TY - ELEC
AU - Dettmann, Carl
PY - 2018
TI - How sticky is the chaos/order boundary?
LA - eng
M3 - Moving Image
AB - 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.
N2 - 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.
UR - https://open.library.ubc.ca/collections/48630/items/1.0372060
ER - End of Reference