Dynamics of "leaking" Hamiltonian systems.

In order to understand the dynamics in more detail, in particular for visualizing the space-filling unstable foliation of closed chaotic Hamiltonian systems, we propose to leak them up. The cutting out of a finite region of their phase space, the leak, through which escape is possible, leads to transient chaotic behavior of nearly all the trajectories. The never-escaping points belong to a chaotic saddle whose fractal unstable manifold can easily be determined numerically. It is an approximant of the full Hamiltonian foliation, the better the smaller the leak is. The escape rate depends sensitively on the orientation of the leak even if its area is fixed. The applications for chaotic advection, for chemical reactions superimposed on hydrodynamical flows, and in other branches of physics are discussed.

[1]  Nenad Pavin,et al.  Bursts in average lifetime of transients for chaotic logistic map with a hole , 1997 .

[2]  N. Chernov,et al.  Ergodic properties of Anosov maps with rectangular holes , 1997 .

[3]  James A. Yorke,et al.  Expanding maps on sets which are almost invariant. Decay and chaos , 1979 .

[4]  Thomas M. Antonsen,et al.  Modeling fractal entrainment sets of tracers advected by chaotic temporally irregular fluid flows using random maps , 1997 .

[5]  P. Grassberger,et al.  Escape and sensitive dependence on initial conditions in a symplectic repeller , 1993 .

[6]  P. Haynes,et al.  The effect of forcing on the spatial structure and spectra of chaotically advected passive scalars , 1999, chao-dyn/9912015.

[7]  P. S. Letelier,et al.  Mixmaster chaos , 2000, gr-qc/0011001.

[8]  H. Buljan,et al.  Bursts in the chaotic trajectory lifetimes preceding controlled periodic motion , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[9]  Conditionally invariant measures for Anosov maps with small holes , 1998 .

[10]  D. Sornette,et al.  Fractal Set of Recurrent Orbits in Billiards , 1990 .

[11]  Grebogi,et al.  Stabilizing chaotic-scattering trajectories using control. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[12]  Tamás Tél,et al.  Advection in chaotically time-dependent open flows , 1998 .

[13]  Artur Lopes,et al.  Open Billiards: Invariant and Conditionally Iinvariant Probabilities on Cantor Sets , 1996, SIAM J. Appl. Math..

[14]  Zoltan Neufeld,et al.  Chaotic advection of reacting substances: Plankton dynamics on a meandering jet , 1999, chao-dyn/9906029.

[15]  Grebogi,et al.  Fractal boundaries for exit in Hamiltonian dynamics. , 1988, Physical review. A, General physics.

[16]  Raymond T. Pierrehumbert,et al.  Tracer microstructure in the large-eddy dominated regime , 1994 .

[17]  Ott,et al.  Fractal dimension in nonhyperbolic chaotic scattering. , 1991, Physical review letters.

[18]  Invariant measures for Anosov maps with small holes , 2000, Ergodic Theory and Dynamical Systems.

[19]  Lopez,et al.  Multifractal structure of chaotically advected chemical fields , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[20]  Cristóbal López,et al.  Smooth-filamental transition of active tracer fields stirred by chaotic advection , 1999, chao-dyn/9906019.

[21]  H. Buljan,et al.  Many-hole interactions and the average lifetimes of chaotic transients that precede controlled periodic motion. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[22]  Alerhand,et al.  Mesoscopic junctions, random scattering, and strange repellers. , 1990, Physical review letters.