Non UBC
DSpace
Dettmann, Carl
2018-09-17T05:00:17Z
2018-03-20T11:10
In dynamical systems with divided phase space, the vicinity of the
boundary between regular and chaotic regions is often "sticky," that
is, trapping orbits from the chaotic region for long times. Here, we
investigate the stickiness in the simplest mushroom billiard, which
has a smooth such boundary, but surprisingly subtle behaviour. As a
measure of stickiness, we investigate P(t), the probability of
remaining in the mushroom cap for at least time t given uniform
initial conditions in the chaotic part of the cap. The stickiness is
sensitively dependent on the radius of the stem r via the Diophantine
properties of rho = (2/pi) arccos r. Almost all rho give rise to
families of marginally unstable periodic orbits (MUPOs) where P(t) ~
C/t, dominating the stickiness of the boundary. After characterising
the set for which rho is MUPO-free, we consider the stickiness in this
case, and where rho also has continued fraction expansion with bounded
partial quotients. We show that t^2 P(t) is bounded, varying
infinitely often between values whose ratio is at least 32/27. When
rho has an eventually periodic continued fraction expansion, that is,
a quadratic irrational, t^2 P(t) converges to a log-periodic function.
In general, we expect less regular behaviour, with upper and lower
exponents lying between 1 and 2. The results may shed light on the
parameter dependence of boundary stickiness in annular billiards and
generic area preserving maps.
https://circle.library.ubc.ca/rest/handle/2429/67193?expand=metadata
38 minutes
video/mp4
Author affiliation: The University of Bristol
Banff (Alta.)
10.14288/1.0372060
eng
Unreviewed
Vancouver : University of British Columbia Library
Banff International Research Station for Mathematical Innovation and Discovery
Attribution-NonCommercial-NoDerivatives 4.0 International
http://creativecommons.org/licenses/by-nc-nd/4.0/
Faculty
BIRS Workshop Lecture Videos (Banff, Alta)
Mathematics
Dynamical systems and ergodic theory
Probability theory and stochastic processes
Dynamical systems
How sticky is the chaos/order boundary?
Moving Image
http://hdl.handle.net/2429/67193