Mathematics – Dynamical Systems
Scientific paper
2010-11-03
J. Phys. A.: Math. Theor. 44, 195102, (2011)
Mathematics
Dynamical Systems
21 pages, 11 figures. Includes discussion of a three-dimensional mushroom
Scientific paper
We investigate mushroom billiards, a class of dynamical systems with sharply divided phase space. For typical values of the control parameter of the system $\rho$, an infinite number of marginally unstable periodic orbits (MUPOs) exist making the system sticky in the sense that unstable orbits approach regular regions in phase space and thus exhibit regular behaviour for long periods of time. The problem of finding these MUPOs is expressed as the well known problem of finding optimal rational approximations of a real number, subject to some system-specific constraints. By introducing a generalized mushroom and using properties of continued fractions, we describe a zero measure set of control parameter values $\rho\in(0,1)$ for which all MUPOs are destroyed and therefore the system is less sticky. The open mushroom (billiard with a hole) is then considered in order to quantify the stickiness exhibited and exact leading order expressions for the algebraic decay of the survival probability function $P(t)$ are calculated for mushrooms with triangular and rectangular stems.
Dettmann Carl P.
Georgiou Orestis
No associations
LandOfFree
Open Mushrooms: Stickiness revisited does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Open Mushrooms: Stickiness revisited, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Open Mushrooms: Stickiness revisited will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-602168