Max Planck Institute for Dynamics and Self-Organization -- Department for Nonlinear Dynamics and Network Dynamics Group
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Thursday, 20.11.2008 17:15 c.t.

Where to place a hole to achieve the fastest escape

by Prof. Dr. Leonid Bunimovich
from Georgia Institute of Technology, Atlanta, USA

Contact person: Marc Timme


Seminarraum Haus 2, 4. Stock (Bunsenstr.)


We consider a natural question in the title which seems to be overlooked in the theory of open dynamical systems. Consider two subsets (holes) A and B in the phase space M of a (closed) dynamical system. Through which of two open systems (with hole a or with hole B) the escape will be faster? Sometimes only a size (measure) of a hole matters, e.g. for rotations of a circle. However, for chaotic dynamical systems escape through a bigger hole could be slower, i.e. the properties of dynamics may play a role (at least) comparable to the role played by the size of a hole. We present an algorithm which allowed for strongly chaotic systems to establish through which hole the escape will be faster. A striking fact is that the corresponding results are valid for all (finite!) times starting with some moment (exactly computable for some chaotic systems with the "strongest" chaos). The last means that survival probability for a system with hole A will be smaller than for the system with the hole B starting with some fixed moment of time. These result indicate that it is possible to make predictions about dynamics for finite times rather than the asymptotic ones (for "sufficiently large" or on time intervals with ends described by some functions of a small parameter in the system, etc). In particular, a role of the distribution of periodic orbits is very important in these results obtained for classical dynamical systems, which raise a question on interpretation of these results for their quantum counterparts.

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