Flow-transverse Decomposition of Vector Fields
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A fundamental problem in algorithmic Conley index theory is to extract isolating neighborhoods and compute their Conley indices. To enable automatic isolating neighborhoods detection, one needs careful decomposition of the vector fields. In this thesis, we study the flow-transverse decomposition of vector fields. In general, flow-transverse decomposition attempts to find a simplicial decomposition of the phase space such that every facet is transverse to the flow. We discuss necessary and sufficient conditions where a planar flow admits a flow-transverse decomposition. We describe a randomized algorithm to construct such a decomposition whenever it is possible.