17 research outputs found
Amplitude distribution of eigenfunctions in mixed systems
We study the amplitude distribution of irregular eigenfunctions in systems
with mixed classical phase space. For an appropriately restricted random wave
model a theoretical prediction for the amplitude distribution is derived and
good agreement with numerical computations for the family of limacon billiards
is found. The natural extension of our result to more general systems, e.g.
with a potential, is also discussed.Comment: 13 pages, 3 figures. Some of the pictures are included in low
resolution only. For a version with pictures in high resolution see
http://www.physik.uni-ulm.de/theo/qc/ or http://www.maths.bris.ac.uk/~maab
No elliptic islands for the universal area-preserving map
A renormalization approach has been used in \cite{EKW1} and \cite{EKW2} to
prove the existence of a \textit{universal area-preserving map}, a map with
hyperbolic orbits of all binary periods. The existence of a horseshoe, with
positive Hausdorff dimension, in its domain was demonstrated in \cite{GJ1}. In
this paper the coexistence problem is studied, and a computer-aided proof is
given that no elliptic islands with period less than 20 exist in the domain. It
is also shown that less than 1.5% of the measure of the domain consists of
elliptic islands. This is proven by showing that the measure of initial
conditions that escape to infinity is at least 98.5% of the measure of the
domain, and we conjecture that the escaping set has full measure. This is
highly unexpected, since generically it is believed that for conservative
systems hyperbolicity and ellipticity coexist
About ergodicity in the family of limacon billiards
By continuation from the hyperbolic limit of the cardioid billiard we show
that there is an abundance of bifurcations in the family of limacon billiards.
The statistics of these bifurcation shows that the size of the stable intervals
decreases with approximately the same rate as their number increases with the
period. In particular, we give numerical evidence that arbitrarily close to the
cardioid there are elliptic islands due to orbits created in saddle node
bifurcations. This shows explicitly that if in this one parameter family of
maps ergodicity occurs for more than one parameter the set of these parameter
values has a complicated structure.Comment: 17 pages, 9 figure
A proof of Kolmogorov\u2019s theorem on invariant tori using canonical transformations defined by the Lie method
In this paper a proof is given of Kolmogorov\u2019s theorem on the existence of invariant tori in nearly integrable Hamiltonian systems. The scheme of proof is that of Kolmogorov, the only difference being in the way canonical transformations near the identity are defined. Precisely, use is made of the Lie method, which avoids any inversion and thus any use of the implicit-function theorem. This technical fact eliminates a spurious ingredient and simplifies the establishment of a central estimate
On the reliability of numerical studies of stochasticity
In the numerical study of classical dynamical systems presenting stochastic behaviour one frequently makes use, in an explicit or an implicit way, of the Birkhoff ergodic theorem. The correct interpretation of the obtained results presents some delicate problems related to the coexistence of many mutually singular invariant measures. In this paper we study this question in an experimental way on some simple model examples
On the reliability of numerical studies of stochasticity I: Existence of time averages
The stochastic properties of classical dynamical systems are often studied by means of numerical computations of orbits up to very large times, so that the accumulation of numerical errors would appear to destroy the reliability of the computations. We discuss this problem on the basis of a theorem of Anosov and Bowen which implies that, if the errors at each step are small enough, for Anosov systems the computations of time averages are reliable even for infinite times. We test numerically from this point of view three classical examples