Chaos Nameplat
  • Volume/Page
  • Keyword
  • DOI
  • Citation
  • Advanced
   
 
 
 

You Tube Flickr Twitter iResearch App Facebook

Search Issue | RSS Feeds RSS
Previous Issue Next Issue

Jun 2012

Volume 22, Issue 2, Articles (02xxxx)

Issue Cover Spotlight Figure

Chaos 22, 026112 (2012); http://dx.doi.org/10.1063/1.3697985 (5 pages)

M. S. Custódio, C. Manchein, and M. W. Beims
Enlarge Image | Read More
Return to All Sections
back to top
RSS Feeds

Introduction to Focus Issue: Statistical mechanics and billiard-type dynamical systems

Edson D. Leonel, Marcus W. Beims, and Leonid A. Bunimovich

Chaos 22, 026101 (2012); http://dx.doi.org/10.1063/1.4730155 (3 pages)

Online Publication Date: 20 June 2012

Full Text: Read Online (HTML) | Download PDF

Show Abstract
Dynamical systems of the billiard type are of fundamental importance for the description of numerous phenomena observed in many different fields of research, including statistical mechanics, Hamiltonian dynamics, nonlinear physics, and many others. This Focus Issue presents the recent progress in this area with contributions from the mathematical as well as physical stand point.
Show PACS
05.20.-y Classical statistical mechanics
05.45.-a Nonlinear dynamics and chaos

Billiards: A singular perturbation limit of smooth Hamiltonian flows

V. Rom-Kedar and D. Turaev

Chaos 22, 026102 (2012); http://dx.doi.org/10.1063/1.4722010 (21 pages) | Cited 1 time

Online Publication Date: 20 June 2012

Full Text: Read Online (HTML) | Download PDF

Show Abstract
Nonlinear multi-dimensional Hamiltonian systems that are not near integrable typically have mixed phase space and a plethora of instabilities. Hence, it is difficult to analyze them, to visualize them, or even to interpret their numerical simulations. We survey an emerging methodology for analyzing a class of such systems: Hamiltonians with steep potentials that limit to billiards.
Show PACS
05.45.-a Nonlinear dynamics and chaos
02.60.Cb Numerical simulation; solution of equations

Many faces of stickiness in Hamiltonian systems

Leonid A. Bunimovich and Luz V. Vela-Arevalo

Chaos 22, 026103 (2012); http://dx.doi.org/10.1063/1.3692974 (7 pages) | Cited 4 times

Online Publication Date: 20 June 2012

Full Text: Read Online (HTML) | Download PDF

Show Abstract
We discuss the phenomenon of stickiness in Hamiltonian systems. By visual examples of billiards, it is demonstrated that one must make a difference between internal (within chaotic sea(s)) and external (in vicinity of KAM tori) stickiness. Besides, there exist two types of KAM-islands, elliptic and parabolic ones, which demonstrate different abilities of stickiness.
Show PACS
05.45.-a Nonlinear dynamics and chaos
02.30.Jr Partial differential equations

Statistical properties of the system of two falling balls

Péter Bálint, Gábor Borbély, and András Némedy Varga

Chaos 22, 026104 (2012); http://dx.doi.org/10.1063/1.3692973 (30 pages) | Cited 1 time

Online Publication Date: 20 June 2012

Full Text: Read Online (HTML) | Download PDF

Show Abstract
We consider the motion of two point masses along a vertical half-line that are subject to constant gravitational force and collide elastically with each other and the floor. This model was introduced by Wojtkowski who established hyperbolicity and ergodicity in case the lower ball is heavier. Here, we investigate the dynamics in discrete time and prove that, for an open set of the external parameter (the relative mass of the lower ball), the system mixes polynomially—modulo logarithmic factors, correlations decay as math(1/n2)—and satisfies the Central Limit Theorem.
Show PACS
05.20.-y Classical statistical mechanics
02.10.De Algebraic structures and number theory

Stable regimes for hard disks in a channel with twisting walls

N. Chernov, A. Korepanov, and N. Simányi

Chaos 22, 026105 (2012); http://dx.doi.org/10.1063/1.3695367 (13 pages) | Cited 2 times

Online Publication Date: 20 June 2012

Full Text: Read Online (HTML) | Download PDF

Show Abstract
We study a gas of N hard disks in a box with semi-periodic boundary conditions. The unperturbed gas is hyperbolic and ergodic (these facts are proved for N = 2 and expected to be true for all N ≥ 2). We study various perturbations by twisting the outgoing velocity at collisions with the walls. We show that the dynamics tends to collapse to various stable regimes, however we define the perturbations, and however small they are.
Show PACS
05.45.Ac Low-dimensional chaos
51.10.+y Kinetic and transport theory of gases

Chaos in the square billiard with a modified reflection law

Gianluigi Del Magno, João Lopes Dias, Pedro Duarte, José Pedro Gaivão, and Diogo Pinheiro

Chaos 22, 026106 (2012); http://dx.doi.org/10.1063/1.3701992 (11 pages) | Cited 2 times

Online Publication Date: 21 June 2012

Full Text: Read Online (HTML) | Download PDF

Show Abstract
The purpose of this paper is to study the dynamics of a square billiard with a non-standard reflection law such that the angle of reflection of the particle is a linear contraction of the angle of incidence. We present numerical and analytical arguments that the nonwandering set of this billiard decomposes into three invariant sets, a parabolic attractor, a chaotic attractor, and a set consisting of several horseshoes. This scenario implies the positivity of the topological entropy of the billiard, a property that is in sharp contrast with the integrability of the square billiard with the standard reflection law.
Show PACS
05.45.Pq Numerical simulations of chaotic systems
05.70.Ce Thermodynamic functions and equations of state
02.60.Cb Numerical simulation; solution of equations

Structure and evolution of strange attractors in non-elastic triangular billiards

Aubin Arroyo, Roberto Markarian, and David P. Sanders

Chaos 22, 026107 (2012); http://dx.doi.org/10.1063/1.4719149 (12 pages) | Cited 2 times

Online Publication Date: 21 June 2012

Full Text: Read Online (HTML) | Download PDF

multimedia

Show Abstract
We study non-elastic billiard dynamics in an equilateral triangular table. In such dynamics, collisions with the walls of the table are not elastic, as in standard billiards; rather, the outgoing angle of the trajectory with the normal vector to the boundary at the point of collision is a uniform factor λ < 1 smaller than the incoming angle. This leads to contraction in phase space for the discrete-time dynamics between consecutive collisions, and hence to attractors of zero Lebesgue measure, which are almost always fractal strange attractors with chaotic dynamics, due to the presence of an expansion mechanism. We study the structure of these strange attractors and their evolution as the contraction parameter λ is varied. For λ∈(0,math), we prove rigorously that the attractor has the structure of a Cantor set times an interval, whereas for larger values of λ gaps arise in the Cantor structure. For λ close to 1, the attractor splits into three transitive components, whose basins of attraction have fractal boundaries.
Show PACS
05.45.Ac Low-dimensional chaos
05.45.Df Fractals

Resonances within chaos

G. Gallavotti, G. Gentile, and A. Giuliani

Chaos 22, 026108 (2012); http://dx.doi.org/10.1063/1.3695370 (6 pages) | Cited 2 times

Online Publication Date: 21 June 2012

Full Text: Read Online (HTML) | Download PDF

Show Abstract
A chaotic system under periodic forcing can develop a periodically visited strange attractor. We discuss simple models in which the phenomenon, quite easy to see in numerical simulations, can be completely studied analytically.
Show PACS
05.45.-a Nonlinear dynamics and chaos
05.45.Xt Synchronization; coupled oscillators

Billiards with a given number of (k,n)-orbits

Sônia Pinto-de-Carvalho and Rafael Ramírez-Ros

Chaos 22, 026109 (2012); http://dx.doi.org/10.1063/1.3697986 (6 pages) | Cited 1 time

Online Publication Date: 21 June 2012

Full Text: Read Online (HTML) | Download PDF

Show Abstract
We consider billiard dynamics inside a smooth strictly convex curve. For each pair of integers (k,n), we focus our attention on the billiard trajectory that traces a closed polygon with n sides and makes k turns inside the billiard table, called a (k,n)-orbit. Birkhoff proved that a strictly convex billiard always has at least two (k,n)-orbits for any relatively prime integers k and n such that 1 ≤ k<n. In this paper, we show that Birkhoff’s lower bound is optimal by presenting examples of strictly convex billiards with exactly two (k,n)-orbits. We generalize the result to billiards with given even numbers of orbits for a finite number of periods.
Show PACS
02.30.Oz Bifurcation theory

Classification of symmetric periodic trajectories in ellipsoidal billiards

Pablo S. Casas and Rafael Ramírez-Ros

Chaos 22, 026110 (2012); http://dx.doi.org/10.1063/1.4706003 (24 pages) | Cited 1 time

Online Publication Date: 21 June 2012

Full Text: Read Online (HTML) | Download PDF

Show Abstract
We classify nonsingular symmetric periodic trajectories (SPTs) of billiards inside ellipsoids of mathn+1 without any symmetry of revolution. SPTs are defined as periodic trajectories passing through some symmetry set. We prove that there are exactly 22n(2n+1-1) classes of such trajectories. We have implemented an algorithm to find minimal SPTs of each of the 12 classes in the 2D case (math2) and each of the 112 classes in the 3D case (math3). They have periods 3, 4, or 6 in the 2D case and 4, 5, 6, 8, or 10 in the 3D case. We display a selection of 3D minimal SPTs. Some of them have properties that cannot take place in the 2D case.
Show PACS
05.45.-a Nonlinear dynamics and chaos

Three unequal masses on a ring and soft triangular billiards

H. A. Oliveira, G. A. Emidio, and M. W. Beims

Chaos 22, 026111 (2012); http://dx.doi.org/10.1063/1.3683465 (6 pages) | Cited 2 times

Online Publication Date: 21 June 2012

Full Text: Read Online (HTML) | Download PDF

Show Abstract
The dynamics of three soft interacting particles on a ring is shown to correspond to the motion of one particle inside a soft triangular billiard. The dynamics inside the soft billiard depends only on the masses ratio between particles and softness ratio of the particles interaction. The transition from soft to hard interactions can be appropriately explored using potentials for which the corresponding equations of motion are well defined in the hard wall limit. Numerical examples are shown for the soft Toda-like interaction and the error function.
Show PACS
05.45.Ac Low-dimensional chaos
05.50.+q Lattice theory and statistics (Ising, Potts, etc.)
FREE

Chaotic and Arnold stripes in weakly chaotic Hamiltonian systems

M. S. Custódio, C. Manchein, and M. W. Beims

Chaos 22, 026112 (2012); http://dx.doi.org/10.1063/1.3697985 (5 pages) | Cited 3 times

Online Publication Date: 21 June 2012

Full Text: Read Online (HTML) | Download PDF

Show Abstract
The dynamics in weakly chaotic Hamiltonian systems strongly depends on initial conditions (ICs) and little can be affirmed about generic behaviors. Using two distinct Hamiltonian systems, namely one particle in an open rectangular billiard and four particles globally coupled on a discrete lattice, we show that in these models, the transition from integrable motion to weak chaos emerges via chaotic stripes as the nonlinear parameter is increased. The stripes represent intervals of initial conditions which generate chaotic trajectories and increase with the nonlinear parameter of the system. In the billiard case, the initial conditions are the injection angles. For higher-dimensional systems and small nonlinearities, the chaotic stripes are the initial condition inside which Arnold diffusion occurs.
Show PACS
05.45.Jn High-dimensional chaos
05.50.+q Lattice theory and statistics (Ising, Potts, etc.)
05.60.-k Transport processes

Quantifying intermittency in the open drivebelt billiard

Carl P. Dettmann and Orestis Georgiou

Chaos 22, 026113 (2012); http://dx.doi.org/10.1063/1.3685522 (9 pages) | Cited 2 times

Online Publication Date: 21 June 2012

Full Text: Read Online (HTML) | Download PDF

Show Abstract
A “drivebelt” stadium billiard with boundary consisting of circular arcs of differing radius connected by their common tangents shares many properties with the conventional “straight” stadium, including hyperbolicity and mixing, as well as intermittency due to marginally unstable periodic orbits (MUPOs). Interestingly, the roles of the straight and curved sides are reversed. Here, we discuss intermittent properties of the chaotic trajectories from the point of view of escape through a hole in the billiard, giving the exact leading order coefficient mathtP(t) of the survival probability P(t) which is algebraic for fixed hole size. However, in the natural scaling limit of small hole size inversely proportional to time, the decay remains exponential. The big distinction between the straight and drivebelt stadia is that in the drivebelt case, there are multiple families of MUPOs leading to qualitatively new effects. A further difference is that most marginal periodic orbits in this system are oblique to the boundary, thus permitting applications that utilise total internal reflection such as microlasers.
Show PACS
05.45.-a Nonlinear dynamics and chaos
02.50.Cw Probability theory

Effect of noise in open chaotic billiards

Eduardo G. Altmann, Jorge C. Leitão, and João Viana Lopes

Chaos 22, 026114 (2012); http://dx.doi.org/10.1063/1.3697408 (7 pages) | Cited 1 time

Online Publication Date: 21 June 2012

Full Text: Read Online (HTML) | Download PDF

Show Abstract
We investigate the effect of white-noise perturbations on chaotic trajectories in open billiards. We focus on the temporal decay of the survival probability for generic mixed-phase-space billiards. The survival probability has a total of five different decay regimes that prevail for different intermediate times. We combine new calculations and recent results on noise perturbed Hamiltonian systems to characterize the origin of these regimes and to compute how the parameters scale with noise intensity and billiard openness. Numerical simulations in the annular billiard support and illustrate our results.
Show PACS
05.45.Ac Low-dimensional chaos
05.45.Pq Numerical simulations of chaotic systems
02.50.Cw Probability theory
05.40.Ca Noise

Lorentz process with shrinking holes in a wall

Péter Nándori and Domokos Szász

Chaos 22, 026115 (2012); http://dx.doi.org/10.1063/1.4717521 (10 pages) | Cited 1 time

Online Publication Date: 21 June 2012

Full Text: Read Online (HTML) | Download PDF

Show Abstract
We ascertain the diffusively scaled limit of a periodic Lorentz process in a strip with an almost reflecting wall at the origin. Here, almost reflecting means that the wall contains a small hole waning in time. The limiting process is a quasi-reflected Brownian motion, which is Markovian, but not strong Markovian. Local time results for the periodic Lorentz process, having independent interest, are also found and used.
Show PACS
05.45.-a Nonlinear dynamics and chaos
02.50.Ga Markov processes
05.40.Jc Brownian motion
FREE

Billiard dynamics: An updated survey with the emphasis on open problems

Eugene Gutkin

Chaos 22, 026116 (2012); http://dx.doi.org/10.1063/1.4729307 (13 pages) | Cited 4 times

Online Publication Date: 25 June 2012

Full Text: Read Online (HTML) | Download PDF

Show Abstract
This is an updated and expanded version of our earlier survey article [E. Gutkin, “Billiard dynamics: a survey with the emphasis on open problems,” Regular Chaotic Dyn. 8, 1–13 (2003)]. Section 1 introduces the subject matter. Sections 2 , 3 , 4 expose the basic material following the paradigm of elliptic, hyperbolic, and parabolic billiard dynamics. In Sec. 5, we report on the recent work pertaining to the problems and conjectures exposed in the survey [E. Gutkin, “Billiard dynamics: a survey with the emphasis on open problems,” Regular Chaotic Dyn. 8, 1–13 (2003)]. Besides, in Sec. 5 we formulate a few additional problems and conjectures. The bibliography has been updated and considerably expanded.
Show PACS
05.45.-a Nonlinear dynamics and chaos
02.40.Hw Classical differential geometry

A two-stage approach to relaxation in billiard systems of locally confined hard spheres

Pierre Gaspard and Thomas Gilbert

Chaos 22, 026117 (2012); http://dx.doi.org/10.1063/1.3697689 (8 pages) | Cited 1 time

Online Publication Date: 25 June 2012

Full Text: Read Online (HTML) | Download PDF

Show Abstract
We consider the three-dimensional dynamics of systems of many interacting hard spheres, each individually confined to a dispersive environment, and show that the macroscopic limit of such systems is characterized by a coefficient of heat conduction whose value reduces to a dimensional formula in the limit of vanishingly small rate of interaction. It is argued that this limit arises from an effective loss of memory. Similarities with the diffusion of a tagged particle in binary mixtures are emphasized.
Show PACS
05.60.-k Transport processes
05.70.Ln Nonequilibrium and irreversible thermodynamics
05.45.-a Nonlinear dynamics and chaos
05.20.-y Classical statistical mechanics

Spreading of energy in the Ding-Dong model

S. Roy and A. Pikovsky

Chaos 22, 026118 (2012); http://dx.doi.org/10.1063/1.3695369 (7 pages) | Cited 3 times

Online Publication Date: 25 June 2012

Full Text: Read Online (HTML) | Download PDF

Show Abstract
We study the properties of energy spreading in a lattice of elastically colliding harmonic oscillators (Ding-Dong model). We demonstrate that in the regular lattice the spreading from a localized initial state is mediated by compactons and chaotic breathers. In a disordered lattice, the compactons do not exist, and the spreading eventually stops, resulting in a finite configuration with a few chaotic spots.
Show PACS
03.65.Ge Solutions of wave equations: bound states
05.50.+q Lattice theory and statistics (Ising, Potts, etc.)
05.45.-a Nonlinear dynamics and chaos
05.60.-k Transport processes

Fermi acceleration in time-dependent rectangular billiards due to multiple passages through resonances

A. P. Itin and A. I. Neishtadt

Chaos 22, 026119 (2012); http://dx.doi.org/10.1063/1.4705101 (8 pages) | Cited 1 time

Online Publication Date: 25 June 2012

Full Text: Read Online (HTML) | Download PDF

Show Abstract
We consider a slowly rotating rectangular billiard with moving boundaries and use canonical perturbation theory to describe the dynamics of a billiard particle. In the process of slow evolution, certain resonance conditions can be satisfied. Correspondingly, phenomena of scattering on a resonance and capture into a resonance happen in the system. These phenomena lead to destruction of adiabatic invariance and to unlimited acceleration of the particle.
Show PACS
05.45.-a Nonlinear dynamics and chaos
03.65.Ta Foundations of quantum mechanics; measurement theory

A consistent approach for the treatment of Fermi acceleration in time-dependent billiards

A. K. Karlis, F. K. Diakonos, and V. Constantoudis

Chaos 22, 026120 (2012); http://dx.doi.org/10.1063/1.3697399 (10 pages) | Cited 2 times

Online Publication Date: 25 June 2012

Full Text: Read Online (HTML) | Download PDF

Show Abstract
The standard description of Fermi acceleration, developing in a class of time-dependent billiards, is given in terms of a diffusion process taking place in momentum space. Within this framework, the evolution of the probability density function (PDF) of the magnitude of particle velocities as a function of the number of collisions n is determined by the Fokker-Planck equation (FPE). In the literature, the FPE is constructed by identifying the transport coefficients with the ensemble averages of the change of the magnitude of particle velocity and its square in the course of one collision. Although this treatment leads to the correct solution after a sufficiently large number of collisions have been reached, the transient part of the evolution of the PDF is not described. Moreover, in the case of the Fermi-Ulam model (FUM), if a standard simplification is employed, the solution of the FPE is even inconsistent with the values of the transport coefficients used for its derivation. The goal of our work is to provide a self-consistent methodology for the treatment of Fermi acceleration in time-dependent billiards. The proposed approach obviates any assumptions for the continuity of the random process and the existence of the limits formally defining the transport coefficients of the FPE. Specifically, we suggest, instead of the calculation of ensemble averages, the derivation of the one-step transition probability function and the use of the Chapman-Kolmogorov forward equation. This approach is generic and can be applied to any time-dependent billiard for the treatment of Fermi-acceleration. As a first step, we apply this methodology to the FUM, being the archetype of time-dependent billiards to exhibit Fermi acceleration.
Show PACS
05.60.Cd Classical transport
02.50.Cw Probability theory
05.10.Gg Stochastic analysis methods (Fokker-Planck, Langevin, etc.)
05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion

The role of dissipation in time-dependent non-integrable focusing billiards

Alexei B. Ryabov and Alexander Loskutov

Chaos 22, 026121 (2012); http://dx.doi.org/10.1063/1.4722744 (7 pages) | Cited 1 time

Online Publication Date: 25 June 2012

Full Text: Read Online (HTML) | Download PDF

Show Abstract
In this study, we compare the dynamical properties of chaotic and nearly integrable time-dependent focusing billiards with elastic and dissipative boundaries. We show that in the system without dissipation the average velocity of particles scales with the number of collisions as math∝nα. In the fully chaotic case, this scaling corresponds to a diffusion process with α ≈ 1/2, whereas in the nearly integrable case, this dependence has a crossover; slow particles accelerate in a slow subdiffusive manner with α<1/2, while acceleration of fast particles is much stronger and their average velocity grows super-diffusively, i.e., α>1/2. Assuming math∝nα for a non-dissipative system, we obtain that in its dissipative counterpart the average velocity approaches to mathfin∝1/δα, where δ is the damping coefficient. So that mathfin∝math in the fully chaotic billiards, and the characteristics exponents α changes with δ from α1>1/2 to α2<1/2 in the nearly integrable systems. We conjecture that in the limit of moderate dissipation the chaotic time-depended billiards can accelerate the particles more efficiently. By contrast, in the limit of small dissipations, the nearly integrable billiards can become the most efficient accelerator. Furthermore, due to the presence of attractors in this system, the particles trajectories will be focused in narrow beams with a discrete velocity spectrum.
Show PACS
05.45.-a Nonlinear dynamics and chaos

Decay of energy and suppression of Fermi acceleration in a dissipative driven stadium-like billiard

André L. P. Livorati, Iberê L. Caldas, and Edson D. Leonel

Chaos 22, 026122 (2012); http://dx.doi.org/10.1063/1.3699465 (8 pages) | Cited 1 time

Online Publication Date: 25 June 2012

Full Text: Read Online (HTML) | Download PDF

Show Abstract
The behavior of the average energy for an ensemble of non-interacting particles is studied using scaling arguments in a dissipative time-dependent stadium-like billiard. The dynamics of the system is described by a four dimensional nonlinear mapping. The dissipation is introduced via inelastic collisions between the particles and the moving boundary. For different combinations of initial velocities and damping coefficients, the long time dynamics of the particles leads them to reach different states of final energy and to visit different attractors, which change as the dissipation is varied. The decay of the average energy of the particles, which is observed for a large range of restitution coefficients and different initial velocities, is described using scaling arguments. Since this system exhibits unlimited energy growth in the absence of dissipation, our results for the dissipative case give support to the principle that Fermi acceleration seems not to be a robust phenomenon.
Show PACS
05.45.-a Nonlinear dynamics and chaos

In-flight and collisional dissipation as a mechanism to suppress Fermi acceleration in a breathing Lorentz gas

Diego F. M. Oliveira and Edson D. Leonel

Chaos 22, 026123 (2012); http://dx.doi.org/10.1063/1.3697392 (11 pages) | Cited 1 time

Online Publication Date: 25 June 2012

Full Text: Read Online (HTML) | Download PDF

Show Abstract
Some dynamical properties for a time dependent Lorentz gas considering both the dissipative and non dissipative dynamics are studied. The model is described by using a four-dimensional nonlinear mapping. For the conservative dynamics, scaling laws are obtained for the behavior of the average velocity for an ensemble of non interacting particles and the unlimited energy growth is confirmed. For the dissipative case, four different kinds of damping forces are considered namely: (i) restitution coefficient which makes the particle experiences a loss of energy upon collisions; and in-flight dissipation given by (ii) F = −ηV2; (iii) F = −ηVμ with μ ≠ 1 and μ ≠ 2 and; (iv) F = −ηV, where η is the dissipation parameter. Extensive numerical simulations were made and our results confirm that the unlimited energy growth, observed for the conservative dynamics, is suppressed for the dissipative case. The behaviour of the average velocity is described using scaling arguments and classes of universalities are defined.
Show PACS
05.45.Ac Low-dimensional chaos
05.45.Pq Numerical simulations of chaotic systems

Dynamics of some piecewise smooth Fermi-Ulam models

Jacopo de Simoi and Dmitry Dolgopyat

Chaos 22, 026124 (2012); http://dx.doi.org/10.1063/1.3695379 (12 pages) | Cited 2 times

Online Publication Date: 25 June 2012

Full Text: Read Online (HTML) | Download PDF

Show Abstract
We find a normal form which describes the high energy dynamics of a class of piecewise smooth Fermi-Ulam ping pong models. Depending on the value of a single real parameter, the dynamics can be either hyperbolic or elliptic. In the first case, we prove that the set of orbits undergoing Fermi acceleration has zero measure but full Hausdorff dimension. We also show that for almost every orbit, the energy eventually falls below a fixed threshold. In the second case, we prove that, generically, we have stable periodic orbits for arbitrarily high energies and that the set of Fermi accelerating orbits may have infinite measure.
Show PACS
05.45.Jn High-dimensional chaos
Return to All Sections
Close

Google Calendar

ADVERTISEMENT

Go to AIP Home   © AIP Publishing LLC
close