Elliptic Point Research Articles

Quasiperiodic and chaotic discrete breathers in a parametrically driven system without linear dispersion

We study a one-dimensional lattice of anharmonic oscillators with only quartic nearest-neighbor interactions, in which discrete breathers (DB's) can be explicitly constructed by an exact separation of their time and space dependence. Introducing parametric periodic driving, we first show how a variety of such DB's can be obtained by selecting spatial profiles from the homoclinic orbits of an invertible map and combining them with initial conditions chosen from the Poincaré surface of section of a simple Duffing's equation. Placing then our initial conditions at the center of the islands of a major resonance, we demonstrate how the corresponding DB can be stabilized by varying the amplitude of the driving. We thus discover around elliptic points a large region of quasiperiodic breathers, which are stable for very long times. Starting with initial conditions close to the elliptic point at the origin, we find that as we approach the main chaotic layer, a quasiperiodic breather either destabilizes by delocalization or turns into a chaotic breather, with an evidently broadbanded Fourier spectrum before it collapses. For some breather profiles stable quasiperiodic breathers exist all the way to the separatrix of the Duffing equation, indicating the presence of large regions of tori around the DB solution in the multidimensional phase space. We argue that these strong localization phenomena are due to the absence of phonon resonances, as there are no linear dispersion terms in our lattices. We also show, however, that these phenomena persist in more realistic physical models, in which weak linear dispersion is included in the equations of motion, with a sufficiently small coefficient.

A landing theorem for periodic rays of exponential maps

For the family of exponential maps z ↦ exp ⁡ ( z ) + κ z\mapsto \exp (z)+\kappa , we show the following analog of a theorem of Douady and Hubbard concerning polynomials. Suppose that g g is a periodic dynamic ray of an exponential map with nonescaping singular value. Then g g lands at a repelling or parabolic periodic point. We also show that there are periodic dynamic rays landing at all periodic points of such an exponential map, with the exception of at most one periodic orbit.

More compact invariant manifolds appearing in the non-linear coupling of oscillators

Near partially elliptic rest points of generic families of vector fields or transformations, many types of normally hyperbolic invariant compact manifolds can appear, diffeomorphic to intersections of quadrics. To cite this article: M. Chaperon et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).

A Primal-Based Penalty Preconditioner for Elliptic Saddle Point Systems

A primal-based penalty preconditioner is presented for a linear set of equations arising from elliptic saddle point problems. We show that the eigenvalues of the preconditioned matrix are positive real and demonstrate that a variant of the preconditioner can be combined with the conjugate gradient algorithm. Our approach is motivated by two basic observations. First, the solution of a problem with constraints is often similar to the solution of a problem where the constraints are penalized. Second, certain methods of solution not available for a constrained problem are possible for its penalized counterpart so motivating a primal-based Schur complement approach. Numerical examples for elliptic two- and three-dimensional problems are presented that confirm theoretical results and demonstrate the effectiveness of the preconditioner.

Simple Power Analysis on Fast Modular Reduction with Generalized Mersenne Prime for Elliptic Curve Cryptosystems

We discuss side channel leakage from modular reduction for NIST recommended domain parameters. FIPS 186-2 has 5 recommended prime fields. These primes have a special form which is referred to as generalized Mersenne prime. These special form primes facilitate especially efficient implementation. A typical implementation of efficient modular reduction with such primes includes conditional reduction. A conditional reduction in modular reduction can constitute an information channel on the secret exponent. Several researchers have produced unified code for elliptic point addition and doubling in order to avoid a simple power analysis (SPA). However, Walter showed that SPA still be possible if Montgomery multiplication with conditional reduction is implemented within the unified code. In this paper we show SPA on the modular reduction with NIST recommended primes, combining with the unified code for elliptic point operations. As Walter stated, our results also indicate that even if the unified codes are implemented for elliptic point operations, underlying field operations should be implemented in constant time. The unified approach in itself can not be a countermeasure for side channel attacks.

One-sided invariant manifolds, recursive folding, and curvature singularity in area-preserving nonlinear maps with nonuniform hyperbolic behavior

Two-dimensional nonlinear models of conservative dynamics are typically nonuniformly hyperbolic in that there are nonhyperbolic trajectories that coexist with a “massive” hyperbolic region. We investigate the influence of nonhyperbolic points on the global geometric structure of a invariant manifolds associated with points of the hyperbolic region. As a case study, we consider a transformation of the Standard Map family and analyze the structure of invariant manifolds in the neighborhood of an isolated parabolic (fixed) point xp. This analysis shows the existence of lobes enclosing the parabolic point, that is, of simply connected regions containing xp whose boundary is formed by two continuous arcs of stable and unstable manifolds that intersect only at two points. From the existence of such regions, we derive that (i) there are points of the hyperbolic region where the local curvature of invariant manifolds is arbitrarily large and (ii) manifolds possess the recursively folding property. Property (ii) means that given an invariant manifold W and established an orientation on it, in the neighborhood of any point of the chaotic region there are nearby arcs of W that are traveled in opposite directions. We propose an archetypal model for which the existence of lobes and the recursive folding property can be derived analytically. The impact of nonuniform hyperbolicity on the evolution of physical processes that occur along with phase space mixing is also addressed.

Nested separatrices in simple shear flows: The effect of localized disturbances on stagnation lines

The effects of localized two-dimensional disturbances on the structure of shear flows featuring a stagnation line are investigated. A simple superposition of a planar Couette flow and Moffatt’s [J. Fluid Mech. 18, 1–18 (1964)] streamfunction for the decay of a disturbance between infinite stationary parallel plates shows that in general the stagnation line is replaced by a chain of alternating elliptic and hyperbolic stagnation points with a separation equal to 2.78 times the half-gap between the plates. The flow structure associated with each saddle point consists of a homoclinic separatrix and two other separatrices which locally diverge but become parallel far from the disturbance. This basic structure repeats to give a sequence of nested separatrices permitting the streamfunction to approach that of simple shear flow far from the disturbance. Using the finite-element method, the specific disturbance caused by a stationary cylinder placed on the stagnation line is considered, and results confirm the existence of the stagnation point chain, with computed separations and velocity damping ratios in very good agreement with those obtained from the Couette-Moffatt superposition. Numerical solutions also illustrate that while Reynolds number greatly affects the stagnation point separation and velocity damping ratio, these two quantities are the same for any pair of adjacent stagnation points in a given chain. Insight gained from the analysis of planar shear flows is applied to the flow in a half-filled horizontal annulus between rotating coaxial cylinders, and is used to explain why only certain flow patterns from the range of mathematically possible structures arise in previous numerical solutions. By way of contrast, the concentric annulus solution is then perturbed to allow for a small eccentricity. The nonuniformity of the intercylinder gap is shown to destroy the chain of stagnation points, but also to unfold additional flow structures not realizable when the gap is uniform.

Modular differential equations of second order with regular singularities at elliptic points for $SL_2(\mathbb {Z})$

We give a definition of the modular differential equations of weight k for a discrete subgroup for Γ ⊂ SL 2 (R); in this paper we set Γ = SL 2 (Z). We solve such equations admitting regular singularities at elliptic points for SL 2 (Z) in terms of the Eisenstein series and the Gauss hypergeometric series. Furthermore, we give a series of such modular differential equations parametrized by an even integer k, and discuss some properties of solution spaces. We find several equations which are solved by a modular form of weight k.

On the construction of the Kolmogorov normal form for the Trojan asteroids

In this paper, we focus on the stability of the Trojan asteroids for the planar restricted three-body problem, by extending the usual techniques for the neighbourhood of an elliptic point to derive results in a larger vicinity. Our approach is based on numerical determination of the frequencies of the asteroid and effective computation of the Kolmogorov normal form for the corresponding torus. This procedure has been applied to the first 34 Trojan asteroids of the IAU Asteroid Catalogue, and it has worked successfully for 23 of them.The construction of this normal form allows computer-assisted proofs of stability. To show this, we have implemented a proof of existence of families of invariant tori close to a given asteroid, for a high order expansion of the Hamiltonian. This proof has been successfully applied to three Trojan asteroids.

Una revisión del mito de Celestina : "Las conversiones", de José Martín Recuerda

José Martín Recuerda writes a peculiar review about the Celestian myth. Hisdefamiliarizing point of view twists a repressive frame. His parabolic point of view shows a postwar period of dictatorship in which he started his career as a dramatic author. Moreover, his grotesque imagery is outstanding, especially because he is defined as a realist author.

Stable products of spheres in the non-linear coupling of oscillators or quasi-periodic motions

For generic families of vector fields or transformations, normally hyperbolic invariant products of spheres appear near partially elliptic rest points. To cite this article: M. Kammerer-Colin de Verdière, C. R. Acad. Sci. Paris, Ser. I 339 (2004).

Nekhoroshev stability estimates for different models of the Trojan asteroids

Estimates of the region of Nekhoroshev stability of Jupiter's Trojan asteroids are obtained by a direct (i.e. without use of the normal form) construction of formal integrals near the Lagrangian elliptic equilibrium points. Formal integrals are constructed in the Hamiltonian model of the planar circular restricted three body problem (PCRTBP), and in a mapping model (Sandor et al. 2002) of the same problem for small orbital eccentricities of the asteroids. The analytical estimates are based on the calculation of the size of the remainder of the formal series by a computer program. An analysis is made of the accumulation of small divisors in the series. The most important divisors introduce competing Fourier terms with sizes growing at similar rates as the order of truncation increases. This makes impossible to improve the estimates by considering nearly resonant forms of the formal integrals for particular near-resonances. Improved estimates were obtained in a mapping model of the PCRTBP. The main source of improvement is the use of better variables (Delaunay). Our best estimate represents a maximum libration amplitude . This is a quite realistic value which demonstrates the usefulness of Nekhoroshev theory.To search for other articles by the author(s) go to: http://adsabs.harvard.edu/abstract_service.html

Power attacks on a side-channel resistant elliptic curve implementation

At ISC 2001 a method for securing elliptic curve point multiplication against side-channel attacks has been proposed by Moller [Lecture Notes in Comput. Sci., vol. 2200, Springer-Verlag, Berlin, 20...

Geometrical Patterns of the Atmospheric Flow Fields

AbstractGeometrical patterns of some fundamental flow fields in the atmosphere are discussed based on different simplifications and assumptions of the fundamental equations. Results show that the streamlines of cyclones and anticyclones on the 2‐D isobaric surface are similar to the manifold of a centric point (or elliptic point). The flow fields of typhoon and tornado are similar to the manifolds of saddle‐focus points in 3‐D space. When the atmospheric long wave evolves and blocking high appears, the manifold corresponds to the case while a saddle and a center point change from coalescence to separation. The geometrical discussions of these flow fields are more clearly intuitive and do help to understand the patterns formation of atmospheric motion and the underlying mechanisms. Differences exist between the theoretical results and the real behavior of the atmospheric flow due to the simplifications and assumptions made.

On the estimate of the arithmetic genus for normal two-dimensional singularities on double coverings

In this paper, we deal with normal two-dimensional singularities with multiplicity two. We call such a singularity double point. The purpose of this paper is to give an estimate of the arithmetic genus for double points in terms of Horikawa's canonical resolution and a $p_{a}$-formula for some class of double points. It is known that, by using the data obtained from the canonical resolution, the geometric genus for double points is formulated, and rational double points and elliptic double points are characterized. We give a characterization of double points with the arithmetic genus two.

Ion acceleration by beating electrostatic waves: domain of allowed acceleration.

The conditions under which a magnetized ion can be accelerated through a nonlinear interaction with a pair of beating electrostatic waves are explored. It has been shown [Benisti et al., Phys. Plasma 5, 3224 (1998)] that the electric field of the beating waves can, under some conditions, accelerate ions from arbitrarily low initial velocity in stark contrast with the well-known nonlinear threshold criteria for ion acceleration by a single wave. It is shown here that the previously found condition is necessary but not sufficient for acceleration to occur. The sufficient and necessary conditions are identified in terms of the location of the critical points of the motion on the Poincaré section. A second-order perturbation analysis was carried out to approximate the location of these critical points and define the domains of allowed and forbidden acceleration. It is shown that for an ion to be significantly energized, the Hamiltonian must be outside the energy barrier defined by the location of the elliptic and hyperbolic critical points. Despite the restriction on the Hamiltonian, an ion with arbitrarily low initial velocity may benefit from this acceleration mechanism.

Time-Reversible Hamiltonian Vector Fields with Symplectic Symmetries

This paper deals with the dynamics of time-reversible Hamiltonian vector fields with two degrees of freedom around an elliptic equilibrium point in presence of 1−1 resonance. The main result says that under certain conditions there are two one-parameter families of reversible periodic solutions terminating at the equilibrium.

Modulus of Analytic Classification for Unfoldings of Generic Parabolic Diffeomorphisms

In this paper we give a complete modulus of analytic classi- fication under weak equivalence for generic analytic 1-parameter unfold- ings of dieomorphisms with a generic parabolic point. The modulus is composed of a canonical parameter associated to the family, together with an unfolding of the Ecalle-Voronin modulus. We then study the fixed points bifurcating from a parabolic point with nontrivial Ecalle- Voronin modulus and show that some of the non-hyperbolic resonant ones are non integrable. In the Addendum it is shown that weak equivalence can be replaced by conjugacy.

Bichromatic optical Lissajous fields

Bichromatic optical Lissajous fields generated via second harmonic generation are described for a variety of Gaussian input fields that contain vector or elliptic singularities. The Lissajous figures repetitively traced out by the electric vector E during each optical cycle are more complicated than the familiar polarization ellipse of monochromatic light and can contain 0–3 self intersections. Eight different characterizations of these figures are discussed, some of which contain unexpected singularities. Six of these characterizations are expressed in terms experimentally measurable bichromatic Stokes parameters. As is the case for the monochromatic polarization ellipse, Lissajous figures can be assigned a handedness (right/left) that depends on the sense in which E traces out the figure. Two different forms of handedness reversal are discussed: these lead to two different Lissajous line singularities. Like the polarization ellipse, Lissajous figures can also be assigned an orientation that leads to a variety of point singularities. An input beam containing a single on axis vector (elliptic) point singularity with absolute charge n is found to produce an output beam that contains 12 n+1 (6 n+1) Lissajous point singularities.

Optical polarization singularities and elliptic stationary points.

Polarization singularities and elliptic stationary points (collectively, elliptic critical points) were measured experimentally via the complex Stokes field S1 + iS2, where S1 and S2 are Stokes parameters. This new, easily implemented method yielded detailed, high-resolution experimental data for all elliptic critical points. These data confirm with high precision the elliptic-field topological sign rule, loop rules, and Stokes singularity relations introduced recently.