Grotunits=360

# The Generalization of the Periodic Orbit Dividing Surface in Hamiltonian
Systems with three or more degrees of freedom - I

MATTHAIOS KATSANIKAS AND STEPHEN WIGGINS  School of Mathematics, University of
Bristol, Fry Building, Woodland Road, Bristol BS8 1UG, United Kingdom
<EMAIL_ADDRESS><EMAIL_ADDRESS>
((to be inserted by publisher))

###### Abstract

We present a method that generalizes the periodic orbit dividing surface
construction for Hamiltonian systems with three or more degrees of freedom. We
construct a torus using as a basis a periodic orbit and we extend this to a
$2n-2$ dimensional object in the $2n-1$ dimensional energy surface. We present
our methods using benchmark examples for two and three degree of freedom
Hamiltonian systems to illustrate the corresponding algorithm for this
construction. Towards this end we use the normal form quadratic Hamiltonian
system with two and three degrees of freedom. We found that the periodic orbit
dividing surface can provide us the same dynamical information as the dividing
surface constructed using normally hyperbolic invariant manifolds. This is
significant because, in general, computations of normally hyperbolic invariant
manifolds are very difficult in Hamiltonian systems with three or more degrees
of freedom. However, our method avoids this computation and the only
information that we need is the location of one periodic orbit.

###### keywords:

phase space; Hamiltonian system, periodic orbits; Dividing surfaces; normally
hyperbolic invariant manifold; Chemical reaction dynamics; Dynamical astronomy

##  1 Introduction

In this paper we construct dividing surfaces from periodic orbits in
Hamiltonian systems with three or more degrees of freedom. This generalises
the periodic orbit dividing surface construction developed in Pechukas &
McLafferty [1973]; Pechukas & Pollak [1977]; Pollak & Pechukas [1978].

A dividing surface (at a fixed value of energy) is a surface that has one less
dimension than that of the energy surface of a Hamiltonian system. This
surface has the property of no-recrossing and orientability. For our purposes
the significance of orientability is that a surface has two well-defined
sides. We underline the fact that we must not confuse the no-recrossing
property of the dividing surfaces with the Poincaré recurrence in the case of
closed and bounded energy surfaces. In the case of Poincaré recurrence the
trajectories starting from an initial point in the phase space (included the
dividing surfaces) will eventually return close to the initial position. This
is independent from the choice of the dividing surface. Dividing surfaces play
a central role in Wigner’s vision for transition state theory Wigner [1938];
Waalkens _et al._ [2007], which is a widely used theory for understanding
chemical reaction dynamics.

In Hamiltonian systems with two degrees of freedom the phase space is four-
dimensional and the energy surface is three-dimensional. This means that the
dividing surface is two-dimensional surface embedded in the three-dimensional
energy surface. This dividing surface is constructed using periodic orbits
(Pechukas [1981], Pollak [1985]) that are 1-dimensional closed curves.

Until now, the construction of a dividing surface in Hamiltonian systems with
three or more degrees can be done from the Normally Hyperbolic Invariant
Manifold (NHIMs) Wiggins _et al._ [2001]; Uzer _et al._ [2002]; Wiggins
[2016]. The phase space is $2n$-dimensional and the energy surface is $2n-1$
dimensional surface. The periodic orbit is one-dimensional and it has not
enough dimensions to construct a dividing surface using the method that we use
in Hamiltonian systems with two degrees of freedom. The Normally Hyperbolic
Invariant Manifold is a $2n-3$ dimensional structure in the $2n-1$ dimensional
energy surface. The dimensions of this object can guarantee the construction
of a $2n-2$ dimensional structure embedded in the $2n-1$ dimensional energy
surface Waalkens _et al._ [2007], Waalkens & Wiggins [2010]. The construction
of a Dividing surface from the NHIM can be done through the computation of the
Poincaré-Birkhoff normal form (NF) theory near an index-one saddle. By this
method we obtain explicit formulas for the Normally Hyperbolic Invariant
Manifolds and Dividing surfaces (Wiggins _et al._ [2001], Uzer _et al._
[2002], Waalkens _et al._ [2007], Toda [2003], Komatsuzaki & Berry [2003]).
But in many situations this method is difficult to apply and it requires
extensive algebraic calculations.

In this paper we propose a new method to construct dividing surfaces without
to know the NHIM, avoiding the extensive and difficult calculations through
the Normal Form theory. We use the periodic orbit, that is actually a
1-dimensional submanifold of the Normally Hyperbolic Invariant Manifold, as a
starting point to construct a dividing surface. We give an introduction and
pseudocode of our algorithm in the general context of the Hamiltonian Systems
with $n$ degrees of freedom in the section 2. We describe our algorithm for
the case of the Hamiltonian systems with two and three degrees of freedom in
the sections 3 and 4 respectively. In these sections, we apply our algorithm
to a quadratic normal form Hamiltonian system with two and three degrees of
freedom and we compare the periodic orbit dividing surfaces that are
constructed from our algorithm with the dividing surfaces that are constructed
from the NHIM. We choose this system because we have analytical formulas for
the periodic orbits and the NHIM (Ezra & Wiggins [2018]). This allows us to
compute the dividing surfaces from the periodic orbits and the NHIM and to
compare these results (see sections 3 and 4). Then we present the algorithm in
the general case of a Hamiltonian system with $n$ degrees of freedom (see
section 5). Finally, we present our conclusions in the last section.

##  2 The skeleton of the Algorithm - Pseudocode

In this section we present a method for the construction of a periodic orbit
dividing surface in Hamiltonian systems with three or more degrees of freedom.
This method generalizes the classical periodic orbit dividing surface that is
valid only in Hamiltonian systems with two degrees of freedom. We consider the
general case of Hamiltonian systems with $n$ degrees of freedom with potential
energy function $V(x_{1},x_{2},...,x_{n})$ with $n\geq 2$ of the form :

$\displaystyle T+V(x_{1},x_{2},...,x_{n})=E$

where $T$ is the kinetic energy.

$\displaystyle
T=p_{x_{1}}^{2}/2m_{1}+p_{x_{2}}^{2}/2m_{2}+...+p_{x_{n}}^{2}/2m_{n}$

where $p_{x_{1}},p_{x_{2}},...p_{x_{n}}$ are the momenta and
$m_{1},m_{2},..m_{n}$ are the corresponding masses.

A basic assumption of our algorithm is that there exists a two dimensional
(2D) subspace in the $2n$ dimensional (D) phase space in which the periodic
orbit can be represented by a closed curve. With this assumption the algorithm
has two versions.

First version.

In this version we assume that the periodic orbit is represented by a closed
curve in a 2D subspace of the phase space that it is not a subspace of the
configuration space (for example in the $(x_{1},p_{x_{1}})$ space). In other
words, the 2D subspace in this version has one coordinate corresponding to a
configuration space variable and the other corresponding to a phase space
variable. We refer to this space as the extended configuration space.

Second version.

In this version we assume that the periodic orbit can be represented as a
closed curve in a 2D subspace of the configuration space. For example the 2D
subspace is described by the coordinates $(x_{1},x_{2})$.

In this section, we describe the two versions of the algorithm for the
construction of the dividing surfaces if we know the location of one periodic
orbit. For this reason we present a diagram (pseudocode) that describes the
two versions of the algorithm (Fig. 1). We see in this diagram that we check
first if the periodic orbit satisfies the conditions that are necessary in
order to use one of two versions or not. Firstly, we check if the periodic
orbit is valid for the second version. If this is true then we follow the
steps of the second version (see Fig. 1) otherwise we check the validity of
the first version of the algorithm. If the periodic orbit satisfies the
conditions for the first version then we follow the steps of this version (see
Fig. 1) otherwise we must use another method.

Figure 1: The pseudocode for the algorithm.

##  3 The Algorithm for Hamiltonian systems with two degrees of freedom

In the previous section we described the difference between the two versions
of our algorithm for the construction of dividing surfaces and the
circumstances when we use one or the other version. Now, we describe in detail
the two versions of our algorithm (for $n=2$) in the case of Hamiltonian
systems with two degrees of freedom with potential energy function $V(x,y)$:

$\displaystyle p_{x}^{2}/2m_{1}+p_{y}^{2}/2m_{2}+V(x,y)=E$

where $p_{x},p_{y}$ are the momenta and $m_{1},m_{2}$ the corresponding
masses.
The algorithm for Hamiltonian systems with two degrees of freedom comprises
the next two algorithms that correspond to the first and second version of our
algorithm. The application of the second version in Hamiltonian systems with
two degrees of freedom give us the same algorithm as the classical algorithm
for periodic orbit dividing surfaces (see Pechukas & McLafferty [1973];
Pechukas & Pollak [1977]; Pollak & Pechukas [1978]; Pechukas & Pollak [1979];
Pollak [1985] and for more details see Ezra & Wiggins [2018]). We describe the
two versions of the algorithm in the first two subsections and we present an
application of the algorithm to the case of a quadratic normal form
Hamiltonian with two degrees of freedom in the third subsection.

###  3.1 The first version of the algorithm

The first version of the algorithm is:

  1. 1.

Locate an unstable periodic orbit PO for a fixed value of Energy $E$.

  2. 2.

Project the PO into the configuration space and consider a 2D subspace of the
phase space in which the projection of the periodic orbit is a closed curve
(for example in the $(y,p_{y})$ space).

  3. 3.

From the projection of the periodic orbit in the configuration space, we
construct a torus that is generated by the Cartesian product of a circle with
small radius and the projection of the periodic orbit in a 2D subspace of the
phase space (for example in the $(y,p_{y})$ space). Actually it is
topologically equivalent to the Cartesian product of two circles $S^{1}\times
S^{1}$. This is a two-dimensional torus. This can be achieved through the
construction of one circle around every point of the periodic orbit in the 2D
subspace of the 3D subspace of the phase space. For example we compute a
circle (with a fixed radius r) in the plane $(x,y)$ around every point of the
periodic orbit in the 3D space $(x,y,p_{y})$. The goal of this step is to
include all coordinates of the configuration space in this torus.

$\displaystyle y_{1,i,j1}=y_{0,i}+rcos(\theta_{j1})$ $\displaystyle
x_{1,i,j1}=x_{0,i}+rsin(\theta_{j1})$ $\displaystyle p_{y,1,i,j1}=p_{y,0,i}$
(4)

$(x_{0,i},y_{0,i},p_{y,0,i}),i=1,...N$ are the points of the periodic orbit in
the 3D subspace $(x,y,p_{y})$. We have the angle
$\theta_{j1}=j1\frac{2\pi}{n_{1}}$ with $j1=1,...,n_{1}$ for the circle that
we need for the construction of the torus.
$x_{1,i,j1},y_{1,i,j1},p_{y,1,i,j1}$ with $i=1,...,N$ and $j1=1,...,n_{1}$ are
the points of the torus that is constructed from the Cartesian product of
projection of the periodic orbit in the 2D subspace $(y,p_{y})$ and a circle
in the $(x,y)$ space in the 3D space $(x,y,p_{y})$.

  4. 4.

For each point $x_{1,i,j1},y_{1,i,j1},p_{y,1,i,j1}$ on this torus we can
calculate the $p_{x,1,i,j1}$ by solving the following equation for a fixed
value of energy (Hamiltonian) $E$:

$\displaystyle
V(x_{1,i,j1},y_{1,i,j1})+\frac{p_{x,1,i,j1}^{2}}{2m_{1}}+\frac{p_{y,1,i,j1}^{2}}{2m_{2}}=E$

Dimensionality and Topology: This algorithm constructs a torus as the product
of the closed curve that represents the projection of the periodic orbit (1D
object) in a 2D subspace of the phase space with one circle in the 3D energy
surface. This torus is a 2-dimensional. Then we obtain the value of the last
momentum from the Hamiltonian of the system.

###  3.2 The second version of the algorithm

The second version of the algorithm is:

  1. 1.

Locate an unstable periodic orbit PO for a fixed value of Energy $E$.

  2. 2.

$(x_{0,i},y_{0,i}),i=1,...N$ are the points of the periodic orbit in the 2D
configuration space $(x,y)$. This is a closed curve in a 2D subspace. Actually
it is topologically equivalent with a circle.

  3. 3.

For each point $x_{0,i},y_{0,i}$ of the periodic orbit we must calculate the
$p_{x,0,i}^{max}$ and $p_{x,0,i}^{min}$ by solving the following equation for
a fixed value of energy (Hamiltonian) $E$ with $p_{y}=0$:

$\displaystyle V(x_{0,i},y_{0,i})+\frac{p_{x,0,i}^{2}}{2m_{1}}=E$

and we find the maximum and minimum values $p_{x,0,i}^{max}$ and
$p_{x,0,i}^{min}$. We choose points $p_{x,0,i}$ in the interval
$p_{x,0,i}^{min}\leq p_{x,0,i}\leq p_{x,0,i}^{max}$. These points can be
uniformly distributed in this interval.

  4. 4.

Now for every point $x_{0,i},y_{0,i},p_{x,0,i}$ we must calculate the
$p_{y,0,i}$ by solving the following equation for a fixed value of energy
(Hamiltonian) E:

$\displaystyle
V(x_{0,i},y_{0,i})+\frac{p_{x,0,i}^{2}}{2m_{1}}+\frac{p_{y,0,i}^{2}}{2m_{2}}=E$

Dimensionality and Topology: This algorithm give us an 1-dimensional object (a
circle or ellipse) as the projection of the periodic orbit (1D object) in the
configuration space. Then we sample the third variable (one of the momenta) in
the interval between its maximum and minimum value. Actually we create an
additional 1D segment and we increase the dimensionality of the initial object
(a circle or ellipse), from 1 to 2 dimensions, which is embedded in the 3D
energy surface. Then we obtain the value of the last momentum from the
Hamiltonian of the system.

###  3.3 Application of the algorithm to the Quadratic Normal Form Hamiltonian
System with two degrees of freedom

Now we will apply the algorithm to the system that is described in subsection
3.3.1. In this system the reaction occurs when $x$ changes sign Ezra & Wiggins
[2018]. The dividing surfaces for this system using the classical algorithm
have been computed (see for example Ezra & Wiggins [2018]). In this subsection
we will use our algorithm to compute the dividing surfaces. This can be done
if we choose the condition $x=0$ that is actually a 3-dimensional surface in
the four dimensional phase space. This will be applied in our algorithm. We
will construct a dividing surface from the periodic orbit PO that is a circle
in the plane $(y,p_{y})$ (see subsection 3.3.2). The first version of our
algorithm is valid for this periodic orbit because we have the combination of
one coordinate of the configuration space and one momentum (see for details
the diagram in section 2). We underline the fact that because one of the
coordinates of the configuration space is $x=0$ (in our case) we have not the
need to construct a two-dimensional torus. This means that the first version
of the algorithm is simplified and we construct the torus using the one circle
that is the projection of the periodic orbit.

####  3.3.1 Hamiltonian Model

The quadratic normal form Hamiltonian system near an index-one saddle Wiggins
[2016] is described by the following Hamiltonian:

$\displaystyle
H=\frac{\lambda}{2}(p_{x}^{2}-x^{2})+\frac{\omega}{2}(p_{y}^{2}+y^{2})$

with $\lambda>0,\omega>0$ and

$\displaystyle H_{1}=\frac{\lambda}{2}(p_{x}^{2}-x^{2})$ $\displaystyle
H_{2}=\frac{\omega}{2}(p_{y}^{2}+y^{2})$

The equations of motion are:

$\displaystyle\dot{x}=\frac{\partial H}{\partial p_{x}}=\lambda p_{x}$
$\displaystyle\dot{p}_{x}=-\frac{\partial H}{\partial x}=\lambda x$
$\displaystyle\dot{y}=\frac{\partial H}{\partial p_{y}}=\omega p_{y}$
$\displaystyle\dot{p}_{y}=-\frac{\partial H}{\partial y}=-\omega y$

The Hamiltonian functions $H_{1}$ and $H_{2}$ are uncoupled. The phase space
of these two subsystems can be studied separately. In this system we can
distribute the total energy between each mode. The
($x,p_{x},y,p_{y})=(0,0,0,0)$) for E=0 is an one-index saddle point of the
full system.

In this system the reaction occurs when the $x$ coordinate changes sign (for
$H_{1}>0$). We choose the condition $x=0$ (Ezra & Wiggins [2018]) to define a
three-dimensional surface in the four-dimensional phase space. From the
equations 3.3.1 we have that $H_{2}\geq 0$ and finally $H=H_{1}+H_{2}\equiv
E>0$. The energy surface is:

$\displaystyle\frac{\lambda}{2}(p_{x}^{2}-x^{2})+\frac{\omega}{2}(p_{y}^{2}+y^{2})=E$

the intersection of $x=0$ with this surface is:

$\displaystyle\frac{\lambda}{2}p_{x}^{2}+\frac{\omega}{2}(p_{y}^{2}+y^{2})=E$

This is the Dividing surface (a two-sphere) that can be constructed by the
NHIM (see for more details in Wiggins [2016] and Ezra & Wiggins [2018]). The
NHIM is the unstable periodic orbit in Hamiltonian systems with two degrees of
freedom. We call this periodic orbit as PO. The equator of this dividing
surface is given by $p_{x}=0$ and we have:

$\displaystyle\frac{\omega}{2}(p_{y}^{2}+y^{2})=E\qquad PO$

This equation represents the periodic orbit (PO) that is a circle in the plane
$(y,p_{y})$.In the next section we will construct, using the algorithm 1, a
two dimensional dividing surface using this periodic orbit.

####  3.3.2 PO-Dividing Surface

We apply the first version (see the introduction of this section) of our
algorithm to the PO (see 3.3.1):

  1. 1.

The periodic orbit 1 is given by 3.3.1 for every fixed value $E$.

  2. 2.

The periodic orbit is a circle and it lies on a plane ($(y,p_{y})$. The radius
of this circle is $\sqrt{\frac{2E}{\omega}}$.

  3. 3.

As we described in the introduction of this section the dividing surface is
embedded in the surface $x=0$ and this means that we have not the need to
construct a two-dimensional torus. This means that this step will be
simplified and we will construct 1-dimensional torus using one circle that is
the projection of the periodic orbit in the $(y,p_{y})$ plane. The equation of
this torus is the equation 3.3.1.

  4. 4.

Step 4 is carried out naturally in this case (with $x=0$). We compute $p_{x}$
from the Hamiltonian.

no-recrossing property:

$\displaystyle
p_{x}=\sqrt{\frac{2}{\lambda}(H_{1}+H_{2}-\frac{\omega}{2}(p_{y}^{2}+y^{2}))}\qquad
p_{x}>0\qquad Forward\quad DS$ $\displaystyle
p_{x}=-\sqrt{\frac{2}{\lambda}(H_{1}+H_{2}-\frac{\omega}{2}(p_{y}^{2}+y^{2}))}\qquad
p_{x}<0\qquad Backward\quad DS$

The new DS that is constructed has the no-recrossing property because
$\dot{x}=\lambda p_{x}$. This is exactly the same dividing surface that we
obtain from the classical algorithm of Pechukas [1981],[Pollak, 1985] (see
equations 12 in Ezra & Wiggins [2018]).

##  4 The Algorithm for Hamiltonian systems with three degrees of freedom

In the section 2 we described the difference between the two versions of our
algorithm for the construction of dividing surfaces and when we use both
versions. Now, we describe in detail the two versions of our algorithm (for
$n=3$) in the case of Hamiltonian systems with three degrees of freedom with a
potential energy $V(x,y,z)$:

$\displaystyle p_{x}^{2}/2m_{1}+p_{y}^{2}/2m_{2}+p_{z}^{2}/2m_{3}+V(x,y,z)=E$

where $p_{x},p_{y},p_{z}$ are the momenta and $m_{1},m_{2},m_{3}$ the
corresponding masses.
When we apply the two versions of the algorithm in Hamiltonian systems with
three degrees of freedom we produce two algorithms, one algorithm for each
version. We describe the two versions of the algorithm in the first two
subsections and we present an application of the algorithm to the case of a
quadratic normal form Hamiltonian with three degrees of freedom in the third
subsection.

###  4.1 The first version of the algorithm

The first version of the algorithm is:

  1. 1.

Locate an unstable periodic orbit PO for a fixed value of Energy $E$.

  2. 2.

Project the PO into the configuration space and we consider a 2D subspace of
the phase space in which the projection of the periodic orbit is a closed
curve (for example in the $(y,p_{y})$ space).

  3. 3.

From the projection of the periodic orbit in the configuration space, we
construct a torus that is generated by the Cartesian product of 2 circles with
small radius and the projection of the periodic orbit in a 2D subspace of the
phase space (for example in the $(y,p_{y})$ space). Actually it is
topologically equivalent with the Cartesian product of three circles
$S^{1}\times S^{1}\times S^{1}$. This is a three-dimensional torus
(Hypertorus). This can be achieved through the construction of one circle
around every point of the periodic orbit in the 2D subspace of a 4D subspace
of the phase space. For example we compute a circle (with a fixed radius r) in
the plane $(x,y)$ around every point of the periodic orbit in the 4D subspace
$(x,y,z,p_{y})$. Then we construct a new circle around every point of the
previous structure in the 4D subspace $(x,y,z,p_{y})$ of the phase space. This
can be done computing a circle (with a fixed radius r) in the plane $(y,z)$.
The goal of this step is to include all coordinates of the configuration space
in this torus.

$\displaystyle y_{1,i,j1}=y_{0,i}+rcos(\theta_{j1})$ $\displaystyle
x_{1,i,j1}=x_{0,i}+rsin(\theta_{j1})$ $\displaystyle z_{1,i,j1}=z_{0,i}$
$\displaystyle p_{y,1,i,j1}=p_{y,0,i}$ (16)

$\displaystyle y_{2,i,j1,j2}=y_{1,i,j1}+rcos(\theta_{j2})$ $\displaystyle
z_{2,i,j1,j2}=z_{1,i,j1}+rsin(\theta_{j2})$ $\displaystyle
x_{2,i,j1,j2}=x_{1,i,j1}$ $\displaystyle p_{y,2,i,j1,j2}=p_{y,1,i,j1}$ (17)

$(x_{0,i},y_{0,i},z_{0,i},p_{y,0,i}),i=1,...N$ are the points of the periodic
orbit in the 4D subspace $(x,y,z,p_{y})$. We have the angle
$\theta_{j1}=j1\frac{2\pi}{n_{1}}$ with $j1=1,...,n_{1}$ for the first circle
and $\theta_{j2}=j2\frac{2k\pi}{n_{1}}$ with $j2=1,...,n_{1}$ for the second
circle that we need for the construction of the torus.

$x_{1,i,j1},y_{1,i,j1},z_{1,i,j1},p_{y,1,i,j1}$ with $i=1,...,N$ and
$j1=1,...,n_{1}$ are the points of the torus that is constructed from the
Cartesian product of projection of the periodic orbit in the 2D subspace
$(y,p_{y})$ and a circle in the $(x,y)$ space in the 4D space $(x,y,z,p_{y})$.
$x_{2,i,j1,j2},y_{2,i,j1,j2},z_{2,i,j1,j2},p_{y,2,i,j1,j2}$ with $i=1,...,N$,
$j1=1,...,n_{1}$ and $j2=1,...,n_{1}$ are the points of the torus that is
constructed from the Cartesian product of the projection of the periodic orbit
in the 2D subspace $(y,p_{y})$, a circle in the $(x,y)$ space and a circle in
the $(y,z)$ space in the 4D space $(x,y,z,p_{y})$.

  4. 4.

For each point $x_{2,i,j1,j2},y_{2,i,j1,j2},z_{2,i,j1,j2},p_{y,2,i,j1,j2}$ on
this torus we must calculate the $p_{x,2,i,j1,j2}^{max}$ and
$p_{x,2,i,j1,j2}^{min}$ by solving the following equation for a fixed value of
energy (Hamiltonian) $E$ with $p_{z}=0$:

$\displaystyle
V(x_{2,i,j1,j2},y_{2,i,j1,j2},z_{2,i,j1,j2})+\frac{p_{x,2,i,j1,j2}^{2}}{2m_{1}}+\frac{p_{y,2,i,j1,j2}^{2}}{2m_{2}}=E$

and we find the maximum and minimum values $p_{x,2,i,j1,j2}^{max}$ and
$p_{x,2,i,j1,j2}^{min}$. We choose points $p_{x,2,i,j1,j2}$ with
$j2=1,...,n_{1}$ in the interval $p_{x,2,i,j1,j2}^{min}\leq
p_{x,2,i,j1,j2}\leq p_{x,2,i,j1,j2}^{max}$. These points can be uniformly
distributed in this interval. Then we obtain the value $p_{z,2,i,j1,j2}$ from
the Hamiltonian:

$\displaystyle
V(x_{2,i,j1,j2},y_{2,i,j1,j2},z_{2,i,j1,j2})+\frac{p_{x,2,i,j1,j2}^{2}}{2m_{1}}+\frac{p_{y,2,i,j1,j2}^{2}}{2m_{2}}+\frac{p_{z,2,i,j1,j2}^{2}}{2m_{3}}=E$

Dimensionality and Topology: This algorithm constructs a torus as the product
of the closed curve that represents the projection of the periodic orbit (1D
object) in a 2D subspace with two circles in the 4D subspace of the 5D energy
manifold. This torus is a 3-dimensional (hypertorus). Then we sample the fifth
variable (one of the momenta) in the interval between its maximum and minimum
value. Actually we create an additional 1D segment and we increase the
dimensionality of the initial torus, from 3 to 4 dimensions, which is embedded
in the 5D energy surface. Then we obtain the value of the last momentum from
the Hamiltonian of the system.

###  4.2 The second version of the algorithm

The second version of the algorithm is:

  1. 1.

Locate an unstable periodic orbit PO for a fixed value of Energy $E$.

  2. 2.

Project the PO into the configuration space and we consider a 2D subspace of
the configuration space in which the projection of the periodic orbit is a
closed curve (for example in the $(x,y)$ space).

  3. 3.

We construct a torus that is generated by the Cartesian product of one circle
with small radius and the projection of the periodic orbit in a 2D subspace of
the configuration space (for example in the $(x,y)$ space). Actually it is the
Cartesian product of two circles $S^{1}\times S^{1}$. This is a two-
dimensional torus. This can be achieved through the construction of one circle
around every point of the periodic orbit in the 2D subspace of a 3D subspace
of the phase space. For example we compute a circle (with a fixed radius r) in
the plane $(y,z)$ around every point of the periodic orbit in the 3D subspace
$(x,y,z)$.

The points of the torus that we constructed are:

$\displaystyle y_{1,i,j1}=y_{0,i}+rcos(\theta_{j1})$ $\displaystyle
z_{1,i,j1}=z_{0,i}+rsin(\theta_{j1})$ $\displaystyle x_{1,i,j1}=x_{0,i}$ (20)

$(x_{0,i},y_{0,i},z_{0,i}),i=1,...N$ are the points of the periodic orbit in
the 3D configuration space $(x,y,z)$. We have the angle
$\theta_{j1}=j1\frac{2\pi}{n_{1}}$ with $j1=1,...,n_{1}$ for the circle that
we need for the construction of the torus.

$x_{1,i,j1},y_{1,i,j1},z_{1,i,j1}$ with $i=1,...,N$ and $j1=1,...,n_{1}$ are
the points of the torus that is constructed from the Cartesian product of
projection of the periodic orbit in the 2D subspace $(x,y)$ and a circle in
the $(y,z)$ space in the 3D space $(x,y,z)$.

  4. 4.

For each point $x_{1,i,j1},y_{1,i,j1},z_{1,i,j1}$ on this torus we must
calculate the $p_{x,1,i,j1}^{max}$ and $p_{x,1,i,j1}^{min}$ by solving the
following equation for a fixed value of energy (Hamiltonian) $E$ with
$p_{y}=p_{z}=0$:

$\displaystyle
V(x_{1,i,j1},y_{1,i,j1},z_{1,i,j1})+\frac{p_{x,1,i,j1}^{2}}{2m_{1}}=E$

and we find the maximum and minimum values $p_{x,1,i,j1}^{max}$ and
$p_{x,1,i,j1}^{min}$. We choose points $p_{x,1,i,j1}$ with $j1=1,...,n_{1}$ in
the interval $p_{x,1,i,j1}^{min}\leq p_{x,1,i,j1}\leq p_{x,1,i,j1}^{max}$.
These points can be uniformly distributed in this interval.

  5. 5.

Now for every point $x_{1,i,j1},y_{1,i,j1},z_{1,i,j1},p_{x,1,i,j1}$ we must
calculate the $p_{y,1,i,j1}^{max}$ and $p_{y,1,i,j1}^{min}$ by solving the
following equation for a fixed value of energy (Hamiltonian) $E$ with
$p_{z}=0$:

$\displaystyle
V(x_{1,i,j1},y_{1,i,j1},z_{1,i,j1})+\frac{p_{x,1,i,j1}^{2}}{2m_{1}}+\frac{p_{y,1,i,j1}^{2}}{2m_{2}}=E$

We choose points $p_{y,1,i,j1}$ with $j1=1,...,n_{1}$ in the interval
$p_{y,1,i,j1}^{min}\leq p_{y,1,i,j1}\leq p_{y,1,i,j1}^{max}$. These points can
be uniformly distributed in this interval. Then we obtain the value
$p_{z,1,i,j1}$ from the Hamiltonian:

$\displaystyle
V(x_{1,i,j1},y_{1,i,j1},z_{1,i,j1})+\frac{p_{x,1,i,j1}^{2}}{2m_{1}}+\frac{p_{y,1,i,j1}^{2}}{2m_{2}}+\frac{p_{z,1,i,j1}^{2}}{2m_{3}}=E$

Dimensionality and Topology: This algorithm constructs a torus as the product
of the closed curve that represents the projection of the periodic orbit (1D
object) in a 2D subspace with one circle in the configuration space. This
torus is a 2-dimensional. Then we sample the fourth variable (one of the
momenta) in the interval between its maximum and minimum value. Actually we
create an additional 1D segment and we increase the dimensionality of the
initial torus, from 2 to 3 dimensions, which is embedded in the 4D subspace of
the 5D energy surface. Then we sample the fifth variable (one of the other two
momenta) in the interval between its maximum and minimum value. This means
that we create an additional 1D segment and we increase the dimensionality of
the initial torus, from 3 to 4 dimensions, which is embedded in the 5D energy
surface. Then we obtain the value of the last momentum from the Hamiltonian of
the system.

###  4.3 Application of the algorithms in the Quadratic Normal Form
Hamiltonian System with three degrees of freedom

Now we will apply the algorithm to the system that is described in subsection
4.3.1. In this system the reaction occurs when $x$ changes sign Ezra & Wiggins
[2018]. This can be done if we choose the condition $x=0$ that is actually a
5-dimensional surface in the six dimensional phase space. This will be applied
in our algorithms (in this way, we consider the intersection of the four
dimensional structure that is obtained by our algorithm with $x=0$). Firstly
we will construct a dividing surface from the periodic orbit PO1 that is a
circle in the plane $(y,p_{y})$ (see subsection 4.3.2). Then we will do the
same for the periodic orbit PO2 that is a circle in the plane $(z,p_{z})$ (see
subsection 4.3.3). In this case we will use version 1 of the algorithm because
the periodic orbits are closed curves in the 2D subspaces of the phase space
that consist of a combination of a coordinate of the configuration space with
one momentum (see the conditions for the choice between the first and second
version in the section 2) . We underline the fact that because one of the
coordinates of the configuration space is $x=0$ (in our case) we have not the
need to construct a three-dimensional torus. This means that the first version
of the algorithm is simplified and we construct the torus using the Cartesian
product of one circle (not two circles) with the projection of the periodic
orbit.

####  4.3.1 Hamiltonian Model

The quadratic normal form Hamiltonian system near an index-one saddle Wiggins
[2016] is described by the following Hamiltonian:

$\displaystyle
H=\frac{\lambda}{2}(p_{x}^{2}-x^{2})+\frac{\omega_{2}}{2}(p_{y}^{2}+y^{2})+\frac{\omega_{3}}{2}(p_{z}^{2}+z^{2})$

with $\lambda>0,\omega_{2}>0,\omega_{3}>0$ and

$\displaystyle H_{1}=\frac{\lambda}{2}(p_{x}^{2}-x^{2})$ $\displaystyle
H_{2}=\frac{\omega_{2}}{2}(p_{y}^{2}+y^{2})$ $\displaystyle
H_{3}=\frac{\omega_{3}}{2}(p_{z}^{2}+z^{2})$

The equations of motion are:

$\displaystyle\dot{x}=\frac{\partial H}{\partial p_{x}}=\lambda p_{x}$
$\displaystyle\dot{p}_{x}=-\frac{\partial H}{\partial x}=\lambda x$
$\displaystyle\dot{y}=\frac{\partial H}{\partial p_{y}}=\omega_{2}p_{y}$
$\displaystyle\dot{p}_{y}=-\frac{\partial H}{\partial y}=-\omega_{2}y$
$\displaystyle\dot{z}=\frac{\partial H}{\partial p_{z}}=\omega_{3}p_{z}$
$\displaystyle\dot{p}_{z}=-\frac{\partial H}{\partial z}=-\omega_{3}z$

The Hamiltonian functions $H_{1}$ ,$H_{2}$ and $H_{3}$ are uncoupled. The
phase space of these three subsystems can be studied separately. In this
system we can distribute the total energy between each mode. The
($x,p_{x},y,p_{y},z,p_{z})=(0,0,0,0,0,0)$) for E=0 is an one-index saddle
point of the full system.

In this system the reaction occurs when the $x$ coordinate changes sign (for
$H_{1}>0$). We choose the condition $x=0$ (Ezra & Wiggins [2018]) to define a
five-dimensional surface in the six-dimensional phase space. From the
equations 4.3.1 we have that $H_{2}\geq 0$ and $H_{3}\geq 0$ and finally
$H=H_{1}+H_{2}+H_{3}\equiv E>0$. The energy surface is:

$\displaystyle\frac{\lambda}{2}(p_{x}^{2}-x^{2})+\frac{\omega_{2}}{2}(p_{y}^{2}+y^{2})+\frac{\omega_{3}}{2}(p_{z}^{2}+z^{2})=E$

the intersection of $x=0$ with this surface is:

$\displaystyle\frac{\lambda}{2}p_{x}^{2}+\frac{\omega_{2}}{2}(p_{y}^{2}+y^{2})+\frac{\omega_{3}}{2}(p_{z}^{2}+z^{2})=E$

This is the dividing surface (a four-sphere) that can be constructed by the
NHIM (see for more details in Wiggins [2016] and Ezra & Wiggins [2018]). The
equator of this dividing is given by $p_{x}=0$ and we have:

$\displaystyle\frac{\omega_{2}}{2}(p_{y}^{2}+y^{2})+\frac{\omega_{3}}{2}(p_{z}^{2}+z^{2})=E\qquad
NHIM$

The previous equation represents the NHIM. The intersection of the NHIM with
$z=0$ is given by the following equation:

$\displaystyle\frac{\omega_{3}}{2}p_{z}^{2}+\frac{\omega_{2}}{2}(p_{y}^{2}+y^{2})=E$

The equator of the NHIM is given by $p_{z}=0$ and we have the following
equation:

$\displaystyle\frac{\omega_{2}}{2}(p_{y}^{2}+y^{2})=E\qquad PO1$

This equation represents the periodic orbit (PO1) that is a circle in the
plane $(y,p_{y})$.

The intersection of the NHIM (see the equation 4.3.1) with $y=0$ is given by
the following equation:

$\displaystyle\frac{\omega_{2}}{2}p_{y}^{2}+\frac{\omega_{3}}{2}(p_{z}^{2}+z^{2})=E$

The equator of the NHIM is given by $p_{y}=0$ and we have the following
equation:

$\displaystyle\frac{\omega_{3}}{2}(p_{z}^{2}+z^{2})=E\qquad PO2$

This equation represents the periodic orbit (PO2) that is a circle in the
plane $(z,p_{z})$. In the next subsections we will construct dividing surfaces
using the periodic orbits PO1 and PO2.

####  4.3.2 PO1-Dividing Surface

We begin by applying the first version of our algorithm to the PO1 (see
4.3.1):

  1. 1.

The PO1 is given by 4.3.1 for every fixed value $E$.

  2. 2.

The periodic orbit is a circle and it lies on a plane ($(y,p_{y})$. The radius
of this circle is $\sqrt{\frac{2E}{\omega_{2}}}$.

  3. 3.

We construct a torus in the 3D subspace $(y,p_{y},z)$ using the product of the
circle of the periodic orbit and a circle with small fixed radius $r>0$. The
equation of this torus (with $x=0$):

$\displaystyle(\sqrt{p_{y}^{2}+y^{2}}-\sqrt{\frac{2E}{\omega_{2}}})^{2}+z^{2}=r^{2}$

  4. 4.

Step 4 is carried out naturally in this case (with $x=0$). We compute the
$p_{z}^{max}$ and $p_{z}^{min}$. We sample points in the interval
$[p_{z}^{min},p_{z}^{max}]$. For every point in the previous interval we
obtain the $p_{x}$ coordinate from the Hamiltonian.

no-recrossing property:

from the equation 3 we have:

$\displaystyle
z^{2}=r^{2}-(\sqrt{p_{y}^{2}+y^{2}}-\sqrt{\frac{2E}{\omega_{2}}})^{2}$

from the equation 4.3.1 (for a fixed value of energy $E$, the numerical value
of the Hamiltonian and for $x=0$) we have:

$\displaystyle\frac{\lambda}{2}p_{x}^{2}=E-\frac{\omega_{2}}{2}(p_{y}^{2}+y^{2})-\frac{\omega_{3}}{2}(p_{z}^{2}+z^{2})$

The equation 4 through the equation 4 and using that $E=H_{1}+H_{2}+H_{3}$
with $H_{2}=\frac{\omega_{2}}{2}(p_{y}^{2}+y^{2})$ (from equation 4.3.1) we
have:

$\displaystyle\frac{\lambda}{2}p_{x}^{2}=H_{1}+H_{3}-\frac{\omega_{3}}{2}p_{z}^{2}+\frac{\omega_{3}}{2}(\sqrt{p_{y}^{2}+y^{2}}-\sqrt{\frac{2E}{\omega_{2}}})^{2}-\frac{\omega_{3}}{2}r^{2}$

From equation 4.3.1 we have that $H_{3}-\frac{\omega_{3}}{2}p_{z}^{2}>0$. We
have also that the quantity
$\frac{\omega_{3}}{2}(\sqrt{p_{y}^{2}+y^{2}}-\sqrt{\frac{2E}{\omega_{2}}})^{2}\geq
0$ is always positive. This means that the condition that we must have for the
equation 4 is:

$\displaystyle H_{1}-\frac{\omega_{3}}{2}r^{2}\geq 0$

This implies that:

$\displaystyle r\leq\sqrt{\frac{2H_{1}}{\omega_{3}}}$

In practise this means that we must choose a small radius r. Using this
condition we have for the dividing surface :

$\displaystyle
p_{x}=\sqrt{\frac{2}{\lambda}(H_{1}+H_{3}-\frac{\omega_{3}}{2}p_{z}^{2}+\frac{\omega_{3}}{2}(\sqrt{p_{y}^{2}+y^{2}}-\sqrt{\frac{2E}{\omega_{2}}})^{2}-\frac{\omega_{3}}{2}r^{2})}\qquad
p_{x}>0\qquad Forward\quad DS$ $\displaystyle
p_{x}=-\sqrt{\frac{2}{\lambda}(H_{1}+H_{3}-\frac{\omega_{3}}{2}p_{z}^{2}+\frac{\omega_{3}}{2}(\sqrt{p_{y}^{2}+y^{2}}-\sqrt{\frac{2E}{\omega_{2}}})^{2}-\frac{\omega_{3}}{2}r^{2})}\qquad
p_{x}<0\qquad Backward\quad DS$

The new DS that is constructed has the no-recrossing property because
$\dot{x}=\lambda p_{x}$.

####  4.3.3 PO2-Dividing Surface

Now we apply the first version of our algorithm to the PO2 (see 4.3.1):

  1. 1.

The PO2 is given by 4.3.1 for every fixed value $E$.

  2. 2.

The periodic orbit is a circle and it lies on a plane $(z,p_{z})$. The radius
of this circle is $\sqrt{\frac{2E}{\omega_{3}}}$.

  3. 3.

We construct a torus in the 3D subspace $(z,p_{z},y)$ using the product of the
circle of the periodic orbit and a circle with small fixed radius $r_{1}>0$.
The equation of this torus (with $x=0$):

$\displaystyle(\sqrt{p_{z}^{2}+z^{2}}-\sqrt{\frac{2E}{\omega_{3}}})^{2}+y^{2}=r_{1}^{2}$

  4. 4.

Step 4 is carried out naturally in this case (with $x=0$). We compute the
$p_{y}^{max}$ and $p_{y}^{min}$. We sample points in the interval
$[p_{y}^{min},p_{y}^{max}]$. For every point in the previous interval we
obtain the $p_{x}$ coordinate from the Hamiltonian.

no-recrossing property:

from the equation 3 we have:

$\displaystyle
y^{2}=r_{1}^{2}-(\sqrt{p_{z}^{2}+z^{2}}-\sqrt{\frac{2E}{\omega_{3}}})^{2}$

from the equation 4.3.1 (for a fixed value of energy $E$, the numerical value
of the Hamiltonian) we have:

$\displaystyle\frac{\lambda}{2}p_{x}^{2}=E-\frac{\omega_{2}}{2}(p_{y}^{2}+y^{2})-\frac{\omega_{3}}{2}(p_{z}^{2}+z^{2})$

The equation 4 through the equation 4 and using that $E=H_{1}+H_{2}+H_{3}$
with $H_{3}=\frac{\omega_{3}}{2}(p_{z}^{2}+z^{2})$ (from equation 4.3.1) we
have:

$\displaystyle\frac{\lambda}{2}p_{x}^{2}=H_{1}+H_{2}-\frac{\omega_{2}}{2}p_{y}^{2}+\frac{\omega_{2}}{2}(\sqrt{p_{z}^{2}+z^{2}}-\sqrt{\frac{2E}{\omega_{3}}})^{2}-\frac{\omega_{2}}{2}r_{1}^{2}$

From equation 4.3.1 we have that $H_{2}-\frac{\omega_{2}}{2}p_{y}^{2}>0$. We
have also that the quantity
$\frac{\omega_{2}}{2}(\sqrt{p_{y}^{2}+y^{2}}-\sqrt{\frac{2E}{\omega_{3}}})^{2}\geq
0$ is always positive. This means that the condition that we must have for the
equation 4 is:

$\displaystyle H_{1}-\frac{\omega_{2}}{2}r_{1}^{2}\geq 0$

This implies that:

$\displaystyle r_{1}\leq\sqrt{\frac{2H_{1}}{\omega_{2}}}$

In practise this means that we must choose a small radius $r_{1}$. Using this
condition we have for the dividing surface :

$\displaystyle
p_{x}=\sqrt{\frac{2}{\lambda}(H_{1}+H_{2}-\frac{\omega_{2}}{2}p_{y}^{2}+\frac{\omega_{2}}{2}(\sqrt{p_{z}^{2}+z^{2}}-\sqrt{\frac{2E}{\omega_{3}}})^{2}-\frac{\omega_{2}}{2}r_{1}^{2})}\qquad
p_{x}>0\qquad Forward\quad DS$ $\displaystyle
p_{x}=-\sqrt{\frac{2}{\lambda}(H_{1}+H_{2}-\frac{\omega_{2}}{2}p_{y}^{2}+\frac{\omega_{2}}{2}(\sqrt{p_{z}^{2}+z^{2}}-\sqrt{\frac{2E}{\omega_{3}}})^{2}-\frac{\omega_{2}}{2}r_{1}^{2})}\qquad
p_{x}<0\qquad Backward\quad DS$

The new DS that is constructed has the no-recrossing property because
$\dot{x}=\lambda p_{x}$.

####  4.3.4 The structure and the comparison of the Dividing surfaces

In this section we constructed the dividing surfaces from the periodic orbits
PO1 and PO2 and from the Normally Hyperbolic Invariant Manifold (NHIM) of the
quadratic normal form Hamiltonian (see section 4.3.1) for
($\lambda=1,\omega_{2}=\sqrt{2},\omega_{3}=1$). These values were used from
Ezra & Wiggins [2018]. The value of the energy was $E=14,H_{1}=4$. We computed
the dividing surfaces from the periodic orbits PO1 and PO2 using the
algorithms and analytical formulas of the previous subsections (see
subsections 4.3.2 and 4.3.3). The dividing surface from the NHIM is obtained
using the algorithm and analytical formulae from Ezra & Wiggins [2018].

We begin by constructing the dividing surface from the periodic orbit 1 (PO1)
in three different cases. For this purpose we construct a torus (using as a
starting point the periodic orbit PO1- see subsection 4.3.2) that is the
cartesian product of the periodic orbit with a circle. The maximum radius of
this circle is $Rmax=\sqrt{\frac{2H_{1}}{\omega_{3}}}$ (see the equation 4).
We used three cases for the construction of a torus using a circle with small
radius r ($r=Rmax/20$), a circle with larger radius r ($r=Rmax/2$) and a
circle with the maximum radius r ($r=Rmax$).

Next we construct the dividing surface from the periodic orbit 2 (PO2). For
this purpose we construct a torus (using as a starting point the periodic
orbit PO2- see subsection 4.3.3) that is the cartesian product of the periodic
orbit with a circle. The maximum radius $r_{1}$ of this circle is
$R1max=\sqrt{\frac{2H_{1}}{\omega_{2}}}$ (see the equation 4). In this paper
we used the maximum radius $r_{1}$ for the construction of the PO2 dividing
surface. Finally we construct the dividing surface from the NHIM.

All dividing surfaces as we mentioned in the previous sections are 4
dimensional structures in the energy surface. All dividing surfaces have $x=0$
(as we mentioned in the previous section and we obtain the $p_{x}$ coordinate
from the Hamiltonian) and this is the reason that we studied the structure of
the dividing surfaces in the 4-dimensional space $(y,z,p_{y},p_{z})$.

We studied first the structure of these dividing surfaces and then we compare
them:

  1. 1.

The structure of the dividing surfaces: All dividing surfaces that are
constructed from the periodic orbits have similar structure in the
4-dimensional space $(y,z,p_{y},p_{z})$. In this subsection, we choose one
representative example of these surfaces, the dividing surface from PO1 with
the associated torus to have r=Rmax. We depict this surface in all 3D
projections of the 4-dimensional space $(y,z,p_{y},p_{z})$. This surface is
constructed as a torus. This topology is very obvious (as we can see in Fig.
6) in $(y,p_{y},p_{z})$ projection. This torus has also an hyperbolic
structure that is represented as an hyperboloid (as we can see in Fig. 5) in
$(y,z,p_{y})$ projection and hyperbolic box structure (as we can see in Fig.
4) in $(y,z,p_{z})$ and $(z,p_{y},p_{z})$ projections. This means that the
dividing surfaces that are constructed from periodic orbits are hyperbolic
tori in the 4-dimensional space $(y,z,p_{y},p_{z})$.

The dividing surface that is constructed from the NHIM is an ellipsoid in all
3D projections of the 4-dimensional space $(y,z,p_{y},p_{z})$. (as we can see
for example in $(y,z,p_{z})$ projection in Fig.7). This means that this
dividing surface is an ellipsoid in the 4-dimensional space.

  2. 2.

Comparison of Dividing surfaces: In Fig. 2 we compare the dividing surfaces
that are constructed from the periodic orbit PO1. For this reason we use the
2D projections $(y,z)$, $(y,p_{y})$ and $(z,p_{z})$. Using these projections
we can understand if one dividing surface is larger in the $y$ or $z$ or
$p_{y}$ or $p_{z}$ direction. This means that these 2D projections are enough
for the comparison of different dividing surfaces without the use of the other
2D projections $(y,p_{z})$, $(z,p_{y})$ and $(p_{y},p_{z})$. As we can see in
the Fig. 2 the dividing surface with associated torus with $r=Rmax/20$ has
similar length in the y-direction with the other two dividing surfaces. The
dividing surfaces form a hyperbolic box in the $(y,z)$ projection that grows
up with the increase of the radius of the one circle that is needed for the
construction of dividing surfaces (Fig. 2). The increase of the radius
increases the extension of the dividing surfaces in the z-direction. This is
the reason that the hyperbolic box of the dividing surface with $r=Rmax/20$ is
smaller than this of the dividing surface with $r=Rmax/2$ and this is smaller
than the hyperbolic box of the dividing surface with $r=Rmax$. In Fig. 2 we
see that the dividing surfaces form a ring structure in the $(y,p_{y})$
projection and an elliptic structure in the $(z,p_{z})$ with increasing
thickness as we increase the radius of the circle that is needed for the
construction of the dividing surfaces (see Fig. 2). As we increase the radius
of the circle that generates the torus that is the basis for the construction
of a periodic orbit dividing surface, this surface becomes more elongated in
the $p_{z}$ and $z$ directions (Fig. 2).

We compare the PO2 dividing surface with the PO1 dividing surface, both of
them associated with the torus with the maximum $r$ and $r_{1}$ respectively.
We compare the two cases of the dividing surfaces in Figs. 2. We observe that
the PO1 dividing surface is more elongated on the y-axis and the PO2 dividing
surface is more elongated on the z-axis (Fig. 2). Furthermore, the PO2
dividing surface form a ring structure in the $(z,p_{z})$ and not in the
$(y,p_{y})$ as the PO1 dividing surface. In addition, the PO2 dividing surface
form an elliptic structure in the $(y,p_{y})$ and not in the $(z,p_{z})$ as
the PO1 dividing surface. All these differences are because of the fact that
the PO1 is a circle in the $(y,p_{y})$ plane and PO2 is a circle in the
$(z,p_{z})$ plane. This means that the combination of these dividing surfaces
give us the dynamical information for every direction in the configuration
space for a small area of interest (the neighbourhood of the periodic orbits).

We compare the dividing surface from the NHIM with the dividing surfaces from
the PO1 and PO2, both of them associated with the torus with the maximum $r$
and $r_{1}$ respectively. We observe in Fig. 2 that the dividing surface from
PO1 and PO2 have the same range of values with the dividing surface from the
NHIM (Fig.3) in the $p_{y}$-direction and $p_{z}$-direction. But the PO1
dividing surface and the PO2 dividing surface have smaller range of values
than this of the dividing surface from the NHIM (Fig. 3) in the $z$-directon
and $y$-direction respectively. This means that the dividing surfaces from PO1
and PO2 are subsets of the dividing surface from the NHIM.

We give an example of trajectories that have initial conditions on one from
dividing surfaces that we constructed from the periodic orbits PO1 and PO2 and
we integrate them for 2 time units. We used as example the dividing surface
that is constructed from the PO1 that is constructed by a torus with small
radius $r=Rmax/20$ (Fig. 8). We observe that this structure has as a central
region a box with a tiny thickness and at the edges it has many sheets in the
configuration space. The structure of this surface is similar with these that
we obtained if the trajectories have initial conditions on other dividing
surfaces that are constructed from the same or the other periodic orbit PO2.

Figure 2: 2D projections of periodic orbit dividing surfaces in $(y,z)$ (first
column), $(y,p_{y})$ (second column) and $(z,p_{z})$ (third column) subspaces
of the phase space. The dividing surfaces are constructed from the periodic
orbit PO1 with the associated torus to have r=Rmax/20 (first row), r=Rmax/2
(second row) and r=Rmax (third row) and from the periodic PO2 with the
associated torus to have r=Rmax (fourth row).

Figure 3: 2D projections of dividing surface that is constructed from the NHIM
in $(y,z)$ (first panel), $(y,p_{y})$ (second panel) and $(z,p_{z})$ (third
panel) subspaces of the phase space.
Figure 4:  The 3D projection $(y,z,p_{z})$ of the dividing surface which is
constructed from the periodic orbit PO1 with the associated torus to have
r=Rmax. The color indicates the values of the third dimension. The same
structure is encountered also in 3D projection $(z,p_{y},p_{z})$. The
viewpoint is in spherical coordinates is $(30^{o},60^{o})$.
Figure 5:  The 3D projection $(y,z,p_{y})$ of the dividing surface which is
constructed from the periodic orbit PO1 with the associated torus to have
r=Rmax. The color indicates the values of the third dimension.The viewpoint is
in spherical coordinates is $(60^{o},20^{o})$.
Figure 6:  The 3D projection $(y,p_{y},p_{z})$ of the dividing surface which
is constructed from the periodic orbit PO1 with the associated torus to have
r=Rmax. The color indicates the values of the third dimension.The viewpoint is
in spherical coordinates is $(20^{o},12^{o})$.
Figure 7:  The 3D projection $(y,z,p_{z})$ of the dividing surface which is
constructed from the NHIM. The color indicates the values of the third
dimension.The viewpoint is in spherical coordinates is $(30^{o},60^{o})$.
Figure 8:  Trajectories in the 3D configuration space that have initial
conditions on the dividing surfaces which is constructed from the periodic
orbit PO1 with the associated torus to have r=Rmax/20. The color indicates the
values of the third dimension. The viewpoint is in spherical coordinates is
$(60^{o},30^{o})$.

##  5 The Algorithm for Hamiltonian systems with $n$ degrees of freedom

In the section 2 we described the difference between the two versions of our
algorithm for the construction of dividing surfaces and when we use the one or
other version. Now, we describe in detail the two versions of our algorithm in
the case of Hamiltonian systems with $n$ degrees of freedom with a potential
energy $V(x_{1},x_{2},...,x_{n})$:

$\displaystyle T+V(x_{1},x_{2},...,x_{n})=E$

where $T$ is the kinetic energy.

$\displaystyle
T=p_{x_{1}}^{2}/2m_{1}+p_{x_{2}}^{2}/2m_{2}+...+p_{x_{n}}^{2}/2m_{n}$

where $p_{x_{1}},p_{x_{2}}...p_{x_{n}}$ are the momenta and
$m_{1},m_{2},...m_{n}$ the corresponding masses.
When we apply the two versions of the algorithm to Hamiltonian systems with
$n$ degrees of freedom we produce two algorithms, one algorithm for each
version.

###  5.1 First Version

The first version of the algorithm is:

  1. 1.

Locate an unstable periodic orbit PO for a fixed value of Energy $E$.

  2. 2.

Project the PO into the configuration space and we consider a 2D subspace of
the phase space in which the projection of the periodic orbit is a closed
curve (for example in the $(x_{2},p_{x_{2}})$ space).

  3. 3.

From the projection of the periodic orbit in the configuration space, we
construct a torus that is generated by the Cartesian product of $n-1$ circles
with small radius and the projection of the periodic orbit in a 2D subspace of
the phase space (for example in the $(x_{2},p_{x_{2}})$ space). Actually
topologically it is equivalent with the Cartesian product of $n$ circles
$S^{1}\times S^{1}\times S^{1}\times...\times S^{1}$. This is a
$n$-dimensional torus. This can be achieved through the construction of one
circle around every point of the periodic orbit in a ($n+1$)D subspace of the
phase space. For example we compute a circle (with a fixed radius r) in the
plane $(x_{1},x_{2})$ around every point of the periodic orbit in the ($n+1$)D
subspace $(x_{1},x_{2},...,x_{n},p_{x_{2}})$. Then we construct a new circle
around every point of the previous structure in other 2D subspace of the
($n+1$)D subspace $(x_{1},...,x_{n},p_{x_{2}})$. This can be done computing a
circle (with a fixed radius r) in the plane $(x_{2},x_{3})$. Then we continue
adding circles until we will have added $n-1$ circles to the initial
projection of the periodic orbit. The target of this step is to include all
coordinates of the configuration space in this torus.

$\displaystyle x_{2,1,i,j1}=x_{2,0,i}+rcos(\theta_{j1})$ $\displaystyle
x_{1,1,i,j1}=x_{1,0,i}+rsin(\theta_{j1})$ $\displaystyle
x_{3,1,i,j1}=x_{3,0,i}$ $\displaystyle...$ $\displaystyle
x_{n,1,i,j1}=x_{n,0,i}$ $\displaystyle p_{x_{2},1,i,j1}=p_{x_{2},0,i}$ (50)

$\displaystyle x_{2,2,i,j1,j2}=x_{2,1,i,j1}+rcos(\theta_{j2})$ $\displaystyle
x_{3,2,i,j1,j2}=x_{3,1,i,j1}+rsin(\theta_{j2})$ $\displaystyle
x_{1,2,i,j1,j2}=x_{1,1,i,j1}$ $\displaystyle...$ $\displaystyle
x_{n,2,i,j1,j2}=x_{n,1,i,j1}$ $\displaystyle
p_{x_{2},2,i,j1,j2}=p_{x_{2},1,i,j1}$ $\displaystyle...$ (51)

$\displaystyle...$ $\displaystyle
x_{2,n-1,i,j1,j2,...,j(n-1)}=x_{2,n-2,i,j1,j2,...,j(n-2)}+rcos(\theta_{j(n-1)})$
$\displaystyle
x_{n,n-1,i,j1,j2...j(n-1)}=x_{n,n-2,i,j1,j2,...,j(n-2)}+rsin(\theta_{j(n-1)})$
$\displaystyle x_{1,n-1,i,j1,j2,...j(n-1)}=x_{1,n-2,i,j1,j2,...,j(n-2)}$
$\displaystyle...$ $\displaystyle
x_{n-1,n-1,i,j1,j2,...,j(n-1)}=x_{n-1,n-2,1,i,j1,j2,...,j(n-2)}$
$\displaystyle
p_{x_{2},n-1,i,j1,j2,...,j(n-1)}=p_{x_{2},n-2,i,j1,j2,...,j(n-2)}$

$(x_{1,0,i},x_{2,0,i},...,x_{n,0,i},p_{x_{2},0,i}),i=1,...N$ are the points of
the periodic orbit in the $(n+1)$D subspace
$(x_{1},x_{2},...,x_{n},p_{x_{2}})$. We have the angle
$\theta_{j1}=j1\frac{2\pi}{n_{1}}$ with $j1=1,...,n_{1}$ for the first circle
and $\theta_{j2}=j2\frac{2k\pi}{n_{1}}$ with $j2=1,...,n_{1}$ for the second
circle and so on the $\theta_{j(n-1)}=j(n-1)\frac{2\pi}{n_{1}}$ with
$j(n-1)=1,...,n_{1}$ for the $n-1$ circle that we need for the construction of
the torus.

$x_{1,1,i,j1},x_{1,1,i,j1},...,x_{n,1,i,j1},p_{x_{2},1,i,j1}$ with $i=1,...,N$
and $j1=1,...,n_{1}$ are the points of the torus that is constructed from the
Cartesian product of projection of the periodic orbit in the 2D subspace
$(x_{2},p_{x_{2}})$ and a circle in the $(x_{1},x_{2})$ space in the $(n+1)$D
space $(x_{1},x_{2},...x_{n},p_{x_{2}})$.
$x_{2,2,i,j1,j2},x_{1,2,i,j1,j2},...x_{n,2,i,j1,j2},p_{x_{2},2,i,j1,j2}$ with
$i=1,...,N$, $j1=1,...,n_{1}$ and $j2=1,...,n_{1}$ are the points of the torus
that is constructed from the Cartesian product of the projection of the
periodic orbit in the 2D subspace $(x_{2},p_{x_{2}})$, and other 2 circles in
the ($n+1$)D space $(x_{1},x_{2},...,x_{n},p_{x_{2}})$. And so on
$x_{1,n-1,i,j1,j2,...j(n-1)},x_{2,n-1,i,j1,j2,...j(n-1)},...x_{n,n-1,i,j1,j2,..j(n-1)},p_{x_{2},n-1,i,j1,j2,...j(n-1)}$
with $i=1,...,N$ and $j1,j2,....j(n-1)=1,...,n_{1}$ are the points of the
torus that is constructed from the Cartesian product of the projection of the
periodic orbit in the 2D subspace $(x_{2},p_{x_{2}})$, and other $n-1$ circles
in the ($n+1$)D space $(x_{1},x_{2},...,x_{n},p_{x_{2}})$.

  4. 4.

For each point
$x_{1,n-1,i,j1,j2,...j(n-1)},x_{2,n-1,i,j1,j2,...j(n-1)},...x_{n,n-1,i,j1,j2,..j(n-1)},p_{x_{2},n-1,i,j1,j2,...j(n-1)}$
with $i=1,...,N$ and $j1,j2,....j(n-1)=1,...,n_{1}$ on this torus we must
calculate the $p_{x_{1},n-1,i,j1,j2,,...j(n-1)}^{max}$ and
$p_{x_{1},n-1,i,j1,j2,,...j(n-1)}^{min}$ by solving the following equation for
a fixed value of energy (Hamiltonian) E with $p_{x_{3}}=...=p_{x_{n}}=0$:

$\displaystyle
V(x_{1,n-1,i,j1,j2,...j(n-1)},x_{2,n-1,i,j1,j2,...j(n-1)},...x_{n,n-1,i,j1,j2,..j(n-1)})+$
$\displaystyle\frac{p_{x_{1},n-1,i,j1,j2,...j(n-1)}^{2}}{2m_{1}}+\frac{p_{x_{2},n-1,i,j1,j2,...j(n-1)}^{2}}{2m_{2}}=E$

and we find the maximum and minimum values
$p_{x_{1},n-1,i,j1,j2,...j(n-1)}^{max}$ and
$p_{x_{1},n-1,i,j1,j2,...j(n-1)}^{min}$. We choose points
$p_{x_{1},n-1,i,j1,j2,...j(n-1)}$ with $j1,j2...j(n-1)=1,...,n_{1}$ in the
interval $p_{x_{1},n-1,i,j1,j2,...j(n-1)}^{min}\leq
p_{x_{1},n-1,i,j1,j2,...j(n-1)}\leq p_{x_{1},n-1,i,j1,j2,...j(n-1)}^{max}$.
These points can be uniformly distributed in this interval. We will repeat the
same procedure to compute the values
$p_{x_{3},n-1,i,j1,j2,...j(n-1)},...,p_{x_{n-1},n-1,i,j1,j2,...j(n-1)}$. In
general for $n_{2}\leq n-1$ for each point
$x_{1,n-1,i,j1,j2,...j(n-1)},x_{2,n-1,i,j1,j2,...j(n-1)},...x_{n,n-1,i,j1,j2,..j(n-1)},p_{x_{1},n-1,i,j1,j2,...j(n-1)},...\\\
p_{x_{n_{2}-1},n-1,i,j1,j2,...j(n-1)}$ with $i=1,...,N$ and
$j1,j2,....j(n-1)=1,...,n_{1}$ we must calculate the
$p_{x_{n_{2}},n-1,i,j1,j2,,...j(n-1)}^{max}$ and
$p_{x_{n_{2}},n-1,i,j1,j2,,...j(n-1)}^{min}$ by solving the following equation
for a fixed value of energy (Hamiltonian) E with
$p_{x_{n_{2}+1}}=...=p_{x_{n}}=0$:

$\displaystyle
V(x_{1,n-1,i,j1,j2,...j(n-1)},x_{2,n-1,i,j1,j2,...j(n-1)},...x_{n,n-1,i,j1,j2,..j(n-1)})+$
$\displaystyle\frac{p_{x_{1},n-1,i,j1,j2,...j(n-1)}^{2}}{2m_{1}}+...+\frac{p_{x_{n_{2}},n-1,i,j1,j2,...j(n-1)}^{2}}{2m_{n_{2}}}=E$

and we find the maximum and minimum values
$p_{x_{n_{2}},n-1,i,j1,j2,...j(n-1)}^{max}$ and
$p_{x_{n_{2}},n-1,i,j1,j2,...j(n-1)}^{min}$. We choose points
$p_{x_{n_{2}},n-1,i,j1,j2,...j(n-1)}$ with $j1,j2,j(n-1)=1,...,n_{1}$ in the
interval $p_{x_{n_{2}},n-1,i,j1,j2,...j(n-1)}^{min}\leq
p_{x_{n_{2}},n-1,i,j1,j2,...j(n-1)}\leq
p_{x_{n_{2}},n-1,i,j1,j2,...j(n-1)}^{max}$. These points can be uniformly
distributed in this interval.

Then we obtain the value $p_{x_{n},n-1,i,j1,j2,...j(n-1)}$ from the
Hamiltonian:

$\displaystyle
V(x_{1,n-1,i,j1,j2,...j(n-1)},x_{2,n-1,i,j1,j2,...j(n-1)},...x_{n,n-1,i,j1,j2,..j(n-1)})+$
$\displaystyle\frac{p_{x_{1},n-1,i,j1,j2,...j(n-1)}^{2}}{2m_{1}}+...+\frac{p_{x_{n},n-1,i,j1,j2,...j(n-1)}^{2}}{2m_{n}}=E$

Dimensionality and Topology: This algorithm constructs a torus as the product
of the closed curve that represents the projection of the periodic orbit (1D
object) in a 2D subspace with $n-1$ circles in the ($n+1$)D subspace of the
energy manifold. This torus is a n-dimensional torus. Then we sample the $n-2$
variables ($n-2$ momenta) in the interval between their maximum and minimum
value. Actually we create $n-2$ additional segments and we increase the
dimensionality of the initial torus, from n to $2n-2$ dimensions, which is
embedded in the $2n-1$ energy surface. Then we obtain the value of the last
momentum from the Hamiltonian of the system.

###  5.2 Second Version

The second version is:

  1. 1.

Locate an unstable periodic orbit PO for a fixed value of Energy $E$.

  2. 2.

Project the PO into the configuration space and we consider a 2D subspace of
the configuration space in which the projection of the periodic orbit is a
closed curve (for example in the $(x_{1},x_{2})$ space).

  3. 3.

From the projection of the periodic orbit in the configuration space, we
construct a torus that is generated by the Cartesian product of $n-2$ circles
with small radius and the projection of the periodic orbit in a 2D subspace of
the configuration space (for example in the $(x_{1},x_{2})$ space). Actually
topologically it is equivalent with the Cartesian product of $n-1$ circles
$S^{1}\times S^{1}\times S^{1}\times...\times S^{1}$. This is a
$n-1$-dimensional torus. This can be achieved through the construction of one
circle around every point of the periodic orbit in a $n$D subspace of the
phase space (configuration space). For example we compute a circle (with a
fixed radius r) in the plane $(x_{2},x_{3})$ around every point of the
periodic orbit in the $n$D configuration space $(x_{1},x_{2},...,x_{n})$. Then
we construct a new circle around every point of the previous structure in
other 2D subspace of the $n$D configuration space $(x_{1},...,x_{n})$. This
can be done computing a circle (with a fixed radius r) in the plane
$(x_{2},x_{4})$. Then we continue adding circles until we will have added
$n-2$ circles to the initial projection of the periodic orbit. The target of
this step is to include all coordinates of the configuration space in this
torus.

$\displaystyle x_{2,1,i,j1}=x_{2,0,i}+rcos(\theta_{j1})$ $\displaystyle
x_{3,1,i,j1}=x_{3,0,i}+rsin(\theta_{j1})$ $\displaystyle
x_{1,1,i,j1}=x_{1,0,i}$ $\displaystyle x_{4,1,i,j1}=x_{4,0,i}$
$\displaystyle...$ $\displaystyle x_{n,1,i,j1}=x_{n,0,i}$

$\displaystyle x_{2,2,i,j1,j2}=x_{2,1,i,j1}+rcos(\theta_{j2})$ $\displaystyle
x_{4,2,i,j1,j2}=x_{4,1,i,j1}+rsin(\theta_{j2})$ $\displaystyle
x_{1,2,i,j1,j2}=x_{1,1,i,j1}$ $\displaystyle x_{3,2,i,j1,j2}=x_{3,1,i,j1}$
$\displaystyle...$ $\displaystyle x_{n,2,i,j1,j2}=x_{n,1,i,j1}$
$\displaystyle...$ (57)

$\displaystyle...$ $\displaystyle
x_{2,n-2,i,j1,j2,...,j(n-2)}=x_{2,n-3,i,j1,j2,...,j(n-3)}+rcos(\theta_{j(n-2)})$
$\displaystyle
x_{n,n-2,i,j1,j2...j(n-2)}=x_{n,n-3,i,j1,j2,...,j(n-3)}+rsin(\theta_{j(n-2)})$
$\displaystyle x_{1,n-2,i,j1,j2,...j(n-2)}=x_{1,n-3,i,j1,j2,...,j(n-3)}$
$\displaystyle...$ $\displaystyle
x_{n-1,n-2,i,j1,j2,...,j(n-2)}=x_{n-1,n-3,1,i,j1,j2,...,j(n-3)}$

$(x_{1,0,i},x_{2,0,i},...,x_{n,0,i}),i=1,...N$ are the points of the periodic
orbit in the $n$D configuration space $(x_{1},x_{2},...,x_{n})$. We have the
angle $\theta_{j1}=j1\frac{2\pi}{n_{1}}$ with $j1=1,...,n_{1}$ for the first
circle and $\theta_{j2}=j2\frac{2k\pi}{n_{1}}$ with $j2=1,...,n_{1}$ for the
second circle and so on the $\theta_{j(n-2)}=j(n-2)\frac{2\pi}{n_{1}}$ with
$j(n-2)=1,...,n_{1}$ for the $n-2$ circle that we need for the construction of
the torus.

$x_{1,1,i,j1},x_{2,1,i,j1},...,x_{n,1,i,j1}$ with $i=1,...,N$ and
$j1=1,...,n_{1}$ are the points of the torus that is constructed from the
Cartesian product of the projection of the periodic orbit in the 2D subspace
$(x_{1},x_{2})$ and a circle in the $n$D space $(x_{1},x_{2},...x_{n})$.
$x_{1,2,i,j1,j2},x_{2,2,i,j1,j2},...x_{n,2,i,j1,j2}$ with $i=1,...,N$,
$j1=1,...,n_{1}$ and $j2=1,...,n_{1}$ are the points of the torus that is
constructed from the Cartesian product of the projection of the periodic orbit
in the 2D subspace $(x_{1},x_{2})$, and other 2 circles in the $n$D space
$(x_{1},x_{2},...,x_{n})$. And so on
$x_{1,n-2,i,j1,j2,...j(n-2)},x_{2,n-2,i,j1,j2,...j(n-2)},...x_{n,n-2,i,j1,j2,..j(n-2)}$
with $i=1,...,N$ and $j1,j2,....j(n-2)=1,...,n_{1}$ are the points of the
torus that is constructed from the Cartesian product of the projection of the
periodic orbit in the 2D subspace $(x_{1},x_{2})$, and other $n-2$ circles in
the $n$D space $(x_{1},x_{2},...,x_{n})$.

  4. 4.

For each point
$x_{1,n-2,i,j1,j2,...j(n-2)},x_{2,n-2,i,j1,j2,...j(n-2)},...x_{n,n-2,i,j1,j2,..j(n-2)}$
with $i=1,...,N$ and $j1,j2,....j(n-2)=1,...,n_{1}$ on this torus we must
calculate the $p_{x_{1},n-2,i,j1,j2,,...j(n-2)}^{max}$ and
$p_{x_{1},n-2,i,j1,j2,,...j(n-2)}^{min}$ by solving the following equation for
a fixed value of energy (Hamiltonian) E with
$p_{x_{2}}=p_{x_{3}}=...=p_{x_{n}}=0$:

$\displaystyle
V(x_{1,n-2,i,j1,j2,...j(n-2)},x_{2,n-2,i,j1,j2,...j(n-2)},...x_{n,n-2,i,j1,j2,..j(n-2)})+$
$\displaystyle\frac{p_{x_{1},n-2,i,j1,j2,...j(n-2)}^{2}}{2m_{1}}=E$

and we find the maximum and minimum values
$p_{x_{1},n-2,i,j1,j2,...j(n-2)}^{max}$ and
$p_{x_{1},n-2,i,j1,j2,...j(n-2)}^{min}$. We choose points
$p_{x_{1},n-2,i,j1,j2,...j(n-2)}$ with $j1,j2...j(n-2)=1,...,n_{1}$ in the
interval $p_{x_{1},n-2,i,j1,j2,...j(n-2)}^{min}\leq
p_{x_{1},n-2,i,j1,j2,...j(n-2)}\leq p_{x_{1},n-2,i,j1,j2,...j(n-2)}^{max}$.
These points can be uniformly distributed in this interval. We will repeat the
same procedure to compute the values
$p_{x_{2},n-2,i,j1,j2,...j(n-2)},...,p_{x_{n-1},n-2,i,j1,j2,...j(n-2)}$. In
general for $n_{2}\leq n-1$ for each point
$x_{1,n-2,i,j1,j2,...j(n-2)},x_{2,n-2,i,j1,j2,...j(n-2)},...x_{n,n-2,i,j1,j2,..j(n-2)},p_{x_{1},n-2,i,j1,j2,...j(n-2)},...\\\
p_{x_{n_{2}-1},n-2,i,j1,j2,...j(n-2)}$ with $i=1,...,N$ and
$j1,j2,....j(n-2)=1,...,n_{1}$ we must calculate the
$p_{x_{n_{2}},n-2,i,j1,j2,,...j(n-2)}^{max}$ and
$p_{x_{n_{2}},n-2,i,j1,j2,,...j(n-2)}^{min}$ by solving the following equation
for a fixed value of energy (Hamiltonian) $E$ with
$p_{x_{n_{2}+1}}=...=p_{x_{n}}=0$:

$\displaystyle
V(x_{1,n-2,i,j1,j2,...j(n-2)},x_{2,n-2,i,j1,j2,...j(n-2)},...x_{n,n-2,i,j1,j2,..j(n-2)})+$
$\displaystyle\frac{p_{x_{1},n-2,i,j1,j2,...j(n-2)}^{2}}{2m_{1}}+...+\frac{p_{x_{n_{2}},n-2,i,j1,j2,...j(n-2)}^{2}}{2m_{n_{2}}}=E$

and we find the maximum and minimum values
$p_{x_{n_{2}},n-2,i,j1,j2,...j(n-2)}^{max}$ and
$p_{x_{n_{2}},n-2,i,j1,j2,...j(n-2)}^{min}$. We choose points
$p_{x_{n_{2}},n-2,i,j1,j2,...j(n-2)}$ with $j1,j2,j(n-2)=1,...,n_{1}$ in the
interval $p_{x_{n_{2}},n-2,i,j1,j2,...j(n-2)}^{min}\leq
p_{x_{n_{2}},n-2,i,j1,j2,...j(n-2)}\leq
p_{x_{n_{2}},n-2,i,j1,j2,...j(n-2)}^{max}$. These points can be uniformly
distributed in this interval.

Then we obtain the value $p_{x_{n},n-2,i,j1,j2,...j(n-2)}$ from the
Hamiltonian:

$\displaystyle
V(x_{1,n-2,i,j1,j2,...j(n-2)},x_{2,n-2,i,j1,j2,...j(n-2)},...x_{n,n-2,i,j1,j2,..j(n-2)})+$
$\displaystyle\frac{p_{x_{1},n-2,i,j1,j2,...j(n-2)}^{2}}{2m_{1}}+...+\frac{p_{x_{n},n-2,i,j1,j2,...j(n-2)}^{2}}{2m_{n}}=E$

Dimensionality and Topology: This algorithm constructs a torus as the product
of the closed curve that represents the projection of the periodic orbit (1D
object) in a 2D subspace of the configuration space with $n-2$ circles in the
$n$D configuration space . This torus is a $n-1$-dimensional torus. Then we
sample the $n-1$ variables ($n-1$ momenta) in the interval between their
maximum and minimum value. Actually we create $n-1$ additional segments and we
increase the dimensionality of the initial torus, from $n-1$ to $2n-2$
dimensions, which is embedded in the $2n-1$ energy surface. Then we obtain the
value of the last momentum from the Hamiltonian of the system.

##  6 Conclusions

We generalized the notion of the periodic orbit dividing surface construction
to Hamiltonian systems with $n$ degrees of freedom. This is very important
because until now this surface was computed only in Hamiltonian systems with
two degrees of freedom though the classical method of Pechukas [1981] and
Pollak [1985]. For Hamiltonian systems with three or more degrees of freedom,
the dividing surfaces could be computed using as a starting point the Normally
Hyperbolic Invariant Manifold (NHIM- see Wiggins [2016], Wiggins [1994]) and
not periodic orbits. In many systems this is very difficult and we need the
Normal form theory to compute this structure. In this method we use the
periodic orbit as a starting point and we construct a torus as a Cartesian
product of $n-1$ circle or $n-2$ circles and a 2D projection of the periodic
orbit avoiding the difficult computation of the NHIM. We compare the dividing
surfaces from the periodic orbits with the dividing surface from the NHIM in a
simple Normal Form quadratic Hamiltonian system with two and three degrees of
freedom. From all these we have the following remarks:

  1. 1.

Our algorithm for the construction of the periodic orbit dividing surfaces is
valid in the cases in which the periodic orbits are closed curves in a 2D
subspace of the configuration space or a 2D subspace of the extended
configuration space (a space that consists of the configuration space plus 1D
space that corresponds to one of the momenta).

  2. 2.

We have two versions of our algorithm for the construction of the dividing
surfaces from periodic orbits. The choice of one of these versions depends
from the fact if the periodic orbit is closed curve in a 2D subspace of the
configuration space or not.

  3. 3.

In Hamiltonian systems with two degrees of freedom the second version of our
algorithm for the construction of dividing surfaces coincides with the
classical method of Pechukas [1981] and Pollak [1985].

  4. 4.

In Hamiltonian systems with three degrees of freedom, the periodic orbit
dividing surfaces have the topology of a hyperbolic torus.

  5. 5.

The periodic orbit dividing surfaces that are constructed from our algorithm
are subsets of the dividing surfaces that are constructed from the NHIM.

  6. 6.

According to the position of the periodic orbits in the phase space the
associated dividing surfaces, that are constructed from our algorithm, give us
the trajectory behaviour for different regions of the configuration space. For
example in our case (a Normal form Hamiltonian system with three degrees of
freedom) the PO1 dividing surface give us more information for the
trajectories in the y-direction and PO2 dividing surface give us more
information in the z-direction.

  7. 7.

The radius of the circles that are needed for the construction of the periodic
orbit dividing surfaces, through our algorithm, is crucial for the efficiency
of the algorithm and the size of the dividing surfaces.

AcknowledgmentsWe acknowledge the support of EPSRC Grant No. EP/P021123/1 and
ONR Grant No. N00014-01-1-0769.

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