Abstract:
In this paper we compare the method of Lagrangian descriptors with the classical
method of Poincaré maps for revealing the phase space structure of two-degree-of-freedom
Hamiltonian systems. The comparison is carried out by considering the dynamics of a twodegree-
of-freedom system having a valley ridge inflection point (VRI) potential energy surface.
VRI potential energy surfaces have four critical points: a high energy saddle and a lower energy
saddle separating two wells. In between the two saddle points is a valley ridge inflection point
that is the point where the potential energy surface geometry changes from a valley to a ridge.
The region between the two saddles forms a reaction channel and the dynamical issue of interest
is how trajectories cross the high energy saddle, evolve towards the lower energy saddle, and
select a particular well to enter. Lagrangian descriptors and Poincaré maps are compared for
their ability to determine the phase space structures that govern this dynamical process.
Keywords:
phase space structure, periodic orbits, stable and unstable manifolds, homoclinic
and heteroclinic orbits, Poincaré maps, Lagrangian descriptors.
Citation:
Rebecca Crossley, Makrina Agaoglou, Matthaios Katsanikas, Stephen Wiggins, “From Poincaré Maps to Lagrangian Descriptors:
The Case of the Valley Ridge Inflection Point Potential”, Regul. Chaotic Dyn., 26:2 (2021), 147–164
\Bibitem{CroAgaKat21}
\by Rebecca Crossley, Makrina Agaoglou, Matthaios Katsanikas, Stephen Wiggins
\paper From Poincaré Maps to Lagrangian Descriptors:
The Case of the Valley Ridge Inflection Point Potential
\jour Regul. Chaotic Dyn.
\yr 2021
\vol 26
\issue 2
\pages 147--164
\mathnet{http://mi.mathnet.ru/rcd1108}
\crossref{https://doi.org/10.1134/S1560354721020040}
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Linking options:
https://www.mathnet.ru/eng/rcd1108
https://www.mathnet.ru/eng/rcd/v26/i2/p147
This publication is cited in the following 7 articles:
Francisco Gonzalez Montoya, “Impenetrable barriers in the phase space of a particle moving around a Kerr rotating black hole”, Physica D: Nonlinear Phenomena, 468 (2024), 134290
Matthaios Katsanikas, Stephen Wiggins, Springer Proceedings in Complexity, Chaos, Fractals and Complexity, 2023, 47
Katsanikas M., Sanjuan B.A., Montoya F.G., Garcia-Garrido V.J., Wiggins S., “Bifurcation Study on a Degenerate Double Van der Waals Cirque Potential Energy Surface Using Lagrangian Descriptors”, Commun. Nonlinear Sci. Numer. Simul., 105 (2022), 106089
Matthaios Katsanikas, Makrina Agaoglou, Stephen Wiggins, Ana M. Mancho, “Phase Space Transport in a Symmetric Caldera Potential with Three Index-1 Saddles and No Minima”, Int. J. Bifurcation Chaos, 32:10 (2022)
Rémi Pédenon-Orlanducci, Timoteo Carletti, Anne Lemaitre, Jérôme Daquin, Nonlinear Systems and Complexity, 36, Nonlinear Dynamics and Complexity, 2022, 221
Katsanikas M., Agaoglou M., Wiggins S., “Bifurcation of Dividing Surfaces Constructed From a Pitchfork Bifurcation of Periodic Orbits in a Symmetric Potential Energy Surface With a Post-Transition-State Bifurcation”, Int. J. Bifurcation Chaos, 31:14 (2021), 2130041
Agaoglou M., Garcia-Garrido V.J., Katsanikas M., Wiggins S., “Visualizing the Phase Space of the Hei2 Van der Waals Complex Using Lagrangian Descriptors”, Commun. Nonlinear Sci. Numer. Simul., 103 (2021), 105993
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