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Dynamics and Bifurcations on the Normally Hyperbolic Invariant Manifold of a Periodically Driven System with Rank-1 Saddle

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Abstract

In chemical reactions, trajectories typically turn from reactants to products when crossing a dividing surface close to the normally hyperbolic invariant manifold (NHIM) given by the intersection of the stable and unstable manifolds of a rank-1 saddle. Trajectories started exactly on the NHIM in principle never leave this manifold when propagated forward or backward in time. This still holds for driven systems when the NHIM itself becomes time-dependent. We investigate the dynamics on the NHIM for a periodically driven model system with two degrees of freedom by numerically stabilizing the motion. Using Poincaré surfaces of section, we demonstrate the occurrence of structural changes of the dynamics, viz., bifurcations of periodic transition state (TS) trajectories when changing the amplitude and frequency of the external driving. In particular, periodic TS trajectories with the same period as the external driving but significantly different parameters — such as mean energy — compared to the ordinary TS trajectory can be created in a saddle-node bifurcation.

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Notes

  1. Note that we use the phrase transition state to refer to the ensemble of states bound indefinitely to the saddle region, i. e., those located on the codimension-2 NHIM. This is distinct from the codimension-1 dividing surface. See footnote 6 of Reference [71] for a more detailed discussion.

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ACKNOWLEDGMENTS

We thank Tobias Mielich, Robin Bardakcioglu, and Matthias Feldmaier for fruitful discussions.

Funding

The German portion of this collaborative work was partially supported by the Deutsche Forschungsgemeinschaft (DFG) through Grant No. MA1639/14-1. The US portion was partially supported by the National Science Foundation (NSF) through Grant No. CHE 1700749. This collaboration has also benefited from support by the European Union’s Horizon 2020 Research and Innovation Program under the Marie Skłodowska-Curie Grant Agreement No. 734557.

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Authors and Affiliations

  1. Institut für Theoretische Physik I, Universität Stuttgart, 70550, Stuttgart, Germany

    Manuel Kuchelmeister, Johannes Reiff & Jörg Main

  2. Department of Chemistry, Johns Hopkins University, 21218 Maryland, Baltimore, United States

    Rigoberto Hernandez

  3. Departments of Chemical & Biomolecular Engineering, and Materials Science and Engineering, Johns Hopkins University, 21218 Maryland, Baltimore, United States

    Rigoberto Hernandez

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  1. Manuel Kuchelmeister
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37D05, 37G15, 37J20, 37M05, 65P30

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Kuchelmeister, M., Reiff, J., Main, J. et al. Dynamics and Bifurcations on the Normally Hyperbolic Invariant Manifold of a Periodically Driven System with Rank-1 Saddle. Regul. Chaot. Dyn. 25, 496–507 (2020). https://doi.org/10.1134/S1560354720050068

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