Phase Space Structure and Transport in a Caldera Potential Energy Surface

Matthaios Katsanikas and Stephen Wiggins

International Journal of Bifurcation and Chaos, Vol. 28, No. 13 (2018) 1830042

https://doi.org/10.1142/S0218127418300422

We study phase space transport in a 2D caldera potential energy surface (PES) using techniques from nonlinear dynamics. The caldera PES is characterized by a flat region or shallow minimum at its center surrounded by potential walls and multiple symmetry related index one saddle points that allow entrance and exit from this intermediate region. We have discovered four qualitatively distinct cases of the structure of the phase space that govern phase space transport. These cases are categorized according to the total energy and the stability of the periodic orbits associated with the family of the central minimum, the bifurcations of the same family, and the energetic accessibility of the index one saddles. In each case, we have computed the invariant manifolds of the unstable periodic orbits of the central region of the potential, and the invariant manifolds of the unstable periodic orbits of the families of periodic orbits associated with the index one saddles. The periodic orbits of the central region are, for the first case, the unstable periodic orbits with period 10 that are outside the stable region of the stable periodic orbits of the family of the central minimum. In addition, the periodic orbits of the central region are, for the second and third cases, the unstable periodic orbits of the family of the central minimum and for the fourth case the unstable periodic orbits with period 2 of a period-doubling bifurcation of the family of the central minimum. We have found that there are three distinct mechanisms determined by the invariant manifold structure of the unstable periodic orbits that govern the phase space transport. The first mechanism explains the nature of the entrance of the trajectories from the region of the low energy saddles into the caldera and how they may become trapped in the central region of the potential. The second mechanism describes the trapping of the trajectories that begin from the central region of the caldera, their transport to the regions of the saddles, and the nature of their exit from the caldera. The third mechanism describes the phase space geometry responsible for the dynamical matching of trajectories originally proposed by Carpenter and described in [Collins et al., 2014] for the two-dimensional caldera PES that we consider.

V. J. Garcia-Garrido, F. Balibrea-Iniesta, S. Wiggins, A. M. Mancho, and C. Lopesino,  Detection of Phase Space Structures of the Cat Map with Lagrangian Descriptors, Regular and Chaotic Dynamics, 23(6), 751-766 (2018).

https://link.springer.com/article/10.1134/S1560354718060096

In this paper we show that Lagrangian Descriptors (LDs), a technique based on Dynamical Systems Theory (DST), are able to reveal the phase space structures present in the well known Arnold’s cat map. This discrete dynamical system, which represents a classical example of an Anosov diffeomorphism that is strongly mixing,  provides us with a benchmark model to test the performance of LDs and their capability to detect fixed points, periodic orbits and their stable and unstable manifolds present in chaotic maps. In this work we show, both from a theoretical and a numerical perspective, how LDs reveal the invariant manifolds of the periodic orbits of the cat map. The application of this methodology in this setting clearly illustrates the chaotic behaviour of the cat map and highlights some technical numerical difficulties that arise in the identification of its phase space structures.

G. S. Ezra and S. Wiggins, Sampling Phase Space Dividing Surfaces Constructed from Normally Hyperbolic Invariant Manifolds (NHIMs), J. Phys. Chem. A., 122(42), 8354–8362 (2018).

https://pubs.acs.org/doi/10.1021/acs.jpca.8b07205

In this paper, we further investigate the construction of a phase space dividing surface (DS) from a normally hyperbolic invariant manifold (NHIM) and the sampling procedure for the resulting dividing surface described in earlier work (Wiggins, S.; J. Chem. Phys. 2016, 144, 054107). Our discussion centers on the relationship between geometrical structures and dynamics for 2 and 3 degree of freedom (DoF) systems, specifically, the construction of a DS from a NHIM. We show that if the equation for the NHIM and associated DS is known (e.g., as obtained from Poincaré–Birkhoff normal form theory), then the numerical procedure described in Wiggins et al. ( J. Chem. Phys. 2016, 144, 054107) gives the same result as a sampling method based upon the explicit form of the NHIM. After describing the sampling procedure in a general context, it is applied to a quadratic Hamiltonian normal form near an index-one saddle where explicit formulas exist for both the NHIM and the DS. It is shown for both 2 and 3 DoF systems that a version of the general sampling procedure provides points on the analytically defined DS with the correct microcanonical density on the constant-energy DS. Excellent agreement is obtained between analytical and numerical averages of quadratic energy terms over the DS for a range of energies.

V. J. Garcia-Garrido, J. Curbelo, A. M. Mancho, S. Wiggins, and C. R. Mechoso,  The Application of Lagrangian Descriptors to 3D Vector Fields, Regular and Chaotic Dynamics, 23(5), 551-568 (2018).

https://doi.org/10.1134/S1560354718050052

Since the 1980s, the application of concepts and ideas from Dynamical Systems Theory to analyze phase space structures has provided a fundamental framework to understand long-term evolution of trajectories in many physical systems. In this context, for the study of fluid transport and mixing the development of Lagrangian techniques that can capture the complex and rich dynamics of time dependent flows has been crucial. Many of these applications have been to atmospheric and oceanic flows in two-dimensional (2D) relevant scenarios. However, the geometrical structures that constitute the phase space structures in time dependent three-dimensional (3D) flows require further exploration. In this paper we explore the capability of Lagrangian Descriptors (LDs), a tool that has been successfully applied to time dependent 2D vector fields, to reveal phase space geometrical structures in 3D vector fields. In particular we show how LDs can be used to reveal phase space structures that govern and mediate phase space transport. We especially highlight the identification of Normally Hyperbolic Invariant Manifolds (NHIMs) and tori. We do this by applying this methodology to three specific dynamical systems: a 3D extension of the classical linear saddle system, a 3D extension of the classical Duffing system, and a geophysical fluid dynamics f-plane approximation model which is described by analytical wave solutions of the 3D Euler equations. We show that LDs successfully identify and recover the template of invariant manifolds that define the dynamics in phase space for these examples.

Article, “Influence of mass and potential energy surface geometry on roaming in Chesnavich’s CH4+ model,” was published 07 September 2018, in The Journal of Chemical Physics (Vol.149, Issue 9).  Vladimir Krajnak and Stephen Wiggins

It may be accessed via the link below:
https://doi.org/10.1063/1.5044532
DOI: 10.1063/1.5044532

Chesnavich’s model Hamiltonian for the reaction CH⁺₄ → CH⁺₃ + H is known to exhibit a range of interesting dynamical phenomena including roaming. The model system consists of two parts: a rigid, symmetric top representing the CH⁺₃ ion and a free H atom. We study roaming in this model with focus on the evolution of geometrical features of the invariant manifolds in phase space that govern roaming under variations of the mass of the free atom m and a parameter a that couples radial and angular motion. In addition, we establish an upper bound on the prominence of roaming in Chesnavich’s model. The bound highlights the intricacy of roaming as a type of dynamics on the verge between isomerisation and nonreactivity as it relies on generous access to the potential wells to allow reactions as well as a prominent area of high potential that aids sufficient transfer of energy between the degrees of freedom to prevent isomerisation.

In July 2018, three of the CHAMPS PDRA’s: Rafael Garcia Meseguer, Matthaios Katsanikas and Vladimir Krajnak, attended the workshop ‘Geometry of Chemical Reaction Dynamics in Gas and Condensed Phases’ at the Telluride Science Research Center in Telluride, CO, USA. The meeting focussed on theoretical aspects of chemical reaction dynamics and forms a platform for discussions about extensions of theory from lower to higher dimensional systems and from gas to condensed phases. Apart from meeting some of the leading researchers in the field and discussing ideas for future work, the PDRA’s also shared their contributions to the field. Rafael gave a talk on the applications of Lagrangian descriptors to finding dividing surfaces and invariant structures, Matthaios shared his findings on phase space transport in the Caldera model and Vladimir reported on the phase space mechanism behind Roaming in Chesnavich’s model and its evolution under parameter variations.

CHAMPS held a research day at Imperial College London on Friday 13th July 2018. This provided an  opportunity for all of the investigators and PDRAs to discuss their research and obtain feedback from the entire CHAMPS team.

A copy of the schedule for the day can be found here: Research Day 13.07.18 Schedule

Professor Stephen Wiggins – Champs Principle Investigator

Professor Dmitry Shalashilin (left) Dr Dmitry Makhov (right)

Dr Matthaios Katsanikas – Champs PDRA

Dr Vladimir Krajnak – Champs PDRA

Professor Dmitry Shalashilin Co-Investigator (left) Professor Darryl Holm – Co-Investigator (right)

Dr Lars Bratholm – Champs PDRA

Professor Dmitry Shalashilin – Champs Co-Investigator

Professor Barry Carpenter – Champs Co-Investigator

Dr Shibabrat Naik – Champs PDRA

Dr Dmitry Makhov – Champs PDRA

Champs PDRAs – Makhov, Matthaios, Krajnak, Garcia-Meseguer, Naik, Bratholm (left to right)