Volume 3, Number 3

Volume 3, Number 3, 1998
On the 70th birthday of J.Moser

Arnold V. I.
The higher-dimensional analogue of a continuous fraction is the polyhedral surface, bounding the convex hull of the semigroup of the integer points in a simplicial cone of the euclidian space. The article describes some conjectures and theorems, extending to such higher-dimensional continouos fraction the Lagrange theorem on quadraticirrationals and the Gauss–Kuzmin statistics.
Citation: Arnold V. I., Higher dimensional continued fractions, Regular and Chaotic Dynamics, 1998, vol. 3, no. 3, pp. 10-17
Dullin H. R.,  Richter P. H.,  Veselov A. P.
An explicit formula for the action variables of the Kovalevskaya top as Abelian integrals of the third kind on the Kovalevskaya curve is found. The linear system of differential equations of Picard–Fuchs type, describing the dependence of these variables on the integrals of the Kovalevskaya system, is presented in explicit form. The results are based on the formula for the actions derived by S.P.Novikov and A.P.Veselov within the theory of algebro-geometric Poisson brackets on the universal bundle of hyperelliptic Jacobians.
Citation: Dullin H. R.,  Richter P. H.,  Veselov A. P., Action variables of the Kovalevskaya top, Regular and Chaotic Dynamics, 1998, vol. 3, no. 3, pp. 18-31
Goriely A.,  Tabor M.
The analysis of complex-time singularities has proved to be the most useful tool for the analysis of integrable systems. Here, we demonstrate its use in the analysis of chaotic dynamics. First, we show that the Melnikov vector, which gives an estimate of the splitting distance between invariant manifolds, can be given explicitly in terms of local solutions around the complex-time singularities. Second, in the case of exponentially small splitting of invariant manifolds, we obtain sufficient conditions on the vector field for the Melnikov theory to be applicable. These conditions can be obtained algorithmically from the singularity analysis.
Citation: Goriely A.,  Tabor M., The role of complex-time singularities in chaotic dynamics, Regular and Chaotic Dynamics, 1998, vol. 3, no. 3, pp. 32-44
Verhulst F.,  Huveneers R.
The dynamics of time-dependent evolution towards symmetry in Hamiltonian systems poses a difficult problem as the analysis has to be global in phasespace. For one and two degrees of freedom systems this leads to the presence of one respectively two global adiabatic invariants and also the persistence of asymmetric features over a long time.
Citation: Verhulst F.,  Huveneers R., Evolution towards symmetry, Regular and Chaotic Dynamics, 1998, vol. 3, no. 3, pp. 45-55
Benettin G.,  Fasso F.,  Guzzo M.
We show that $L_4$ and $L_5$ in the spatial restricted circular three-body problem are Nekhoroshev-stable for all but a few values of the reduced mass up to the Routh critical value. This result is based on two extensions of previous results on Nekhoroshev-stability of elliptic equilibria, namely to the case of "directional quasi-convexity", a notion introduced here, and to a (non-convex) steep case. We verify that the hypotheses are satisfied for $L_4$ and $L_5$ by means of numerically constructed Birkhoff normal forms.
Citation: Benettin G.,  Fasso F.,  Guzzo M., Nekhoroshev-stability of $L_4$ and $L_5$ in the spatial restricted three-body problem, Regular and Chaotic Dynamics, 1998, vol. 3, no. 3, pp. 56-72
Treschev D. V.,  Zubelevich O. E.
An estimate for the difference of the frequencies on two invariant curves, bounding a resonance zone of an area-preserving close to integrable map, is obtained. Analogous results for Hamiltonian systems are presented.
Citation: Treschev D. V.,  Zubelevich O. E., Invariant tori in Hamiltonian systems with two degrees of freedom in a neighborhood of a resonance, Regular and Chaotic Dynamics, 1998, vol. 3, no. 3, pp. 73-81
Sevryuk M. B.
For any collection of $n \geqslant 2$ numbers $\omega_1,\ldots,\omega_n$, we prove the existence of an infinitely differentiable Hamiltonian system of differential equations $X$ with $n$ degrees of freedom that possesses the following properties: 1) $0$ is an elliptic (provided that all the $\omega_i$ are different from zero) equilibrium of system $X$ with eigenfrequencies $\omega_1,\ldots,\omega_n$; 2) system $X$ is linear up to a remainder flat at $0$; 3) the measure of the union of the invariant $n$-tori of system $X$ that lie in the $\varepsilon$-neighborhood of $0$ tends to zero as $\varepsilon\to 0$ faster than any prescribed function. Analogous statements hold for symplectic diffeomorphisms, reversible flows, and reversible diffeomorphisms. The results obtained are discussed in the context of the standard theorems in the KAM theory, the well-known Russmann and Anosov–Katok theorems, and a recent theorem by Herman.
Citation: Sevryuk M. B., Invariant sets of degenerate Hamiltonian systems near equilibria, Regular and Chaotic Dynamics, 1998, vol. 3, no. 3, pp. 82-92
Chenciner A.
Nous étudions les propriétés du problème des n corps qui proviennent de l'homogénéité du potentiel et retrouvons dans un cadre conceptuel commun divers résultats de Sundman, McGehee et Saari. Les résultats ne sont pas nouveaux mais il nous a semblé que cette présentation les éclaire agréablement. Nous considérons des potentiels de type newtonien, homogènes de degre $2\kappa$ en la configuration. Pour n'être pas obligés de distinguer divers cas dans les inégalités, nous supposerons, ce qui inclut le cas newtonien, que $-1<\kappa<0$.
Citation: Chenciner A., Collisions totales, Mouvements Complètement Paraboliques et Réduction des Homothéties Dans le Problème des $n$ corps, Regular and Chaotic Dynamics, 1998, vol. 3, no. 3, pp. 93-106
Celletti A.,  Chierchia L.
Birkhoff periodic orbits associated to spin-orbit resonances in Celestial Mechanics and in particular to the Moon–Earth and Mercury–Sun systems are considered. A general method (based on a quantitative version of the Implicit Function Theorem) for the construction of such orbits with particular attention to "effective estimates" on the size of the perturbative parameters is presented and tested on the above mentioned systems. Lyapunov stability of the periodic orbits (for small values of the perturbative parameters) is proved by constructing KAM librational invariant surfaces trapping the periodic orbits.
Citation: Celletti A.,  Chierchia L., Construction of stable periodic orbits for the spin-orbit problem of celestial mechanics, Regular and Chaotic Dynamics, 1998, vol. 3, no. 3, pp. 107-121
Lenz K. E.,  Lomeli H. E.,  Meiss J. D.
A natural generalization of the Henon map of the plane is a quadratic diffeomorphism that has a quadratic inverse. We study the case when these maps are volume preserving, which generalizes the the family of symplectic quadratic maps studied by Moser. In this paper we obtain a characterization of these maps for dimension four and less. In addition, we use Moser's result to construct a subfamily of in n dimensions.
Citation: Lenz K. E.,  Lomeli H. E.,  Meiss J. D., Quadratic volume preserving maps: an extension of a result of Moser, Regular and Chaotic Dynamics, 1998, vol. 3, no. 3, pp. 122-131
Pedroni M.,  Vanhaecke P.
In this paper we generalize the Mumford system which describes for any fixed $g$ all linear flows on all hyperelliptic Jacobians of dimension $g$. The phase space of the Mumford system consists of triples of polynomials, subject to certain degree constraints, and is naturally seen as an affine subspace of the loop algebra of $\mathfrak{sl}(2)$. In our generalizations to an arbitrary simple Lie algebra $\mathfrak{g}$ the phase space consists of $\mathrm{dim}\,\mathfrak{g}$ polynomials, again subject to certain degree constraints. This phase space and its multi-Hamiltonian structure is obtained by a Poisson reduction along a subvariety $N$ of the loop algebra $\mathfrak{g}((\lambda-1))$ of $\mathfrak{g}$. Since $N$ is not a Poisson subvariety for the whole multi-Hamiltonian structure we prove an algebraic. Poisson reduction theorem for reduction along arbitrary subvarieties of an affine Poisson variety; this theorem is similar in spirit to the Marsden–Ratiu reduction theorem. We also give a different perspective on the multi-Hamiltonian structure of the Mumford system (and its generalizations) by introducing a master symmetry; this master symmetry can be described on the loop algebra $\mathfrak{g}((\lambda-1))$ as the derivative in the direction of $\lambda$ and is shown to survive the Poisson reduction. When acting (as a Lie derivative) on one of the Poisson structures of the system it produces a next one, similarly when acting on one of the Hamiltonians (in involution) or their (commuting) vector fields it produces a next one. In this way we arrive at several multi-Hamiltonian hierarchies, built up by a master symmetry.
Citation: Pedroni M.,  Vanhaecke P., A Lie algebraic generalization of the Mumford system, its symmetries and its multi-hamiltonian structure, Regular and Chaotic Dynamics, 1998, vol. 3, no. 3, pp. 132-160
Jalnapurkar S. M.,  Marsden J. E.
In this paper, we obtain feedback laws to asymptotically stabilize relative equilibria of mechanical systems with symmetry. We use a notion of stability "modulo the group action" developed by Patrick [1992]. We deal with both internal instability and instability of the rigid motion. The methodology is that of potential shaping, but the system is allowed to be internally underactuated, i.e., have fewer internal actuators than the dimension of the shape space.
Citation: Jalnapurkar S. M.,  Marsden J. E., Stabilization of relative equilibria II, Regular and Chaotic Dynamics, 1998, vol. 3, no. 3, pp. 161-179
Simó C.
Area preserving maps close to integrable but not satisfying the twist condition are studied. The existence of invariant curves is proved, but they are no longer graphs with respect to the angular variable. Beyond the generic, codimension 1 case, several higher codimension cases are studied. Meandering curves, higher order meandering and labyrinthic curves show up. Several examples illustrate that this behavior occurs in very simple families of maps.
Citation: Simó C., Invariant curves of analytic perturbed nontwist area preserving maps, Regular and Chaotic Dynamics, 1998, vol. 3, no. 3, pp. 180-195

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