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Invariant manifolds mediating collinear hydrogen exchange reaction

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posted on 02.12.2019 by Mohammed Alharthi
We discuss the phase space structure for the collinear hydrogen exchange system crosses an energy barrier. Above the reaction threshold, the system must pass through a Symmetric Stretch Periodic Orbit (SSPO) where the dynamics are structuring a Dividing Surface (DS) that separates reactants and products. At low energy, the SSPO can serve as a dividing surface that satisfies the no-recrossing assumption of Transition State Theory (TST). As the energy increases, saddle-node bifurcations occur on both sides of the SSPO. Above the bifurcation energy, trajectories appear that recross the central DS. The region of recrossing trajectories is bounded by the stable manifolds of the additional Unstable Periodic Orbits (UPOs). We investigate the fractal structure of the TST is violating islands of the DS and how it is determined by the invariant manifolds of the additional periodic orbits. We demonstrate that the various layers are all bounded by the same stable manifold. The second part is devoted to the study of the phase space structure in a region where the stable manifold initiated. It appears that some of the ensemble trajectories do not cross the DS. We demonstrate the area of those trajectories validating the boundary of various layers from those trajectories that do not. Following this, we make use of the symmetry and demonstrate that these various layers appeared on all versions of the DS and are bounded by the symmetric invariant manifolds belonging to the UPOs, which existed on both sides of the SSPO. Finally, we demonstrate those invariant manifolds intersecting each other, before intersecting the DS, which leads to more complicated behaviour.


Saudi Arabia, Ministry of Higher Education

Taif University (Ta'if, Saudi Arabia)



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  • Mathematical Sciences


Loughborough University

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© Mohammed Alharthi

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A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of the degree of Doctor of Philosophy of Loughborough University.




Thomas Bartsch ; Anatoly Neishtadt

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