Smooth analog of standard mapping

Twenty years ago Bullett [1] published an article [1] where he found the invariant curves of standard mapping, having replaced the sinusoidal force by its smooth analog, a piecewise linear saw. His studies discovered an unexpected effect: at certain values of the perturbation parameter, unsplit separatrices of integer and fractional resonances arise among global invariant curves, while chaotic layers, which are usually attendant to these separatrices, are absent. Interestingly, the system remains nonintegrable in this case and the separatrices persist, confining momentum global diffusion under the condition of strong local chaos. For reasons not well understood, this important effect and its consequences had gone largely unnoticed until Ovsyannikov [2] independently proved a similar theorem for integer resonances in terms of the same model of symmetric piecewise linear 2D mapping. Since then, piecewise linear maps and their related continuous systems have become a subject of extensive research. Both Bullett and Ovsyannikov restricted analysis to the invariant curves of the new type, since the two branches of split separatrices form chaotic trajectories that are impossible to treat analytically. To the author’s knowledge, examples of persisting separatrices other than those given in [1, 2] have not been reported. In this work, the author presents numerical and analytical results that directly or indirectly concern the effect of separatrix persistence in the absence of attendant dynamic chaos. Issues remaining to be understood are noted.     

Numerical exploration of a family of strictly convex billiards with boundary of classC 2

We are interested in the possible existence of strictly convex ergodic billiards. Such billiards are searched for by means of numerical investigation. The boundary of a billiard is built with four arcs of classC . Adjacent arcs have equal curvatures at connecting points. The surface of section of the billiards is explored. It seems as if symmetric billiards always have invariant curves (islands). Asymmetric billiards have been found which look ergodic. They are built with an arc of an ellipse, two arcs of circles, and one-half of a Descartes oval.     

Exponentially small splittings in Hamiltonian systems

Families of area preserving analytical maps, depending on a small parameter epsilon, are considered, with the case epsilon=0 corresponding to an integrable map. The asymptotic formulas for the splittings of separatrices are derived by the method of analytical continuation of the separatrices to the complex domain. The main terms of the asymptotics are exponentially small with respect to the size of the perturbation. As epsilon tends to zero, the intersection angle of the separatrices can oscillate. The exponent and the oscillatory multiplier of the asymptotic formulas are determined by the position of poles of the homoclinic (heteroclinic) orbit of the limiting flow. Pre-exponential coefficients in the asymptotic formulas contain a multiplier obtained by the numerical study of separatrices of "model" maps in the complex domain.     

Time-Domain Mapping of Electromagnetic Ray Movement Inside 2D Anisotropic Region

The paper presents the extension of the time-domain mapping method applied to 2D billiard problem inside an anisotropic region bounded by ellipse [1]. In this paper, it has been considered the ray movement inside 2D anisotropic region bounded by arbitrary differentiable curve. It has been proved that the problem can be one-to-one mapped onto 2D mathemeatical billiard problem inside the region possessing isotropic properties by linear transformation of group velocity hodograph and boundary with the same coefficient, which is equal to anisotropy of the ray group velocity, simultaneously. The main features of the ray movement inside 2D anistropic region are discussed.     

Veiled assonance between geodesics on ellipsoid and billiard in its focal ellipse

We introduce new special ellipsoidal confocal coordinates in ⁿ (n ≥ 3) and apply them to the geodesic problem on a triaxial ellipsoid in ³ as well as the billiard problem in its focal ellipse.

Using such appropriate coordinates we show that these different dynamical systems have the same common analytic first integral. This fact is not evident because there exists a geometrical spatial gap between the geodesic and billiard flows under consideration, and this separating gap just “veils” the resemblance of the two systems.

In short, a geodesic on the ellipsoid and a billiard trajectory inside its focal ellipse are in a “veiled assonance”—under the same initial data they will be tangent to the same confocal hyperboloid. But this assonance is rather incomplete: the dynamical systems in question differ by their intrinsic action angle-variables, thereby the different dynamics arise on the same phase space (i.e. the same phase curves in the same phase space bear quite different rotation numbers).

Some results of this work have been published before in Russian (Tabanov, 1993) and presented to the International Geometrical Colloquium (Moscow, May 10–14, 1993) and the International Symposium on Classical and Quantum Billiards (Ascona, Switzerland, July 25–30, 1994).     

Algebraic fidelity decay for local perturbations

From a reflection measurement in a rectangular microwave billiard with randomly distributed scatterers the scattering and the ordinary fidelity was studied. The position of one of the scatterers is the perturbation parameter. Such perturbations can be considered as local since wave functions are influenced only locally, in contrast to, e.g., the situation where the fidelity decay is caused by the shift of one billiard wall. Using the random-plane-wave conjecture, an analytic expression for the fidelity decay due to the shift of one scatterer has been obtained, yielding an algebraic 1/t decay for long times. A perfect agreement between experiment and theory has been found, including a predicted scaling behavior concerning the dependence of the fidelity decay on the shift distance. The only free parameter has been determined independently from the variance of the level velocities.     

A Galaxy-like perturbation of the Robertson–Walker metric

The standard procedure for finding analytic perturbations in General Relativity suffers from the drawback that it is cumbersome to use beyond linear order perturbations. We present an alternate method of finding perturbations and provide an example in the form of a spherically symmetric over-dense region in an open Robertson–Walker background. The perturbation has several properties which are similar to those of a spiral galaxy.     

Numerical simulation on tunnel splitting of Bose-Einstein condensate in multi-well potentials

The low-energy-level macroscopic wave functions of the Bose-Einstein condensate (BEC) trapped in a symmetric double-well and a periodic potential are obtained by solving the Gross-Pitaevskii equation numerically. The ground state tunnel splitting is evaluated in terms of the even and odd wave functions corresponding to the global ground and excited states respectively. We show that the numerical result is in good agreement with the analytic level splitting obtained by means of the periodic instanton method.     

Approximating Multi-Dimensional Hamiltonian Flows by Billiards

The behavior of a point particle traveling with a constant speed in a region , undergoing elastic collisions at the regions’s boundary, is known as the billiard problem. Various billiard models serve as approximation to the classical and semi-classical motion in systems with steep potentials (e.g. for studying classical molecular dynamics, cold atom’s motion in dark optical traps and microwave dynamics). Here we develop methodologies for examining the validity and accuracy of this approximation. We consider families of smooth potentials , that, in the limit , become singular hard-wall potentials of multi-dimensional billiards. We define auxiliary billiard domains that asymptote, as to the original billiards, and provide, for regular trajectories, asymptotic expansion of the smooth Hamiltonian solution in terms of these billiard approximations. The asymptotic expansion includes error estimates in the C ^r norm and an iteration scheme for improving this approximation. Applying this theory to smooth potentials that limit to the multi-dimensional close to ellipsoidal billiards, we predict when the billiard’s separatrix splitting (which appears, for example, in the nearly flat and nearly oblate ellipsoids) persists for various types of potentials.     

Optical Ray Movement Inside Dielectric Sphere

The paper presents results of numerical investigations of electromagnetic (optical) ray propagation inside a sphere, made of optically uniaxial crystal. The types of caustics formed by the ray trajectory inside the sphere are cleared up. It has been demonstrated the difference between types of caustics formed by the extraordinary ray trajectory inside the sphere made of optically positive and negative crystals. Results are discussed on the basis of one-to-one mapping of 2D billiard problem inside isotropic region bounded by ellipse onto anisotropic one.     

A Proof of the Exponentially Small Transversality of the Separatrices for the Standard Map

In 1984 V. F. Lazutkin [Laz84, LST89] obtained an asymptotic formula for the separatrix splitting angle for the standard map. The difficulty of this problem is related to the exponential smallness of the splitting with respect to a perturbation parameter. Lazutkin's proof was based on two conjectures. Probably, the original form of those conjectures was incorrect, but Lazutkin's method was very efficient and inspired a large number of studies on the exponentially small splitting of separatrices. The consequent works [Laz91, Laz92, GLS94] and [Gel96] prepared the base to fill all the gaps of the original proof. The present paper contains a complete and self-contained proof of a refined version of the original formula (formula (1.7) of the present paper). In this form the formula was obtained in [GLS94]. The proof is inspired by the ideas of Lazutkin's original paper [Laz84]. Received: 16 February 1998 / Accepted: 20 August 1998     

A peculiar Maxwell’s Demon observed in a time-dependent stadium-like billiard

The dynamics of a driven stadium-like billiard is considered using the formalism of discrete mappings. The model presents a resonant velocity that depends on the rotation number around fixed points and external boundary perturbation which plays an important separation rule in the model. We show that particles exhibiting Fermi acceleration (initial velocity is above the resonant one) are scaling invariant with respect to the initial velocity and external perturbation. However, initial velocities below the resonant one lead the particles to decelerate therefore unlimited energy growth is not observed. This phenomenon may be interpreted as a specific Maxwell’s Demon which may separate fast and slow billiard particles.     

Hard Chaos in Magnetic Billiards¶(On the Euclidean Plane)

This work presents local and global results on the stability of the dynamics of classical magnetic billiard systems (with homogeneous magnetic field) established on the Euclidean plane. In the first part of the paper our previous results concerning the properties of the stability matrices on curved Riemannian manifolds are rederived in a simpler, elementary way in the Euclidean case. As applications, the stability regions for special symmetric orbits are determined analytically and numerically. Using a new technique, necessary conditions for hard chaos and lower estimations for the Lyapunov exponent are given for planar magnetic billiards with dispersing and focusing boundary segments, too. It is also shown that in the investigated billiard types hard chaos is structurally stable below a certain threshold magnetic field. Received: 15 November 1996 / Accepted: 8 January 1997     

Numerical study of a billiard in a gravitational field

Billiards have always been used as models for mechanical systems. In this paper we describe a very simple billiard which, over a range of one continous parameter only, exhibits the characteristics of Hamiltonian systems having two degrees of freedom and a discontinuity. The relationship between this billiard and the well-known one-dimensional self-gravitating system (with N = 3) is given. This billiard consists of a mass point moving in a symmetric wedge of angle 2θ under the influence of a constant gravitational field. For θ<45° KAM and chaotic regions coexist in the phase space. A specific family of curves, related to collisions at the wedge vertex, limits the expansion of near-integrable regions. For θ=45°, the motion is strictly integrable. Finally, for θ>;45°, complete chaos is obtained, suggesting K-system behavior. The general properties of the mapping and some numerical results obtained are discussed. Of special interest are invariant curves which cross a line of discontinuity, and a new “universality” class for Lyapunov numbers.     

Drastic facilitation of the onset of global chaos

We show that the onset of global chaos in a time periodically perturbed Hamiltonian system may occur at unusually small magnitudes of perturbation if the unperturbed system possesses more than one separatrix. The relevant scenario is the combination of the overlap in the phase space between resonances of the same order and their overlap in energy with chaotic layers associated with separatrices of the unperturbed system. We develop the asymptotic theory and verify it in simulations.     

Classical irreversibility and mapping of the isosceles triangle billiard

Mapping of the two-dimensional isosceles triangle billiard onto the circular one-dimensional motion of two mass points is described. The singular nature of trajectories directly incident on an acute vertex is discussed in the framework of the present mapping. For an obtuse-angled isosceles triangle, dynamical equations in two-particle space applied to an orbit along a hypotenuse incident on the obtuse vertex suggests irreversible behavior at the critical angle =2/3. Thus it is found that the nonsingular motion of a finite smooth-walled disk on this trajectory exhibits irreversibility. A finite spherical smooth-walled particle moving in a uniform right cylinder whose cross section includes this critical vertex angle likewise exhibits irreversibility. Each such example comprises an irreversible orbit for a single-particle Hamiltonian.     

Calculation of the higher-order perturbations in the theory of gamma transport

The higher-order perturbations in the application of perturbation theory to the -transport equation are discussed. An analytic expression is obtained for the perturbation flux of arbitrary order arising from a variation in the electron density of the medium, regardless of its geometry. This expression can be used in the constant-cross-section approximation to find the perturbation due to a change in the chemical composition of a plane layer. The numerical calculation is illustrated with an example.The author thanks S. B. Shikhov for interest in this study.     

Is the size of <Emphasis Type="Italic">θ</Emphasis> <Subscript>13</Subscript> related to the smallness of the solar mass splitting?

θ₁₃ is small compared to the other neutrino mixing angles. The solar mass splitting is about two orders smaller than the atmospheric splitting. We indicate how both could arise from a perturbation of a more symmetric structure. The perturbation also affects the solar mixing angle and can tweak alternate mixing patterns such as tribimaximal, bimaximal, or other variants to viability. For real perturbations only normal mass ordering with the lightest neutrino mass less than 10^?2 eV can accomplish this goal. Both mass orderings can be accommodated by going over to complex perturbations if the lightest neutrino is heavier. The CP-phase in the lepton sector, fixed by θ₁₃ and the lightest neutrino mass, distinguishes different options.     

Static and time-dependent perturbations of the classical elliptical billiard

The elliptical billiard problem defines a two-dimensional integrable discrete dynamical system. Integrability not being a robust property, we study some static and time-dependent perturbations of this problem. For the static case, we observe the transition from integrability to chaos, on some perturbations of the ellipse. Then we study time-dependent perturbations, supposing that the boundary deforms periodically with the time, remaining always an ellipse. We investigate numerically the now four-dimensional phase space, looking mainly at the question of whether or not the velocity of a given trajectory may increase indefinitely.