Shortest periodic billiard trajectories in convex bodies

We show that the length of any periodic billiard trajectory in any convex body is always at least 4 times the inradius of K; the equality holds precisely when the width of K is twice its inradius, e.g., K is centrally symmetric, in which case we prove that the shortest periodic trajectories are all bouncing ball (2-link) orbits.     

Effect of the phase on the dynamics of a perturbed bouncing ball system

The phase control method is a non-feedback control technique which has been used for different purposes in continuous periodically driven dynamical systems. One of the main goals of this paper is to apply this control technique to the bouncing ball system, which can be seen as a paradigmatic periodically driven discrete dynamical system, and has a rather simple physical interpretation. The main idea is to apply a periodic control signal including a phase difference with respect to the periodic forcing of the initial system and to analyze its effect on the dynamics of the bouncing ball system. The numerical simulations we have carried out clearly show the strong effect of the phase of the control signal in suppressing or generating chaotic behavior and in changing the period of a periodic orbit. We have also analyzed the effect of the phase in the phenomenon of the crisis-induced intermittency, showing how the phase enhances the size of the attractor near a crisis and can induce the intermittent behavior. Finally we have analyzed the scaling behavior of the crisis by varying the phase difference between the perturbation and the external forcing.     

On the model of non-holonomic billiard

In this paper we develop a new model of non-holonomic billiard that accounts for the intrinsic rotation of the billiard ball. This model is a limit case of the problem of rolling without slipping of a ball without slipping over a quadric surface. The billiards between two parallel walls and inside a circle are studied in detail. Using the three-dimensional-point-map technique, the non-integrability of the non-holonomic billiard within an ellipse is shown.     

Escape distribution for an inclined billiard

Hénon [8] used an inclined billiard to investigate aspects of chaotic scattering which occur in satellite encounters and in other situations. His model consisted of a piecewise mapping which described the motion of a point particle bouncing elastically on two disks. A one parameter family of orbits, named h-orbits, was obtained by starting the particle at rest from a given height. We obtain an analytical expression for the escape distribution of the h-orbits, which is also compared with results from numerical simulations. Finally, some discussion is made about possible applications of the h-orbits in connection with Hill’s problem.     

Lower estimates for the number of closed trajectories of generalized billiards

The mathematical study of periodic billiard trajectories is a classical question that goes back to George Birkhoff. A billiard is the motion of a particle in the absence of field of force. Trajectories of such a particle are geodesics. A billiard ball rebounds from the boundary of a given domain making the angle of incidence equal the angle of reflection. Let k be a fixed integer. Birkhoff proved a lower estimate for the number of closed billiard trajectories of length k in an arbitrary plane domain. We give a general definition of a closed billiard trajectory when the billiard ball rebounds from a submanifold of a Euclidean space. Besides, we show how in this case one can apply the Morse inequalities using the natural symmetry (a closed polygon may be considered starting at any of its vertices and with the reversed direction). Finally, we prove the following estimate. Let M be a smooth closed m-dimensional submanifold of a Euclidean space, and let p > 2 be a prime integer. Then M has at least

Invariants of Isospectral Deformations and Spectral Rigidity

We introduce a notion of weak isospectrality for continuous deformations. Consider the Laplace–Beltrami operator on a compact Riemannian manifold with Robin boundary conditions. Given a Kronecker invariant torus Λ of the billiard ball map with a Diophantine vector of rotation we prove that certain integrals on Λ involving the function in the Robin boundary conditions remain constant under weak isospectral deformations. To this end we construct continuous families of quasimodes associated with Λ. We obtain also isospectral invariants of the Laplacian with a real-valued potential on a compact manifold for continuous deformations of the potential. These invariants are obtained from the first Birkhoff invariant of the microlocal monodromy operator associated to Λ. As an application we prove spectral rigidity of the Robin boundary conditions in the case of Liouville billiard tables of dimension two in the presence of a (?/2?)² group of symmetries.     

On the covering radius of lattice zonotopes and its relation to view-obstructions and the lonely runner conjecture

The goal of this paper is twofold; first, we show the equivalence between certain problems in geometry, such as view-obstructions, billiard ball motions, and the estimation of covering radii of lattice zonotopes. Second, we utilize the latter interpretation and provide upper bounds of said radii by virtue of the Flatness Theorem. Our results allow us to specify how rational dependencies in the view-direction influence the obstruction parameter. These problems are similar in nature to the famous Lonely Runner Problem for which we draw analogous conclusions.     

Variational approach to homoclinic orbits in twist maps and an application to billiard systems

We study in this article the topological entropy of billiard systems on a convex domain of the Euclidean plane. We restrict our attention to those systems whose boundary curve has positive curvature and show that for generic billiard ball systems satisfying this condition the topological entropy is positive.     

Two interacting hard disks within a circular cavity: towards a quasi-equilibrium quantal equation of states

Two interacting hard disks confined in a circular cavity are investigated. Each disk shows a free motion except when bouncing elastically with its partner and with the boundary wall. According to the analysis of Lyapunov exponents, this system is classically nonintegrable and almost chaotic because of the (short-range) interaction between the disks. The system can be quantized by incorporating the excluded volume effect for the wave function. Eigenvalues and eigenfunctions are obtained by tuning the relative size between the disks and the billiard. The pressure P is defined as the derivative of each eigenvalue with respect to the cavity volume V. Since the energy spectra of eigenvalues versus the disk size show a multitude of level repulsions, P–V characteristics shows the anomalous pressure fluctuations accompanied by many van der Waals-like peaks in each of excited eigenstates taken as a quasi-equilibrium. For each eigenstate, we calculate the expectation values of the square distance between two disks, and point out their relationship with the pressure fluctuations. Role of Bose and Fermi statistics is also investigated.     

On the chaotic behavior of non-flat billiards

We study the problem of the motion of a particle on a non-flat billiard. The particle is subject to the gravity and to a small amplitude periodic (or almost periodic) forcing and is reflected with respect to the normal axis when it hits the boundary of the billiard. We prove that the unperturbed problem has an impact homoclinic orbit and give a Melnikov type condition so that the perturbed problem exhibit chaotic behavior in the sense of Smale’s horseshoe.     

Billiard and the five-gap theorem

Let us consider the interval [0,1) as a billiard table rectangle with perimeter 1 and a sequence F(m)∈[0,1),m∈N∪{0}, of successive rebounds of a billiard ball against the sides of a billiard rectangle. We prove that if I is an open segment of a billiard rectangle, then the differences between the successive values of m for which the F(m) lies in I, take at most one even and at most four distinct odd values.     

Artin Billiard: Exponential Decay of Correlation Functions

The hyperbolic Anosov C-systems have an exponential instability of their trajectories and as such represent the most natural chaotic dynamical systems. The C-systems defined on compact surfaces of the Lobachevsky plane of constant negative curvature are especially interesting. An example of such a system was introduced in a brilliant article published in 1924 by the mathematician Emil Artin. The dynamical system is defined on the fundamental region of the Lobachevsky plane, which is obtained by identifying points congruent with respect to the modular group, the discrete subgroup of the Lobachevsky plane isometries. The fundamental region in this case is a hyperbolic triangle. The geodesic trajectories of the non-Euclidean billiard are bounded to propagate on the fundamental hyperbolic triangle. Here, we present Artin’s results, calculate the correlation functions/observables defined on the phase space of the Artin billiard, and show that the correlation functions decay exponentially with time. We use the Artin symbolic dynamics, differential geometry, and the group theory methods of Gelfand and Fomin.     

Estimates for correlations in billiards with large arcs

Bunimovich billiards are ergodic and mixing. However, if the billiard table contains very large arcs on its boundary then if there exist trajectories experience infinitely many collisions in the vicinity of periodic trajectories on the large arc. The hyperbolicity is nonuniform and the mixing rate is very slow. The corresponding dynamics are intermittent between regular and chaotic, which makes them particularly interesting in physical studies. The study of mixing rates in intermittent chaotic systems is mo...     

An Ellipsoidal Billiard with a Quadratic Potential

There exists an infinite hierarchy of integrable generalizations of the geodesic flow on an n-dimensional ellipsoid. These generalizations describe the motion of a point in the force fields of certain polynomial potentials. In the limit as one of semiaxes of the ellipsoid tends to zero, one obtains integrable mappings corresponding to billiards with polynomial potentials inside an (n-1)-dimensional ellipsoid.In this paper, for the first time we give explicit expressions for the ellipsoidal billiard with a quadratic (Hooke) potential, its representation in Lax form, and a theta function solution. We also indicate the generating function of the restriction of the potential billiard map to a level set of an energy type integral. The method we use to obtain theta function solutions is different from those applied earlier and is based on the calculation of limit values of meromorphic functions on generalized Jacobians.     

平面映射的周期解分支

本文运用一些技巧对Taylor映射在4相似文献   

Complex dynamic behaviour of an economic cycle model

In this paper, we investigate the dynamics of a nonlinear economic cycle model. The necessary and sufficient conditions are given to guarantee the existence and stability of the fixed point. It is also shown that the system undergoes a Neimark–Sacker bifurcation by using center manifold theorem and bifurcation theory. Furthermore, Marotto’s chaos is proved when certain conditions are satisfied. Numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviour, such as the period-10, -16, -20 orbits, attracting invariant cycles, quasi-periodic orbits, 10-coexisting chaotic attractors, and boundary crisis. Specifically, we have stabilized the chaotic orbits at an unstable fixed point using the feedback control method.     

Unbounded Orbits for Semicircular Outer Billiard

We show that the outer billiard around a semicircle has an open ball escaping to infinity. Submitted: April 19, 2008.; Accepted: January 11, 2009.     

Boundary Behaviour of -Harmonic Functions and Non-Isotropic Hausdorff Measure

 In this paper we consider weighted boundary behaviour of -harmonic functions on the unit ball of . In particular, we show that a measure on the unit sphere of admits a component along the non-isotropic Hausdorff measure on that sphere and that this component gives rise to certain weighted boundary behaviour of the -harmonic extension of the original measure. (Received 24 July 2000; in revised form 10 August 2001)     

A Verified Optimization Technique to Locate Chaotic Regions of Hénon Systems

We present a new verified optimization method to find regions for Hénon systems where the conditions of chaotic behaviour hold. The present paper provides a methodology to verify chaos for certain mappings and regions. We discuss first how to check the set theoretical conditions of a respective theorem in a reliable way by computer programs. Then we introduce optimization problems that provide a model to locate chaotic regions. We prove the correctness of the underlying checking algorithms and the optimization model. We have verified an earlier published chaotic region, and we also give new chaotic places located by the new technique.     

Chaos in temporarily destabilized regular systems with the slow passage effect

We provide evidences for chaotic behaviour in temporarily destabilized regular systems. In particular, we focus on time-continuous systems with the slow passage effect. The extreme sensitivity of the slow passage phase enables the existence of long chaotic transients induced by random pulsatile perturbations, thereby evoking chaotic behaviour in an initially regular system. We confirm the chaotic behaviour of the temporarily destabilized system by calculating the largest Lyapunov exponent. Moreover, we show that the newly obtained unstable periodic orbits can be easily controlled with conventional chaos control techniques, thereby guaranteeing a rich diversity of accessible dynamical states that is usually expected only in intrinsically chaotic systems. Additionally, we discuss the biological importance of presented results.