# 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
chapter, 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
chapter, 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 [32], [30], [33], [34], [35].

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 (missing reference). 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 trajectories 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], [42], [43].

The method of Lagrangian descriptors (LDs) 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 LDs 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 [44], [45], [46]. In this
chapter, 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 the 2 DoF Barbanis system.

# Barbanis 2 DoF Model

## Model system: coupled harmonic 2 DoF Hamiltonian

As pointed out in the Introduction, our focus is to adopt a well-understood model system which is a 2 degrees-of-freedom coupled harmonic oscillator with the Hamiltonian

\begin{aligned} \mathcal{H}(x,y,p_x,p_y) =& T(p_x, p_y) + V_{\rm B}(x,y) \\ =& \frac{1}{2}p_x^2 + \frac{1}{2}p_y^2 + \frac{1}{2}\omega_x^2 x^2 + \frac{1}{2}\omega_y^2 y^2 + \delta x y^2 \label{eqn:Hamiltonian_Barbanis} \end{aligned}where $\omega_x, \omega_y, \delta$ are the harmonic oscillator frequencies of the $x$ and $y$
degree-of-freedom, and the coupling strength, respectively. We will fix
the parameters as $\omega_x
= 1.0, \omega_y = 1.1, \delta = -0.11$ in this study. The two
degrees-of-freedom potential is also referred to as *Barbanis*
potential, and has been investigated as a model of galactic
motion ([47], [18]), dynamical
tunneling and molecular spectra in physical
chemistry ([48], [49], [50]), structural
mechanics and ship capsize ([9], [10]).

The equilibria of the Hamiltonian vector field are located at

$$\left(-\frac{\omega_y^2}{2\delta}, \pm \frac{1}{\sqrt{2}}\frac{\omega_x \omega_y}{\delta}, 0, 0 \right) \qquad \text{and} \qquad \left(0, 0, 0, 0 \right)$$and are at total energy $E_c = \frac{\omega_x^2 \omega_y^4}{8 \delta^2}$ and $0$ respectively. The energy of the two index-1 saddles (as defined and shown in
App. 5.2.1) located at positive and negative
y-coordinates and positive x-coordinate for $\delta < 0$ will be
referred to as *critical energy*, $E_c$. In our discussion, we will
refer to the total energy of a trajectory or initial condition in terms
of the excess energy, $\Delta E = E_c - e$, which can be negative or
positive to denote energy below or above the critical energy. For the
parameters used in this study, the index-1 saddle equilibrium points are
located at $\left( 5.5, \pm 7.071, 0,
0 \right)$ and have energy, $E_c = 15.125$.

The contours of the coupled harmonic 2 DoF potential energy function in \eqref{eqn:Hamiltonian_Barbanis} is shown in Fig. fig:1 along with the 3D view of the surface. We note here that the potential has steep walls for $x < 0$ when $\delta < 0$ and steep drop-off beyond the bottlenecks around the index-1 saddles. This leads to unphysical motion in the sense of trajectories approaching $-\infty$ with ever increasing acceleration even for finite values of the configuration space coordinates [19].

In Fig. fig:2 we show the *Hill's region*, as
defined in App. 5, for the model
system \eqref{eqn:Hamiltonian_Barbanis}. It is important to note here that
even though Hill's region is shown on the configuration space, it
captures the dynamical picture, that is the *phase space perspective*,
of the Hamiltonian. This visualization of the energetically accessible
and forbidden realm is the first step towards introducing
two-dimensional surfaces to explore trajectory behavior. The complete
description of the unstable periodic orbit and its invariant manifolds
is described in App. 5.1 along with the visualization in the 3D
space.

(SPLIT SET OF FIGURES INTO TWO, AS WELL AS CAPTIONS.)

**fig:1**

*Potential energy function underlying the coupled harmonic Hamiltonian \eqref{eqn:Hamiltonian_Barbanis} as isopotential contour and surface. The index-1 saddles are shown as red crosses in both the plots. Parameters used are $\omega_x = 1.0, \omega_y = 1.1, \delta = -0.11$.*

**fig:2**

*Hill's region for energy below and above the energy of the index-1 saddle. Parameters used are $\omega_x = 1.0, \omega_y = 1.1, \delta = -0.11$.*

Since this model system is conservative 2 DoF Hamiltonian, that is the phase space is $\mathbb{R}^4$, the energy surface is three dimensional, the dividing surface is two dimensional, and the normally hyperbolic invariant manifold (NHIM), referred to as the unstable periodic orbit, is one dimensional [51]. Now, if we consider the intersection of a two dimensional surface with the three-dimensional energy surface, we would obtain the one-dimensional energy boundary on the surface of section. We will focus our study by using the isoenergetic two-dimensional surface \begin{equation} U_{xp_x, +} = \left\{(x,y,p_x,p_y) \; | \; y = 0, \; p_y(x,y,p_x;e) > 0 \right\} \label{eqn:sos_Uxpx} \end{equation}

where the sign of the momentum coordinate enforces a directional crossing of the surface. Due to the form of the vector field \eqref{eqn:two_dof_Barbanis} and choice of $\delta < 0$, this directionality condition implies motion towards positive $y$-coordinate.

In this article, detecting the phase space structures will constitute finding the intersection of the NHIM and its invariant manifolds with a two dimensional surface (for example, Eqn. \eqref{eqn:sos_Uxpx}).

## Results

We begin by noting that two-dimensional Poincaré surface of section have sufficient dimensionality to capture trajectories on a three dimensional energy surface, however for high dimensional systems trajectories can go "around" the two dimensional surface. One approach available in the literature is to use high dimensional Poincaré sections which can "catch" trajectories but are hard to visualize on paper or in the virtual 3D space. Even when gets around this issue, using suitable projective geometry, the fact that the qualitative analysis based on Poincaré sections depends on trajectories returning to this surface can not be circumvented since trajectories on and inside the spherical cylinders will not return to the Poincaré surface of section.

**fig:3**

*Top row: Poincar\'e surface of section, $U_{xp_x}$~\eqref{eqn:sos_Uxpx}, at excess energy (a) $\Delta E = -0.125$, (b) $\Delta E = 0.000$, (c) $\Delta E = 0.125$ where the intersection of the surface of section with the energy surface is shown in green. Bottom row: Lagrangian descriptor on the surface of section, $U_{xp_x}$ \eqref{eqn:sos_Uxpx}, for the excess energies (d) $\Delta E = -0.125$, (e) $\Delta E = 0.000$, (f) $\Delta E = 0.125$ and the integration time $\tau = 50$. The intersection of the surface of section with the cylindrical manifolds of the NHIM \textemdash unstable periodic orbit for this system \textemdash associated with the index-1 saddle equilibrium point in the bottleneck is shown in cyan (stable) and magenta (unstable) curves. The magenta and cyan curves in $p_x > 0$ correspond to the invariant manifolds of unstable periodic orbit at $y > 0$ index-1 saddle, and the ones in $p_x < 0$ correspnd to the invariant manifolds of unstable periodic orbit at $y < 0$ index-1 saddle.*

## Coupled harmonic 2 DoF system

As discussed in aforementioned
literature [52], [42], [53], points with
minimum Lagrangian descriptor (LD) values and singularity are on the
invariant manifolds. In addition, LD plots show dynamical correspondence
with Poincaré sections (in the sense that regions with regular and
chaotic dynamics are distinct in both Poincaré section and LD plots)
while also depicting the geometry of manifold
intersections [53], [42], [54]. This
correspondence in the LD features and Poincaré section is confirmed in
Fig. fig:3 where we show the Poincaré
surface of section
Eqn. \eqref{eqn:sos_Uxpx} of trajectories and LD contour maps on the
same isoenergetic two-dimensional surface for negative and positive
excess energies. It can be seen that the chaotic dynamics as marked by
the sea of points in Poincaré section is revealed as the tangle of
invariant manifolds which are points of minima and singularity in the LD
plots. As shown by the one dimensional slices of the LD plots, there are
multiple such minima and singularities and as the excess energy is
increased to positive values, there are regions of discontinuities along
the one dimensional slice. Next, as the energy is increased and the
bottleneck opens at critical energy $E_c$, trajectories that leave the
potential well and do not return to the surface of section are not
observed on the Poincaré section while the LD contour maps clearly
identifies these regions as discontinuities in the LD values. These
regions lead to escape because they are inside the cylindrical manifolds
of the unstable periodic orbit associated with the index-1 saddle
equilibrium point [10]. These regions on the isoenergetic
two-dimensional surface are also referred to as *reactive islands* in
chemical reaction dynamics [55], [56], [57]. The
escape regions or reactive islands that appear over the integration time
interval can also be identified by using the forward and backward LD
contour maps where these regions appear as discontinuities. In
Fig. fig:3 (f), we show these for
$\Delta E =
0.125$ and $\tau = 50$ along with the intersection of the cylindrical
manifolds' intersections that are computed using differential correction
and numerical continuation. The detailed comparison and extension to
high dimensional systems is not the focus of this study and will be
discussed in forthcoming work. Thus LD maps also provide a quick and
reliable approach for detecting regions that will lead to escape within
the observed time, or in the computational context, the integration
time.

To detect the NHIM in this case, unstable periodic orbit associated with the index-1 saddles (marked by cross in Fig. fig:1 ), we define an isoenergetic two dimensional surface that is parametrized by the $y$-coordinate and placed near the $x$-coordinate of the saddle equilibrium that has the negative $y$-coordinate. This can be expressed as a parametric two dimensional surface

\begin{aligned} % U_{xy}^+ = & \left\{(x,y,p_x,p_y) \; | \; p_x = 0, \; p_y(x,y;e) > 0 \right\}, \\ U_{xp_x}^+(k) = & \left\{(x,y,p_x,p_y) \, | \, y = k, p_y(x,y,p_x;e) > 0 \right\} \label{eqn:sos_xpx_k} % U_{yp_y}^+ = & \left\{(x,y,p_x,p_y) \; | \; x = k_x, \; p_y(x,y,p_x;e) > 0 \right\}, \end{aligned}for total energy, $e$, which is above the critical energy, $E_c$, $k$ is the $y$-coordinate. The variable integration time LD contour maps are shown in Fig. fig:4 ) along with the projection of the low dimensional slices \eqref{eqn:sos_xpx_k} in the configuration space and the NHIM. The points on NHIM, which is an unstable periodic orbit for 2 DoF, on this surface is the coordinate with maximum (for variable integration time) LD value. The full visualization of the NHIM as the black ellipse, $\mathbb{S}^1$, is in Fig. fig:4 (d) and has been computed using differential correction and numerical continuation (details in App. 5.1 ) and shows clearly that points on this unstable periodic orbit are detected by the LD contour map.

**fig:4**

*Lagrangian descriptor computed for variable integration time on two dimensional slices \eqref{eqn:sos_xpx_k} near the bottleneck that detect the NHIM and its invariant manifolds associated with the index-1 saddle. The two dimensional surfaces are shown in the top figure in (d) projected as orange lines on the configuration space and the unstable periodic orbit as black line connecting the isopotential contour corresponding to $\Delta E = 0.125$ with the Hill's region shown in grey.*

The two dimensional slices represent low dimensional probe of the unstable periodic orbit and the movie of a rotating view can be found here.

## Conclusions

In this article, we discussed a trajectory diagnostic method as a low dimensional probe of high dimensional invariant manifolds in 2 DoF nonlinear Hamiltonian systems. This trajectory diagnostic --- Lagrangian descriptor (LD) --- can represent a geometric property of interest in a system with escape/transition and features, that is minima or maxima, in its contour map identify points on the high dimensional invariant manifolds.

Comparing the points on the NHIM in 2 DoF system obtained using the LD method with differential correction and numerical continuation, we also verified the method for a nonlinear autonomous system following our previous work on decoupled and coupled 2 DoF linear system [17]. The LD method based detection of NHIM is simple to implement and quickly provides a lay of the dynamical land which is a preliminirary step in applying phase space transport to problems in physics and chemistry. This method can also be used to set up starting guess for other numerical procedures which rely on good initial guess or can also be used in conjunction with machine learning methods for rendering the smooth pieces of NHIM [45], [58].

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