Phase Space Structure and Transport in a Caldera Potential Energy Surface

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. We have found that there are three distinct mechanisms determined by the invariant manifold structure of the unstable periodic orbits 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.Comment: 24 pages, International Journal of Bifurcation and Chaos (in press

Finding NHIM in 2 and 3 degrees-of-freedom with H\'enon-Heiles type potential

We present the capability of Lagrangian descriptors for revealing the high dimensional phase space structures that are of interest in nonlinear Hamiltonian systems with index-1 saddle. These phase space structures include normally hyperbolic invariant manifolds and their stable and unstable manifolds, and act as codimenision-1 barriers to phase space transport. The method is applied to classical two and three degrees-of-freedom Hamiltonian systems which have implications for myriad applications in physics and chemistry.Comment: 15 pages, 6 figures. This manuscript is better served as dessert to the main course: arXiv:1903.1026

Finite-time Lagrangian transport analysis: Stable and unstable manifolds of hyperbolic trajectories and finite-time Lyapunov exponents

We consider issues associated with the Lagrangian characterisation of flow structures arising in aperiodically time-dependent vector fields that are only known on a finite time interval. A major motivation for the consideration of this problem arises from the desire to study transport and mixing problems in geophysical flows where the flow is obtained from a numerical solution, on a finite space-time grid, of an appropriate partial differential equation model for the velocity field. Of particular interest is the characterisation, location, and evolution of "transport barriers" in the flow, i.e. material curves and surfaces. We argue that a general theory of Lagrangian transport has to account for the effects of transient flow phenomena which are not captured by the infinite-time notions of hyperbolicity even for flows defined for all time. Notions of finite-time hyperbolic trajectories, their finite time stable and unstable manifolds, as well as finite-time Lyapunov exponent (FTLE) fields and associated Lagrangian coherent structures have been the main tools for characterizing transport barriers in the time-aperiodic situation. In this paper we consider a variety of examples, some with explicit solutions, that illustrate, in a concrete manner, the issues and phenomena that arise in the setting of finite-time dynamical systems. Of particular significance for geophysical applications is the notion of "flow transition" which occurs when finite-time hyperbolicity is lost, or gained. The phenomena discovered and analysed in our examples point the way to a variety of directions for rigorous mathematical research in this rapidly developing, and important, new area of dynamical systems theory

Dynamics of the Morse Oscillator: Analytical Expressions for Trajectories, Action-Angle Variables, and Chaotic Dynamics

We consider the one degree-of-freedom Hamiltonian system defined by the Morse potential energy function (the "Morse oscillator"). We use the geometry of the level sets to construct explicit expressions for the trajectories as a function of time, their period for the bounded trajectories, and action-angle variables. We use these trajectories to prove sufficient conditions for chaotic dynamics, in the sense of Smale horseshoes, for the time-periodically perturbed Morse oscillator using a Melnikov type approach

On the dynamical origin of asymptotic t^2 dispersion of a nondiffusive tracer in incompressible laminar flows

Using an elementary application of Birkhoff's ergodic theorem, necessary and sufficient conditions are given for the existence of asymptotically t^2 dispersion of a distribution of nondiffusive passive tracer in a class of incompressible laminar flows. Nonergodicity is shown to be the dynamical mechanism giving rise to this behavior

Negligibility of small divisor effects in the normal form theory for nearly-integrable Hamiltonians with decaying non-autonomous perturbations

The paper deals with the problem of the existence of a normal form for a nearly-integrable real-analytic Hamiltonian with aperiodically time-dependent perturbation decaying (slowly) in time. In particular, in the case of an isochronous integrable part, the system can be cast in an exact normal form, regardless of the properties of the frequency vector. The general case is treated by a suitable adaptation of the finite order normalization techniques usually used for Nekhoroshev arguments. The key point is that the so called "geometric part" is not necessary in this case. As a consequence, no hypotheses on the integrable part are required, apart from analyticity. The work, based on two different perturbative approaches developed by A.Giorgilli et al., is a generalisation of the techniques used by the same authors to treat more specific aperiodically time-dependent problems.Comment: 16 page

Homoclinic Orbits In Slowly Varying Oscillators

We obtain existence and bifurcation theorems for homoclinic orbits in three-dimensional flows that are perturbations of families of planar Hamiltonian systems. The perturbations may or may not depend explicitly on time. We show how the results on periodic orbits of the preceding paper are related to the present homoclinic results, and apply them to a periodically forced Duffing equation with weak feedback

Influence of mass and potential energy surface geometry on roaming in Chesnavich's CH4+_4^+ model

Chesnavich's model Hamiltonian for the reaction CH$_4^+ \rightarrow$ CH$_3^+$ 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