Congratulation to CHAMPS PDRA Lars Bratholm for being awarded a Alan Turing Institute Fellowship
We are please that Dr. Lars Bratholm is in the first group of Alan Turing Institute fellows at the University of Bristol. This is a notable accomplishment and we anticipate that this will serve to further develop links between CHAMPS and the Alan Turing Institute.
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.
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.
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.
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.
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.