# American Institute of Mathematical Sciences

September  2012, 32(9): 3029-3042. doi: 10.3934/dcds.2012.32.3029

## The efficient approximation of coherent pairs in non-autonomous dynamical systems

Received  February 2012 Revised  March 2012 Published  April 2012

The aim of this paper is the construction of numerical tools for the efficient approximation of transport phenomena in non-autonomous dynamical systems. We focus on transfer operator methods which have been developed in the last years for the treatment of non-autonomous dynamical systems. For instance Froyland et al. [11] proposed a method for the approximation of so-called coherent pairs -- these pairs of sets represent time-dependent slowly mixing structures -- by thresholding singular vectors of a normalized transfer operator over a fixed time-interval. In principle such transfer operator methods involve long term simulations of trajectories on the whole state space. In our main result we show that transport phenomena over a fixed (long) time horizon imply the existence of almost invariant sets over shorter time intervals if the transport process is slow enough. This fact is used to formulate an algorithm that preselects part of state space as a candidate for containing one of the sets of a coherent pair. By this we significantly reduce the related numerical effort.
Citation: Michael Dellnitz, Christian Horenkamp. The efficient approximation of coherent pairs in non-autonomous dynamical systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (9) : 3029-3042. doi: 10.3934/dcds.2012.32.3029
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