Estimating stable and unstable sets and their role as transport barriers in stochastic flows

Navigating phase space transport with the origin-fate map

We introduce and demonstrate the usage of the origin-fate map (OFM) as a tool for the detailed investigation of phase space transport in reactant-product-type systems. For these systems, which exhibit clearly defined start and end states, it is possible to build a comprehensive picture of the lobe dynamics by considering backward and forward integration of sets of initial conditions to index their origin and fate. We illustrate the method and its utility in the study of a two degrees of freedom caldera potential with four exits, demonstrating that the OFM not only recapitulates results from classical manifold theory but even provides more detailed information about complex lobe structures. The OFM allows the detection of dynamically significant transitions caused by the creation of new lobes and is also able to guide the prediction of the position of unstable periodic orbits (UPOs). Further, we compute the OFM on the periodic orbit dividing surface (PODS) associated with the transition state of a caldera entrance, which allows for a powerful analysis of reactive trajectories. The intersection of the manifolds corresponding to this UPO with other manifolds in the phase space results in the appearance of lobes on the PODS, which are directly classified by the OFM. This allows computations of branching ratios and the exploration of a fractal cascade of lobes as the caldera is stretched, which results in fluctuations in the branching ratio and chaotic selectivity. The OFM is found to be a simple and very useful tool with a vast range of descriptive and quantitative applications.