equations, viscosity solutions, variational solutions, calculus of variations Citation: Bernard P., Semi-concave Singularities and the Hamilton–Jacobi Equation, Regular and Chaotic Dynamics, 2013, vol. 18, no. 6, pp. 674-685 DOI:10.1134/S1560354713060075
 Neishtadt A. I.,  Vasiliev A. A.,  Artemyev A. V. Capture into Resonance and Escape from it in a Forced Nonlinear Pendulum Abstract We study the dynamics of a nonlinear pendulum under a periodic force with small amplitude and slowly decreasing frequency. It is well known that when the frequency of the external force passes through the value of the frequency of the unperturbed pendulum’s oscillations, the pendulum can be captured into resonance. The captured pendulum oscillates in such a way that the resonance is preserved, and the amplitude of the oscillations accordingly grows. We consider this problem in the frames of a standard Hamiltonian approach to resonant phenomena in slow-fast Hamiltonian systems developed earlier, and evaluate the probability of capture into resonance. If the system passes through resonance at small enough initial amplitudes of the pendulum, the capture occurs with necessity (so-called autoresonance). In general, the probability of capture varies between one and zero, depending on the initial amplitude. We demonstrate that a pendulum captured at small values of its amplitude escapes from resonance in the domain of oscillations close to the separatrix of the pendulum, and evaluate the amplitude of the oscillations at the escape. Keywords: autoresonance, capture into resonance, adiabatic invariant, pendulum Citation: Neishtadt A. I.,  Vasiliev A. A.,  Artemyev A. V., Capture into Resonance and Escape from it in a Forced Nonlinear Pendulum, Regular and Chaotic Dynamics, 2013, vol. 18, no. 6, pp. 686-696 DOI:10.1134/S1560354713060087
 Arnaud M. $C^1$-generic Billiard Tables have a Dense Set of Periodic Points Abstract We prove that the set of periodic points of a generic $C^1$-billiard table is dense in the phase space. Keywords: billiard map, closing lemma, periodic points, twist map Citation: Arnaud M., $C^1$-generic Billiard Tables have a Dense Set of Periodic Points, Regular and Chaotic Dynamics, 2013, vol. 18, no. 6, pp. 697-702 DOI:10.1134/S1560354713060099
 Féjoz J. On Action-angle Coordinates and the Poincaré Coordinates Abstract This article is a review of two related classical topics of Hamiltonian systems and celestial mechanics. The first section deals with the existence and construction of action-angle coordinates, which we describe emphasizing the role of the natural adiabatic invariants "$\oint_\gamma pdq$". The second section is the construction and properties of the Poincaré coordinates in the Kepler problem, adapting the principles of the former section, in an attempt to use known first integrals more directly than Poincaré did. Keywords: Hamiltonian system, Lagrangian fibration, action-angle coordinates, Liouville–Arnold theorem, adiabatic invariants, Kepler problem, two-body problem, Poincaré coordinates, planetary problem, first integral, integrability, perturbation theory Citation: Féjoz J., On Action-angle Coordinates and the Poincaré Coordinates, Regular and Chaotic Dynamics, 2013, vol. 18, no. 6, pp. 703-718 DOI:10.1134/S1560354713060105
 Niederman L. Generic Super-exponential Stability of Elliptic Equilibrium Positions for Symplectic Vector Fields Abstract In this article, we consider linearly stable elliptic fixed points (equilibrium) for a symplectic vector field and prove generic results of super-exponential stability for nearby solutions. We will focus on the neighborhood of elliptic fixed points but the case of linearly stable isotropic reducible invariant tori in a Hamiltonian system should be similar. More specifically, Morbidelli and Giorgilli have proved a result of stability over superexponentially long times if one considers an analytic Lagrangian torus, invariant for an analytic Hamiltonian system, with a diophantine translation vector which admits a sign-definite torsion. Then, the solutions of the system move very little over times which are super-exponentially long with respect to the inverse of the distance to the invariant torus. The proof proceeds in two steps: first one constructs a high-order Birkhoff normal form, then one applies the Nekhoroshev theory. Bounemoura has shown that the second step of this construction remains valid if the Birkhoff normal form linked to the invariant torus or the elliptic fixed point belongs to a generic set among the formal series. This is not sufficient to prove this kind of super-exponential stability results in a general setting. We should also establish that the most strongly non resonant elliptic fixed point or invariant torus in a Hamiltonian system admits Birkhoff normal forms fitted for the application of the Nekhoroshev theory. Actually, the set introduced by Bounemoura is already very large but not big enough to ensure that a typical Birkhoff normal form falls into this class. We show here that this property is satisfied generically in the sense of the measure (prevalence) through infinitedimensional probe spaces (that is, an infinite number of parameters chosen at random) with methods similar to those developed in a paper of Gorodetski, Kaloshin and Hunt in another setting. Keywords: Hamiltonian systems, perturbation of integrable systems, effective stability Citation: Niederman L., Generic Super-exponential Stability of Elliptic Equilibrium Positions for Symplectic Vector Fields, Regular and Chaotic Dynamics, 2013, vol. 18, no. 6, pp. 719-731 DOI:10.1134/S1560354713060117
 Hu X.,  Ou Y. An Estimation for the Hyperbolic Region of Elliptic Lagrangian Solutions in the Planar Three-body Problem Abstract It is well known that the linear stability of elliptic Lagrangian solutions depends on the mass parameter $\beta = 27(m_1m_2+m_2m_3+m_3m_1)/(m_1+m_2+m_3)^2 \in [0,9]$ and the eccentricity $e \in [0,1)$. Based on new techniques for evaluating the hyperbolicity and the recently developed trace formula for Hamiltonian systems [9], we identify regions for $(\beta,e)$ such that elliptic Lagrangian solutions are hyperbolic. Consequently, we have proven that the elliptic relative equilibrium of square central configurations is hyperbolic with any eccentricity. Keywords: central configurations, elliptic relative equilibrium, linear stability, hyperbolicity, $n$-body problem Citation: Hu X.,  Ou Y., An Estimation for the Hyperbolic Region of Elliptic Lagrangian Solutions in the Planar Three-body Problem, Regular and Chaotic Dynamics, 2013, vol. 18, no. 6, pp. 732-741 DOI:10.1134/S1560354713060129
 Chaperon M. A Forgotten Theorem on ${\bf Z}^k \times {\bf R}^m$-action Germs and Related Questions Abstract Two weakly hyperbolic smooth ${\bf Z}^k \times {\bf R}^m$-action germs are smoothly conjugate if and only if they are formally conjugate, and such. Keywords: Hyperbolic, action germ, conjugacy, invariant manifold, normal form, flow, Lyapunov function, Cauchy problem, holomorphic vector field, first integral Citation: Chaperon M., A Forgotten Theorem on ${\bf Z}^k \times {\bf R}^m$-action Germs and Related Questions, Regular and Chaotic Dynamics, 2013, vol. 18, no. 6, pp. 742-773 DOI:10.1134/S1560354713060130
 Bolotin S. V.,  Negrini P. Shilnikov Lemma for a Nondegenerate Critical Manifold of a Hamiltonian System Abstract Let $M$ be a normally hyperbolic symplectic critical manifold of a Hamiltonian system. Suppose $M$ consists of equilibria with real eigenvalues. We prove an analog of the Shilnikov lemma (strong version of the $\lambda$-lemma) describing the behavior of trajectories near $M$. Using this result, trajectories shadowing chains of homoclinic orbits to $M$ are represented as extremals of a discrete variational problem. Then the existence of shadowing periodic orbits is proved. This paper is motivated by applications to the Poincaré’s second species solutions of the 3 body problem with 2 masses small of order $\mu$. As $\mu \to 0$, double collisions of small bodies correspond to a symplectic critical manifold $M$ of the regularized Hamiltonian system. Thus our results imply the existence of Poincaré’s second species (nearly collision) periodic solutions for the unrestricted 3 body problem. Keywords: Hamiltonian system, symplectic map, generating function, heteroclinic orbit Citation: Bolotin S. V.,  Negrini P., Shilnikov Lemma for a Nondegenerate Critical Manifold of a Hamiltonian System, Regular and Chaotic Dynamics, 2013, vol. 18, no. 6, pp. 774-800 DOI:10.1134/S1560354713060142
 Eliasson H.,  Fayad B.,  Krikorian R. KAM-tori Near an Analytic Elliptic Fixed Point Abstract We study the accumulation of an elliptic fixed point of a real analytic Hamiltonian by quasi-periodic invariant tori. We show that a fixed point with Diophantine frequency vector $\omega_0$ is always accumulated by invariant complex analytic KAM-tori. Indeed, the following alternative holds: If the Birkhoff normal form of the Hamiltonian at the invariant point satisfies a Rüssmann transversality condition, the fixed point is accumulated by real analytic KAM-tori which cover positive Lebesgue measure in the phase space (in this part it suffices to assume that $\omega_0$ has rationally independent coordinates). If the Birkhoff normal form is degenerate, there exists an analytic subvariety of complex dimension at least $d+1$ passing through 0 that is foliated by complex analytic KAM-tori with frequency $\omega_0$. This is an extension of previous results obtained in [1] to the case of an elliptic fixed point. Keywords: Hamiltonian dynamics, elliptic fixed points, normal forms, KAM theory, invariant tori, Russmann’s condition, Herman’s conjecture, stability Citation: Eliasson H.,  Fayad B.,  Krikorian R., KAM-tori Near an Analytic Elliptic Fixed Point, Regular and Chaotic Dynamics, 2013, vol. 18, no. 6, pp. 801-831 DOI:10.1134/S1560354713060154
 Borisov A. V.,  Kilin A. A.,  Mamaev I. S. The Problem of Drift and Recurrence for the Rolling Chaplygin Ball Abstract We investigate the motion of the point of contact (absolute dynamics) in the integrable problem of the Chaplygin ball rolling on a plane. Although the velocity of the point of contact is a given vector function of variables of the reduced system, it is impossible to apply standard methods of the theory of integrable Hamiltonian systems due to the absence of an appropriate conformally Hamiltonian representation for an unreduced system. For a complete analysis we apply the standard analytical approach, due to Bohl and Weyl, and develop topological methods of investigation. In this way we obtain conditions for boundedness and unboundedness of the trajectories of the contact point. Keywords: nonholonomic constraint, absolute dynamics, bifurcation diagram, bifurcation complex, drift, resonance, invariant torus Citation: Borisov A. V.,  Kilin A. A.,  Mamaev I. S., The Problem of Drift and Recurrence for the Rolling Chaplygin Ball, Regular and Chaotic Dynamics, 2013, vol. 18, no. 6, pp. 832-859 DOI:10.1134/S1560354713060166
 Pinzari G. Aspects of the Planetary Birkhoff Normal Form Abstract The discovery of the Birkhoff normal form for the planetary many-body problem opened new insights and hopes for the comprehension of the dynamics of this problem. Remarkably, it allowed to give a direct proof (after the proof in [18]) of the celebrated Arnold’s Theorem [5] on the stability of planetary motions. In this paper, after reviewing the story of the proof of this theorem, we focus on technical aspects of this normal form. We develop an asymptotic formula for it that may turn to be useful in applications. Then we provide two simple applications to the three-body problem: we prove that the “density” of the Kolmogorov set of the spatial three-body problem does not depend on eccentricities and the mutual inclination but depends only on the planets’ masses and the separation among semi-axes (going in the direction of an assertion by V.I. Arnold [5]) and, using Nehorošhev Theory [33], we prove, in the planar case, stability of all planetary actions over exponentially-long times, provided meanmotion resonances are excluded. We also briefly discuss difficulties for full generalization of the results in the paper. Keywords: averaging theory, Birkhoff normal form, Nehoroshev theory, planetary many-body problem, Arnold’s Theorem on the stability of planetary motions, properly-degenerate KAM theory, steepness Citation: Pinzari G., Aspects of the Planetary Birkhoff Normal Form, Regular and Chaotic Dynamics, 2013, vol. 18, no. 6, pp. 860-906 DOI:10.1134/S1560354713060178

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