Seminar

Understanding the transport properties of higher-dimensional

systems is of great importance in a wide variety of applications,

e.g., for celestial mechanics, particle accelerators, or the

dynamics of atoms and molecules.  A prototypical class of model

systems are billiards for which a Poincaré section leads to

discrete-time map.  For the dynamics in three-dimensional

billiards a four-dimensional symplectic map is obtained which is

challenging to visualize. By means of the recently introduced 3D

phase-space slices an intuitive representation of the

organization of the mixed phase space with regular and chaotic

dynamics is obtained. Of particular interest for applications are

constraints to classical transport between different regions of

phase space which manifest in the statistics of Poincaré

recurrence times. For a 3D paraboloid billiard we observe a slow

power-law decay caused by long-trapped trajectories which we

analyze in phase space and in frequency space. Consistent with

previous results for 4D maps we find that: (i) Trapping takes

place close to regular structures outside the Arnold web. (ii)

Trapping is not due to a generalized island-around-island

hierarchy. (iii) The dynamics of sticky orbits is governed by

resonance channels which extend far into the chaotic sea. We find

clear signatures of partial transport barriers. Moreover, we

visualize the geometry of stochastic layers in resonance channels

explored by sticky orbits.



Reference:

 3D Billiards: Visualization of Regular Structures and

 Trapping of Chaotic Trajectories

 M. Firmbach, S. Lange, R. Ketzmerick, and A. Bäcker,

 Phys. Rev. E 98, 022214 (2018)

 https://doi.org/10.1103/PhysRevE.98.022214