# Three degree-of-freedom Barbanis system

## Introduction

It is well-known now that the paradigm of escape from a potential well
and the topology of phase space structures that mediate such escape are
used in a broad array of problems such as isomerization of molecular
clusters [1], reaction rates in chemical
physics [2], [3], ionization of a hydrogen atom
under electromagnetic field in atomic physics [4], transport
of defects in solid state and semiconductor physics [5],
buckling modes in structural mechanics [6], [7], ship
motion and capsize [8], [9], [10], escape and
recapture of comets and asteroids in celestial
mechanics [11], [12], [13], and
escape into inflation or re-collapse to singularity in
cosmology [14]. As such a method that can identify the high
dimensional phase space structures using low dimensional surface as
probes can aid in quantifying the escape rates. These low dimensional
surfaces has been shown to be of as *reactive islands* in chemical
physics and lead to insights into sampling rare transition
events [15], [16]. However,
to benchmark the methodology, we first applied it to linear systems
where the closed-form analytical expression of the phase space
structures is known [17]. As the next step, in this
article, we will focus on nonlinear Hamiltonian systems which have been
extensively studied as "built by hand" models of galactic dynamics and
for demonstrating quantum dynamical
tunneling [18], [19], [20], [21], [22], [23], [24], [25], [26], [27].
The nonlinear Hamiltonian systems considered here have an underlying
Hénon-Heiles type potential with the simplest form of nonlinearity, and
show regular, quasi-periodic, and chaotic trajectories along with
bifurcations of periodic orbits. A Hénon-Heiles type potential has a
well with bottlenecks connecting the region of bounded motion (trapped
region) to unbounded motion (escape off to infinity), and have
rotational symmetry. In addition, these Hénon-Heiles type potentials are
studied as first benchmark nonlinear systems in applying new phase space
transport methods to astrophysical and molecular motion. In this
article, we will present verification of a method that uses trajectory
diagnostic on a low dimensional surface for revealing the phase space
structures in 4 or more dimensions.

Conservative dynamics on an open potential well has received considerable attention because the phase space structures, normally hyperbolic invariant manifolds (NHIM) and its invariant manifolds, explain the intricate fractal structure of ionization rates [28], [29], [30]. Furthermore, the discrepancies in observed and predicted ionization rates in atomic systems has also been explained by accounting for the topology of the phase space structures. These have been connected with the breakdown of ergodic assumption that is the basis for using ionization and dissociation rate formulae [31]. This rich literature on chaotic escape of electrons from atoms sets a precedent for applying new methods for finding NHIM and its invariant manifolds in Hamiltonian with open potential wells (missing reference).

As we noted earlier, trajectory diagnostic methods which can probe phase space to detect the high dimensional invariant manifolds have potential to be of use in many degrees-of-freedom models. One such method is the Lagrangian descriptors (LDs) that can reveal phase space structures by encoding geometric property of trajectories (such as, phase space arc length, configuration space distance or displacement, cumulative action or kinetic energy) initialised on a two dimensional surface [32], [33], [34], [35]. The method was originally developed in the context of Lagrangian transport in time-dependent two dimensional fluid mechanics. However, it has also been successful in locating transition state trajectory in chemical reactions [36], [37], [38]. Besides, also being applicable to both Hamiltonian and non-Hamiltonian systems, as well as to systems with arbitrary time-dependence such as stochastic and dissipative forces, and geophysical data from satellite and numerical simulations [39], [40], [41], [35], [42].

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 (NHIM) 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.

The method of Lagrangian descriptor (LD) is straightforward to implement
computationally and it provides a "high resolution" method for exploring
the influence of high dimensional phase space structure on trajectory
behaviour. The method of LD takes an *opposite* approach to that of
classical Lyapunov exponent type calculations by emphasizing the initial
conditions of trajectories, rather than their advected locations that is
involved in calculating normalized rate of divergence. This is achieved
by considering a two dimensional section of the full phase space and
discretizing with a dense grid of initial conditions. Even though the
trajectories wander off in the phase space, as the initial conditions
evolve in time, there is no loss in resolution of the two dimensional
section. In contrast to inferring the phase space structures from
Poincaré sections, LD plots do not suffer from loss of resolution since
the affects of the structure are encoded in the initial conditions and
there is no need for the trajectory to return to the section. Our
objective is to clarify the use of Lagrangian descriptors as a
diagnostic on two dimensional sections of high dimensional phase space
structures. This diagnostic is also meant to be used as the preliminary
step in computing the NHIM, their stable and unstable manifolds using
other computational
means [43], [44], [45]. In this
article, we will present the method's capability to detect the high
dimensional phase space structures such as the NHIM, their stable, and
unstable manifolds in 2 and 3 DoF Hamiltonian systems.

## Development of the Problem [[sec:model_prob_2dof]]{#sec:model_prob_2dof label="sec:model_prob_2dof"}

The model system to consider is the coupled harmonic potential in 3 dimensions and underlying a 3 degrees-of-freedom system in [46], [47]. The Hamiltonian is given by

\begin{align} \mathcal{H}(x,y,z,p_x,p_y,p_z) = T(p_x, p_y, p_z) + V_{\rm BC}(x,y,z) = \frac{1}{2}p_x^2 + \frac{1}{2}p_y^2 + \frac{1}{2}p_z^2 + \frac{1}{2}\omega_x^2 x^2 + \frac{1}{2}\omega_y^2 y^2 + \frac{1}{2}\omega_z^2 z^2 - \epsilon x^2y - \eta x^2 z \label{eqn:Hamiltonian_BC_3dof} \end{align}where $\omega_x^2, \omega_y^2, \omega_z^2, \epsilon, \eta$ are the parameters related to the coupled harmonic 3 dimensional potential energy function [47]. In this study, we will fix the parameters to be $\omega_x^2 = 0.9, \omega_y^2 = 1.6, \omega_z^2 = 0.4, \epsilon = 0.08, \eta = 0.01$. The two index-1 saddle equilibria (as shown in the App. 6{reference-type="ref" reference="sect:coupled_3dof"}) of the Hamiltonian vector field [eqn:three_dof_Barbanis]{reference-type="eqref" reference="eqn:three_dof_Barbanis"} are located at

\begin{equation} \left(\pm \frac{\omega_x\omega_y\omega_z}{\sqrt{2(\epsilon^2\omega_z^2 + \eta^2\omega_y^2)}}, \frac{\epsilon \omega_x^2\omega_z^2}{2(\epsilon^2\omega_z^2 + \eta^2\omega_y^2)}, \frac{\eta \omega_x^2\omega_y^2}{2(\epsilon^2\omega_z^2 + \eta^2\omega_y^2)}, 0, 0, 0 \right) \label{eqn:eq_pt_BC_3dof} \end{equation}and the total energy is $$E_c = \frac{1}{8} \omega_x^2 \frac{\omega_x^2 \omega_y^2 \omega_z^2}{ \left( \epsilon^2 \omega_z^2 + \eta^2 \omega_y^2 \right)}.$$ The equilibrium point at $(0,0,0,0,0,0)$ is stable and has total energy $0$. For the parameters used in this study, the equilibrium points are located at $\left( \pm 10.290, 5.294, 2.647, 0, 0, 0 \right)$ and $\left( 0, 0, 0, 0, 0, 0 \right)$ and have total energy, $E_c \approx 23.824$ and $E = 0$, respectively.

We show the isopotential contours of the potential energy function at fixed value of $z_{\rm eq}$ in Fig. [fig:Barbanis_Contopoulos_3dof]{reference-type="ref" reference="fig:Barbanis_Contopoulos_3dof"} along with the Hill's regions for positive excess energy, $\Delta E = 6.000$ and projected on the configuration space coordinates at the equilibrium point.

{width="25.00000%"}\ {width="70.00000%"} [[fig:Barbanis_Contopoulos_3dof]]{#fig:Barbanis_Contopoulos_3dof label="fig:Barbanis_Contopoulos_3dof"}

Fig. 2. (a) Potential energy function underlying the coupled harmonic Hamiltonian~\eqref{eqn:Hamiltonian_BC*3dof} at $z*{\rm eq} = 2.647$ as isopotential contour and surface. (b) Hill's region for excess energy, $\Delta E = 6.000$ and projected on the configuration space coordinates at the equilibrium point. We note here that the potential energy surface and the Hill's region is plotted by fixing one of the configuration coordinates at the equilibrium point.

Since this model system is conservative 3 DoF Hamiltonian, that is the
phase space is $\mathbb{R}^6$, the energy surface is five dimensional,
the dividing surface is four dimensional, and the normally hyperbolic
invariant manifold (NHIM) is three dimensional, or precisely 3-sphere,
and its invariant manifolds are four dimensional, or precisely
$\mathbb{R}^1 \times \mathbb{S}^3$ or *spherical
cylinders* [48]. Now, if we consider the intersection of
a two-dimensional section with the five dimensional energy surface in
$\mathbb{R}^6$, we would obtain the one-dimensional energy boundary on
the surface. We will focus our study near the bottleneck by considering
the isoenergetic two dimensional surfaces

In this 3 DoF system, detecting points on the three dimensional NHIM and four dimensional invariant manifolds will constitute finding their intersection with the above two dimensional surfaces.

## Revealing Phase Space Structures

The Lagrangian descriptor based approach for detecting NHIM in 2 DoF system can now be applied to the 3 DoF system [eqn:Hamiltonian_BC_3dof]{reference-type="eqref" reference="eqn:Hamiltonian_BC_3dof"}. On the five dimensional energy surface, the phase space structures such as the NHIM and its invariant manifolds are three and four dimensional, respectively [48]. As noted earlier, direct visualization techniques will fall short in 4 or more DoF systems even if they are successful in 2 and 3 DoF. So, LD based approach can be used to detect points on a NHIM and its invariant manifolds using low dimensional probe which are based on trajectory diagnostic on an isoenergetic two dimensional surface.

It is to be noted that the increase in phase space dimension, leads to a polynomial scaling in the number of coordinate pairs (that is $2N(2N-1)(N-1)$ coordinate pairs for $N$ DoF system) and is thus, impractical to present the procedure on all the combination of coordinates. We will present the results for the three configuration space coordinates by combining each with its corresponding momentum coordinate.

On these isoenergetic surfaces, we compute the variable integration time Lagrangian descriptor for small excess energy, $\Delta E \approx 0.176$, or total energy $E = 24.000$, and show the contour maps in Fig. [fig:Barbanis3dof_M_pxpypz]{reference-type="ref" reference="fig:Barbanis3dof_M_pxpypz"}. The maxima identifying the points on the NHIM and its invariant manifolds can be visualized using one dimensional slices for constant momenta. This indicates clearly the initial conditions in the phase space (points on the isoenergetic two dimensional surfaces in $\mathbb{R}^6$, for example [eqn:Barbanis3dof_uxpx]{reference-type="eqref" reference="eqn:Barbanis3dof_uxpx"}) that do not leave the saddle region.

{width="33.00000%"}\ {width="33.00000%"}\ {width="33.00000%"}
Fig. 5. Detecting points on the NHIM using variable integration time Lagrangian descriptor on the two dimensional surfaces (a) $U_{xp_x}^+$~\eqref{eqn:Barbanis3dof*uxpx}, (b) $U*{yp_y}^+$~\eqref{eqn:Barbanis3dof_uypy}, and (c) $U_{zp_z}^+$~\eqref{eqn:Barbanis3dof_uzpz} at excess energy $\Delta E \approx 0.176$ or total energy $E = 24.000$. For this energy value, the saddle region, as defined in Eqn.~\eqref{eqn:var_time_qs}, is taken to be $q_s = [9,12] \times [2.5,7.5] \times [1,4]$ and $\tau = 50$.

## Implications for reaction dynamics

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