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Phase Space Transport in a Symmetric Caldera Potential with Three Index-1 Saddles and No Minima

Matthaios Katsanikas

https://orcid.org/0000-0002-1098-458X

Research Center for Astronomy and Applied Mathematics, Academy of Athens, Soranou Efesiou 4, Athens, GR-11527, Greece

School of Mathematics, University of Bristol, Fry Building, Woodland Road, Bristol, BS8 1UG, United Kingdom

E-mail Address: mkatsan@academyofathens.gr

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Makrina Agaoglou

Instituto de Ciencias Matemáticas, CSIC, C/Nicolás Cabrera 15, Campus Cantoblanco, 28049 Madrid, Spain

E-mail Address: makrina.agaoglou@icmat.es

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Stephen Wiggins

School of Mathematics, University of Bristol, Fry Building, Woodland Road, Bristol, BS8 1UG, United Kingdom

E-mail Address: s.wiggins@bristol.ac.uk

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Ana M. Mancho

Instituto de Ciencias Matemáticas, CSIC, C/Nicolás Cabrera 15, Campus Cantoblanco, 28049 Madrid, Spain

E-mail Address: a.m.mancho@icmat.es

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https://doi.org/10.1142/S0218127422300233Cited by:16 (Source: Crossref)

Abstract

We apply the method of Lagrangian Descriptors (LDs) to a symmetric Caldera-type potential energy surface which has three index-1 saddles surrounding a relatively flat region that contains no minimum. Using this method we show the phase space transport mechanism that is responsible for the existence and nonexistence of the phenomenon of dynamical matching for this form of Caldera potential energy surface.

Keywords:

References

Agaoglou, M., Aguilar-Sanjuan, B., García-Garrido, V. J., García-Meseguer, R., González-Montoya, F., Katsanikas, M., Krajňák, V., Naik, S. & Wiggins, S. [2019] Chemical Reactions: A Journey into Phase Space (Bristol, UK, Zenodo), https://doi.org/10.5281/zenodo.3568210. Google Scholar
Agaoglou, M., Aguilar-Sanjuan, B., García-Garrido, V. J., González-Montoya, F., Katsanikas, M., Krajňák, V., Naik, S. & Wiggins, S. [2020] Lagrangian Descriptors: Discovery and Quantification of Phase Space Structure and Transport (Zenodo), https://doi.org/10.5281/zenodo.3958985. Google Scholar
Athanassoula, E., Romero-Gómez, M., Bosma, A. & Masdemont, J. [2009] “ Rings and spirals in barred galaxies — II. Ring and spiral morphology,” Monthly Not. Roy. Astron. Soc. 400, 1706–1720. Crossref, Web of Science, Google Scholar
Balibrea-Iniesta, F., Lopesino, C., Wiggins, S. & Mancho, A. M. [2016] “ Lagrangian descriptors for stochastic differential equations: A tool for revealing the phase portrait of stochastic dynamical systems,” Int. J. Bifurcation and Chaos 26, 1630036-1–20. Link, Web of Science, Google Scholar
Balibrea-Iniesta, F., Xie, J., García-Garrido, V. J., Bertino, L., Mancho, A. M. & Wiggins, S. [2019] “ Lagrangian transport across the upper arctic waters in the Canada basin,” Quart. J. Roy. Meteorol. Soc. 145, 76–91. Crossref, Web of Science, Google Scholar
Carpenter, B. K. [1985] “ Trajectories through an intermediate at a fourfold branch point. Implications for the stereochemistry of biradical reactions,” J. Amer. Chem. Soc. 107, 5730–5732. Crossref, Web of Science, Google Scholar
Collins, P., Kramer, Z., Carpenter, B., Ezra, G. & Wiggins, S. [2014] “ Nonstatistical dynamics on the Caldera,” J. Chem. Phys. 141, 034111. Crossref, Web of Science, Google Scholar
Crossley, R., Agaoglou, M., Katsanikas, M. & Wiggins, S. [2021] “ From Poincaré maps to Lagrangian descriptors: The case of the valley ridge inflection point potential,” Regul. Chaot. Dyn. 26, 147–164. Crossref, Web of Science, Google Scholar
García-Garrido, V. J., Curbelo, J., Mancho, A. M., Wiggins, S. & Mechoso, C. R. [2018] “ The application of Lagrangian descriptors to 3D vector fields,” Regul. Chaot. Dyn. 23, 551–568. Crossref, Web of Science, Google Scholar
Geng, Y., Katsanikas, M., Agaoglou, M. & Wiggins, S. [2021a] “ The bifurcations of the critical points and the role of depth in a symmetric Caldera potential energy surface,” Int. J. Bifurcation and Chaos 31, 2130034-1–11. Link, Web of Science, Google Scholar
Geng, Y., Katsanikas, M., Agaoglou, M. & Wiggins, S. [2021b] “ The influence of a pitchfork bifurcation of the critical points of a symmetric Caldera potential energy surface on dynamical matching,” Chem. Phys. Lett. 768, 138397. Crossref, Web of Science, Google Scholar
Katsanikas, M. & Wiggins, S. [2018] “ Phase space structure and transport in a Caldera potential energy surface,” Int. J. Bifurcation and Chaos 28, 1830042-1–20. Link, Web of Science, Google Scholar
Katsanikas, M. & Wiggins, S. [2019] “ Phase space analysis of the nonexistence of dynamical matching in a stretched Caldera potential energy surface,” Int. J. Bifurcation and Chaos 29, 1950057-1–9. Link, Web of Science, Google Scholar
Katsanikas, M., García-Garrido, V. J., Agaoglou, M. & Wiggins, S. [2020a] “ Phase space analysis of the dynamics on a potential energy surface with an entrance channel and two potential wells,” Phys. Rev. E 102, 012215. Crossref, Web of Science, Google Scholar
Katsanikas, M., García-Garrido, V. J. & Wiggins, S. [2020b] “ Detection of dynamical matching in a Caldera Hamiltonian system using Lagrangian descriptors,” Int. J. Bifurcation and Chaos 30, 2030026-1–16. Link, Web of Science, Google Scholar
Katsanikas, M., García-Garrido, V. J. & Wiggins, S. [2020c] “ The dynamical matching mechanism in phase space for Caldera-type potential energy surfaces,” Chem. Phys. Lett. 743, 137199. Crossref, Web of Science, Google Scholar
Katsanikas, M. & Wiggins, S. [2022] “ The nature of reactive and non-reactive trajectories for a three dimensional Caldera potential energy surface,” Physica D 435, 133293. Crossref, Web of Science, Google Scholar
Kelley, A. [1967] “ On the Lyapunov subcenter manifold,” J. Math. Anal. Appl. 18, 472–478. Crossref, Web of Science, Google Scholar
Lopesino, C., Balibrea-Iniesta, F., Wiggins, S. & Mancho, A. M. [2015] “ Lagrangian descriptors for two dimensional, area preserving, autonomous and nonautonomous maps,” Commun. Nonlin. Sci. Numer. Simul. 27, 40–51. Crossref, Web of Science, Google Scholar
Lopesino, C., Balibrea-Iniesta, F., García-Garrido, V. J., Wiggins, S. & Mancho, A. M. [2017] “ A theoretical framework for Lagrangian descriptors,” Int. J. Bifurcation and Chaos 27, 1730001-1–25. Link, Web of Science, Google Scholar
Madrid, J. A. J. & Mancho, A. M. [2009] “ Distinguished trajectories in time dependent vector fields,” Chaos 19, 013111. Crossref, Web of Science, Google Scholar
Mancho, A. M., Wiggins, S., Curbelo, J. & Mendoza, C. [2013] “ Lagrangian descriptors: A method for revealing phase space structures of general time dependent dynamical systems,” Commun. Nonlin. Sci. Numer. Simul. 18, 3530–3557. Crossref, Web of Science, Google Scholar
Mendoza, C. & Mancho, A. M. [2010] “ The hidden geometry of ocean flows,” Phys. Rev. Lett. 105, 038501. Crossref, Web of Science, Google Scholar
Moser, J. [1958] “ On the generalization of a theorem of A. Lyapunov,” Commun. Pure Appl. Math. 11, 257–271. Crossref, Web of Science, Google Scholar