246 research outputs found
The characteristic exponents of the falling ball model
We study the characteristic exponents of the Hamiltonian system of () point masses freely falling in the vertical half line
under constant gravitation and colliding with each other and
the solid floor elastically. This model was introduced and first studied
by M. Wojtkowski. Hereby we prove his conjecture: All relevant characteristic
(Lyapunov) exponents of the above dynamical system are nonzero, provided that
(i. e. the masses do not increase as we go up) and
Counting Berg partitions
We call a Markov partition of a two dimensional hyperbolic toral automorphism
a Berg partition if it contains just two rectangles. We describe all Berg
partitions for a given hyperbolic toral automorphism. In particular there are
exactly (k + n + l + m)/2 nonequivalent Berg partitions with the same
connectivity matrix (k, l, m, n)
Well-posed infinite horizon variational problems on a compact manifold
We give an effective sufficient condition for a variational problem with
infinite horizon on a compact Riemannian manifold M to admit a smooth optimal
synthesis, i. e. a smooth dynamical system on M whose positive
semi-trajectories are solutions to the problem. To realize the synthesis we
construct a well-projected to M invariant Lagrange submanifold of the
extremals' flow in the cotangent bundle T*M. The construction uses the
curvature of the flow in the cotangent bundle and some ideas of hyperbolic
dynamics
Track billiards
We study a class of planar billiards having the remarkable property that
their phase space consists up to a set of zero measure of two invariant sets
formed by orbits moving in opposite directions. The tables of these billiards
are tubular neighborhoods of differentiable Jordan curves that are unions of
finitely many segments and arcs of circles. We prove that under proper
conditions on the segments and the arcs, the billiards considered have non-zero
Lyapunov exponents almost everywhere. These results are then extended to a
similar class of of 3-dimensional billiards. Finally, we find that for some
subclasses of track billiards, the mechanism generating hyperbolicity is not
the defocusing one that requires every infinitesimal beam of parallel rays to
defocus after every reflection off of the focusing boundary.Comment: 7 figure
Fluctuation Theorem, non linear response and the regularity of time reversal symmetry
The Gallavotti - Cohen Fluctuation Theorem (FT) implies an infinite set of
identities between correlation functions that can be seen as a generalization
of Green Kubo formula to the nonlinear regime. As an application, we discuss a
perturbative check of the FT relation through these identities for a simple
Anosov reversible system; we find that the lack of differentiability of the
time reversal symmetry implies a violation of the Gallavotti - Cohen
fluctuation relation. Finally, a brief comparison with Lebowitz - Spohn FT is
reported.Comment: 7 page
Hyperbolic outer billiards : a first example
We present the first example of a hyperbolic outer billiard. More precisely
we construct a one parameter family of examples which in some sense correspond
to the Bunimovich billiards.Comment: 11 pages, 8 figures, to appear in Nonlinearit
Acceleration of bouncing balls in external fields
We introduce two models, the Fermi-Ulam model in an external field and a one
dimensional system of bouncing balls in an external field above a periodically
oscillating plate. For both models we investigate the possibility of unbounded
motion. In a special case the two models are equivalent
Classification of biological micro-objects using optical coherence tomography: in silico study
We report on the development of a technique for differentiating between biological micro-objects using a rigorous, full-wave model of OCT image formation. We model an existing experimental prototype which uses OCT to interrogate a microfluidic chip containing the blood cells. A full-wave model is required since the technique uses light back-scattered by a scattering substrate, rather than by the cells directly. The light back-scattered by the substrate is perturbed upon propagation through the cells, which flow between the substrate and imaging system’s objective lens. We present the key elements of the 3D, Maxwell equation-based computational model, the key findings of the computational study and a comparison with experimental results
Entropy production in Gaussian thermostats
We show that an arbitrary Anosov Gaussian thermostat on a surface is
dissipative unless the external field has a global potential
Linear stability in billiards with potential
A general formula for the linearized Poincar\'e map of a billiard with a
potential is derived. The stability of periodic orbits is given by the trace of
a product of matrices describing the piecewise free motion between reflections
and the contributions from the reflections alone. For the case without
potential this gives well known formulas. Four billiards with potentials for
which the free motion is integrable are treated as examples: The linear
gravitational potential, the constant magnetic field, the harmonic potential,
and a billiard in a rotating frame of reference, imitating the restricted three
body problem. The linear stability of periodic orbits with period one and two
is analyzed with the help of stability diagrams, showing the essential
parameter dependence of the residue of the periodic orbits for these examples.Comment: 22 pages, LaTex, 4 Figure
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