4,281 research outputs found
Introducing symplectic billiards
In this article we introduce a simple dynamical system called symplectic
billiards. As opposed to usual/Birkhoff billiards, where length is the
generating function, for symplectic billiards symplectic area is the generating
function. We explore basic properties and exhibit several similarities, but
also differences of symplectic billiards to Birkhoff billiards.Comment: 41 pages, 16 figure
Estimating Lyapunov exponents in billiards
Dynamical billiards are paradigmatic examples of chaotic Hamiltonian
dynamical systems with widespread applications in physics. We study how well
their Lyapunov exponent, characterizing the chaotic dynamics, and its
dependence on external parameters can be estimated from phase space volume
arguments, with emphasis on billiards with mixed regular and chaotic phase
spaces. We show that in the very diverse billiards considered here the leading
contribution to the Lyapunov exponent is inversely proportional to the chaotic
phase space volume, and subsequently discuss the generality of this
relationship. We also extend the well established formalism by Dellago, Posch,
and Hoover to calculate the Lyapunov exponents of billiards to include external
magnetic fields and provide a software implementation of it
Stickiness in mushroom billiards
We investigate dynamical properties of chaotic trajectories in mushroom
billiards. These billiards present a well-defined simple border between a
single regular region and a single chaotic component. We find that the
stickiness of chaotic trajectories near the border of the regular region occurs
through an infinite number of marginally unstable periodic orbits. These orbits
have zero measure, thus not affecting the ergodicity of the chaotic region.
Notwithstanding, they govern the main dynamical properties of the system. In
particular, we show that the marginally unstable periodic orbits explain the
periodicity and the power-law behavior with exponent observed in the
distribution of recurrence times.Comment: 7 pages, 6 figures (corrected version with a new figure
Dynamical tunneling in mushroom billiards
We study the fundamental question of dynamical tunneling in generic
two-dimensional Hamiltonian systems by considering regular-to-chaotic tunneling
rates. Experimentally, we use microwave spectra to investigate a mushroom
billiard with adjustable foot height. Numerically, we obtain tunneling rates
from high precision eigenvalues using the improved method of particular
solutions. Analytically, a prediction is given by extending an approach using a
fictitious integrable system to billiards. In contrast to previous approaches
for billiards, we find agreement with experimental and numerical data without
any free parameter.Comment: 4 pages, 4 figure
Unbounded Orbits for Outer Billiards
Outer billiards is a basic dynamical system, defined relative to a planar
convex shape. This system was introduced in the 1950's by B.H. Neumann and
later popularized in the 1970's by J. Moser. All along, one of the central
questions has been: is there an outer billiards system with an unbounded orbit.
We answer this question by proving that outer billiards defined relative to the
Penrose Kite has an unbounded orbit. The Penrose kite is the quadrilateral that
appears in the famous Penrose tiling. We also analyze some of the finer orbit
structure of outer billiards on the penrose kite. This analysis shows that
there is an uncountable set of unbounded orbits. Our method of proof relates
the problem to self-similar tilings, polygon exchange maps, and arithmetic
dynamics.Comment: 65 pages, computer-aided proof. Auxilliary program, Billiard King,
available from author's website. Latest version is essentially the same as
earlier versions, but with minor improvements and many typos fixe
The dynamics of digits: Calculating pi with Galperin's billiards
In Galperin billiards, two balls colliding with a hard wall form an analog
calculator for the digits of the number . This classical, one-dimensional
three-body system (counting the hard wall) calculates the digits of in a
base determined by the ratio of the masses of the two particles. This base can
be any integer, but it can also be an irrational number, or even the base can
be itself. This article reviews previous results for Galperin billiards
and then pushes these results farther. We provide a complete explicit solution
for the balls' positions and velocities as a function of the collision number
and time. We demonstrate that Galperin billiard can be mapped onto a
two-particle Calogero-type model. We identify a second dynamical invariant for
any mass ratio that provides integrability for the system, and for a sequence
of specific mass ratios we identify a third dynamical invariant that
establishes superintegrability. Integrability allows us to derive some new
exact results for trajectories, and we apply these solutions to analyze the
systematic errors that occur in calculating the digits of with Galperin
billiards, including curious cases with irrational number bases.Comment: 30 pages, 13 figure
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HYPERBOLICITY AND CERTAIN STATISTICAL PROPERTIES OF CHAOTIC BILLIARD SYSTEMS
In this thesis, we address some questions about certain chaotic dynamical systems. In particular, the objects of our studies are chaotic billiards. A billiard is a dynamical system that describe the motions of point particles in a table where the particles collide elastically with the boundary and with each other.
Among the dynamical systems, billiards have a very important position. They are models for many problems in acoustics, optics, classical and quantum mechanics, etc.. Despite of the rather simple description, billiards of different shapes of tables exhibit a wide range of dynamical properties from being complete integrable to chaotic. A very important and also very interesting type of billiards is chaotic (or hyperbolic) billiards. In a hyperbolic billiard system, two nearby trajectories in the phase space can be separated exponentially fast in future.
In the first two Chapters, we prove the Central Limit Theorem and the Almost Sure Invariance Principle for a class of billiard systems with flat points. They are two among the important statistical properties for chaotic systems. In the last chapter, we introduce a random perturbation to a wide class of billiards and prove that even if the original system is completely integrable, the perturbed system can be chaotic even under arbitrarily small random perturbation
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