2D Generating Surfaces and Dividing Surfaces in Hamiltonian Systems with Three Degrees of Freedom

Abstract

In our previous work, we developed two methods for generalizing the construction of a periodic orbit dividing surface for a Hamiltonian system with three or more degrees of freedom. Starting with a periodic orbit, we extend it to form a torus or cylinder, which then becomes a higher-dimensional object within the energy surface (see [Katsanikas & Wiggins, 2021a, 2021b, 2023a, 2023b]). In this paper, we present two methods to construct dividing surfaces not from periodic orbits but by using 2D surfaces (2D geometrical objects) in a Hamiltonian system with three degrees of freedom. To illustrate the algorithm for this construction, we provide benchmark examples of three-degree-of-freedom Hamiltonian systems. Specifically, we employ the uncoupled and coupled cases of the quadratic normal form of a Hamiltonian system with three degrees of freedom.

Full text

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International Journal of Bifurcation and Chaos
©World Scientific Publishing Company
2D Generating surfaces and dividing surfaces in Hamiltonian
systems with three degrees of freedom
Matthaios Katsanikas
Research Center for Astronomy and Applied Mathematics, Academy of Athens, Soranou Efesiou 4,
Athens, GR-11527, Greece.
School of Mathematics, University of Bristol, Fry Building, Woodland Road, Bristol, BS8 1UG, United
Kingdom.
mkatsan@academyofathens.gr
Stephen Wiggins
School of Mathematics, University of Bristol, Fry Building, Woodland Road, Bristol, BS8 1UG, United
Kingdom.
Department of Mathematics, United States Naval Academy, Chauvenet Hall, 572C Holloway Road,
Annapolis, MD 21402-5002, USA.
s.wiggins@bristol.ac.uk
Received (to be inserted by publisher)
In previous work we developed two methods for generalizing the construction of a periodic orbit
dividing surface for a Hamiltonian system with three or more degrees of freedom. Starting with a
periodic orbit, we extend it to form a torus or cylinder, which then becomes a higher-dimensional
object within the energy surface (see [Katsanikas & Wiggins, 2021a,b, 2023a,b]). In this paper,
we present two methods to construct dividing surfaces not from periodic orbits but using 2D
surfaces (2D geometrical objects) in a Hamiltonian system with three degrees of freedom. To
illustrate the algorithm for this construction, we provide benchmark examples of three-degree-
of-freedom Hamiltonian systems. Specifically, we employ the uncoupled and coupled cases of the
quadratic normal form of a Hamiltonian system with three degrees of freedom.
Keywords: Chemical reaction dynamics; phase space; Hamiltonian system, periodic orbit; Di-
viding surface; normally hyperbolic invariant manifold; Dynamical Astronomy;
1. Introduction
Dividing surfaces play a crucial role in understanding the dynamics of Hamiltonian systems and have
applications in various fields, including chemistry and dynamical astronomy. The classical method for
constructing dividing surfaces, which relied on periodic orbits, was limited to Hamiltonian systems with
two degrees of freedom [Pechukas & McLafferty, 1973; Pechukas & Pollak, 1977; Pollak & Pechukas, 1978;
Pechukas, 1981; Pollak, 1985]).
For Hamiltonian systems with three or more degrees of freedom, the computation of dividing sur-
faces becomes more challenging. One approach involves using Normally Hyperbolic Invariant Manifolds
(NHIMs) in combination with normal form theory. However, the computation of NHIMs can be computa-
tionally intensive and difficult ([Wiggins et al., 2001; Uzer et al., 2002; Waalkens et al., 2007; Toda, 2003;
1

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2M. Katsanikas and S. Wiggins
Komatsuzaki & Berry, 2003]).
In our previous research, we proposed two methods (see [Katsanikas & Wiggins, 2021a,b, 2023a,b]) that
avoid the need to compute NHIMs and instead utilize periodic orbits. These methods offer an alternative
approach to identifying dividing surfaces in Hamiltonian systems. One of these methods was applied to
detect the phenomenon of dynamical matching in a 3D extension of the caldera potential, as discussed in
[Katsanikas & Wiggins, 2022; Wiggins & Katsanikas, 2023]. This phenomenon has been extensively studied
in 2D caldera-type Hamiltonian systems, as outlined in the references [Katsanikas & Wiggins, 2018, 2019;
Katsanikas et al., 2022b, 2020; Geng et al., 2021; Katsanikas et al., 2022a,c].These developments can provide
valuable insights into the the study of phase space transport in Hamiltonian systems with multiple degrees
of freedom and, in particular, shed light on the phenomenon of dynamical matching in different contexts.
In this paper, we construct the dividing surfaces in Hamiltonian systems with three degrees of freedom
using not 1D geometrical objects (periodic orbits) but 2D objects. As we explained above, in previous
approaches the dividing surfaces are constructed using periodic orbits or NHIMs. This means that the
starting object for the construction of a dividing surface is a closed, orientable, invariant co-dimension
2 submanifold of the energy level. The starting object in a Hamiltonian system with three degrees of
freedom is the NHIM (a 3D closed, orientable, invariant co-dimension 2 submanifold of the energy level).
In previous papers, we presented two methods to construct dividing surfaces in Hamiltonian systems with
three degrees of freedom using not 3D closed, orientable, and invariant objects but 1D closed, orientable,
and invariant objects (periodic orbits). In this sense, we extended the notion of the starting objects to
lower-dimensional objects. Now in this paper, we show that we can construct dividing surfaces (with the
no-recrossing property) using 2D closed and orientable objects (without knowing or requiring that they are
invariant). In this way, we extended the notion of the starting surfaces for constructing dividing surfaces to
lower-dimensional and non-invariant objects in Hamiltonian systems with three degrees of freedom. These
objects are 2D closed and orientable submanifolds that can be the starting objects of the construction of
dividing surfaces that have the no-recrossing property. We refer to these objects as 2D generating surfaces.
We describe the algorithms for the construction of the 2D generating surfaces in the next section (see
section 2). Secondly, we present the construction of the dividing surfaces using these objects (see section
3). This extension will be very useful in future studies for the detection of geometrical objects from which
we construct dividing surfaces, using methods of machine learning in Hamiltonian systems. This is because
it is easier to detect 2-dimensional surfaces than 1-dimensional invariant objects like periodic orbits. Then
we give an example of this construction for the case of the quadratic normal form Hamiltonian system
with three degrees of freedom (see section 4) and we describe their geometrical structure (see section 5).
Furthermore, in section 6, we present the construction of these objects in a case of the coupled quadratic
normal form Hamiltonian system with three degrees of freedom. Finally, we present our conclusions in the
last section (section 7).
2. Methods for the construction of 2D generating surfaces
As we referred to in the introduction, we will construct 2D closed and orientable geometrical objects (that
we named 2D generating surfaces) that are the basis of the construction of dividing surfaces. In this section
we describe two methods (subsections 2.1 and 2.2 respectively) of the construction of the 2D generating
surfaces in Hamiltonian systems with three degrees of freedom. We consider the general case of Hamiltonian
systems with 3 degrees of freedom with potential energy function V(x1, x2, x3) of the form :
T+V(x1, x2, x3) = E
(1)
where Tis the kinetic energy.
T=p2
x1/2m1+p2
x2/2m2+p2
x3/2m3
(2)
where px1, px2, px3are the momenta and m1, m2, m3are the corresponding masses.

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2D Generating surfaces and Dividing surfaces in Hamiltonian systems with three degrees of freedom 3
2.1. The algorithm of the first method for the construction of 2D generating
surface
In this section, we will present the first method for the construction of 2D generating surfaces. The con-
struction of these surfaces is based on the periodic orbits. The first method has one necessary condition in
order to be applied that is that one at least 2D projection of the periodic orbit, in the phase space, to be
a closed curve. This algorithm has two versions. The first corresponds to the case in which the projection
of the periodic orbit is not a closed curve in the configuration space and the second version to the case in
which this projection is a closed curve in the configuration space.
The first version has two steps:
(1) Locate an unstable periodic orbit PO for a fixed value of Energy E. We check if there is a 2D projection
of the periodic orbit, in the phase space (but not in the configuration space), to be a closed curve (for
example in the (y, py) space).
(2) From the projection of the periodic orbit in the phase space, 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 phase space (for example in the (y, py) space). Actually it is topologically equivalent
with the Cartesian product of two circles S1×S1. 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
(x, y) around every point of the periodic orbit in the 4D subspace (x, y , py). The goal of this step is to
include two coordinates of the configuration space in this torus.
y1,i,j1=y0,i +rcos(θj1)
x1,i,j1=x0,i +rsin(θj1)
py,1,i,j1=py ,0,i (3)
(x0,i, y0,i , py,0,i ), i = 1, ...N are the points of the periodic orbit in the 3D subspace (x, y, py). We have
the angle θj1=j12π
n1with j1=1, ..., n1for the circle that we need for the construction of the torus.
x1,i,j1, y1,i,j 1, py,1,i,j1with i= 1, ..., N and j1 = 1, ..., n1are the points of the torus that is constructed
from the Cartesian product of projection of the periodic orbit in the 2D subspace (y, py) and a circle
in the (x, y) space in the 3D space (x, y, py).
The second version is:
(1) Locate an unstable periodic orbit PO for a fixed value of Energy E. We check if there is a 2D projection
of the periodic orbit, in the configuration space, to be a closed curve (for example in the (x, y) space).
(2) 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 S1×S1. 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:
y1,i,j1=y0,i +rcos(θj1)
z1,i,j1=z0,i +rsin(θj1)
x1,i,j1=x0,i (4)
(x0,i, y0,i , z0,i), i = 1, ...N are the points of the periodic orbit in the 3D configuration space (x, y, z).
We have the angle θj1=j12π
n1with j1 = 1, ..., n1for the circle that we need for the construction of the
torus.

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4M. Katsanikas and S. Wiggins
x1,i,j1, y1,i,j 1, z1,i,j1with i= 1, ..., N and j1=1, ..., n1are 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).
In both of versions, the resulting object is a 2D torus in a 3D subspace of the phase space.
2.2. The algorithm of the second method for the construction 2D generating
surfaces
In this section, we will present the second method for the construction of 2D generating surfaces. The
construction of these surfaces is based on the periodic orbits. The second method has not the condition of
the first method. This means that we don’t need to look for a 2D projection of the periodic orbit to be a
closed curve.
The algorithm has two steps:
(1) We start with a fixed value of energy E and find an unstable periodic orbit in the phase space of the
dynamical system. This could be done using numerical techniques like continuation methods, shooting
methods, or other approaches that search for closed orbits. Once we have the periodic orbit in phase
space, we project it into the configuration space. This means we consider the coordinates (positions)
of the orbit without their conjugate momenta. For example,we consider (x, y).
(2) We construct a torus or a cylinder that is generated by the Cartesian product of one circle with small
radius rand the projection of the periodic orbit. Actually it is the Cartesian product of two circles
S1×S1or a line (or curve) with one circle R×S1. This is a two-dimensional torus or cylinder. This
can be achieved through the construction of one circle around every point of the periodic orbit in the
2D subspace of the configuration 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 2D subspace of the 3D space (x, y, z).
The points of the torus or cylinder that we constructed are:
y1,i,j1=y0,i +rcos(θj1)
z1,i,j1=z0,i +rsin(θj1)
x1,i,j1=x0,i (5)
(x0,i, y0,i , z0,i), i = 1, ...N are the points of the periodic orbit in the 3D configuration space (x, y, z).
We have the angle θj1=j12π
n1with j1 = 1, ..., n1for the circle that we need for the construction of the
2D generating surface.
x1,i,j1, y1,i,j 1, z1,i,j1with i= 1, ..., N and j1 = 1, ..., n1are the points of the torus or cylinder 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 of the 3D space (x, y, z).
3. The algorithms for the construction of the dividing surfaces
In this section, we will present the algorithms for the construction of dividing surfaces from 2D generating
surfaces. In the first subsection 3.1, we will describe the algorithms for the construction of the dividing
surfaces that are constructed through the first method of construction of 2D generating surfaces (see the
subsection 2.1). In the second subsection 3.2, we will describe the algorithms for the construction of the
dividing surfaces that are constructed through the second method of construction of 2D generating surfaces
(see the subsection 2.2).
3.1. The algorithm of the first method for the construction of the dividing
surfaces
In the previous section (see the subsection 2.1) we described the two versions of the first method of the
construction of 2D generating surfaces in Hamiltonian systems with three degrees of freedom. In this

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2D Generating surfaces and Dividing surfaces in Hamiltonian systems with three degrees of freedom 5
subsection, we will present the corresponding two versions for the construction of the associated dividing
surfaces.
The algorithm of the first version is:
(1) Using as a base the 2D generating surface (the 2D torus), that was constructed through the first version
of the method that was presented in the subsection 2.1, we construct the cartesian product of the 2D
torus with a circle in the 4D space that contains the coordinates of the configuration space plus one
momentum (for example (x, y, z, py)). Actually, it is topologically equivalent to the Cartesian product
of three circles S1×S1×S1. This is a three-dimensional torus (Hypertorus). This can be done by
constructing a circle around every point of the 2D generating surface (for example in the plane (y, z)
of the 4D subspace (x, y, z, py) of the phase space).
y2,i,j1,j 2=y1,i,j1+rcos(θj2)
z2,i,j1,j 2=z1,i,j1+rsin(θj2)
x2,i,j1,j 2=x1,i,j1
py,2,i,j1,j 2=py,1,i,j 1(6)
θj2=j22π
n1with j2=1, ..., n1
1for the circle that we need for the construction.
x2,i,j1,j 2, y2,i,j1,j 2, z2,i,j1,j 2, py,2,i,j 1,j2with i= 1, ..., N ,j1 = 1, ..., n1and j2 = 1, ..., n1are the points
of the torus that is constructed from the Cartesian product of the projection of the periodic orbit in
the 2D subspace (y, py), a circle in the (x, y) space and a circle in the (y, z) space in the 4D space
(x, y, z, py).
(2) For each point x2,i,j1,j2, y2,i,j 1,j2, z2,i,j 1,j2, py,2,i,j1,j 2on this torus we must calculate the pmax
x,2,i,j1,j 2and
pmin
x,2,i,j1,j 2by solving the following equation for a fixed value of energy (Hamiltonian) Ewith pz= 0:
V(x2,i,j1,j 2, y2,i,j1,j 2, z2,i,j1,j 2) + p2
x,2,i,j1,j 2
2m1
+p2
y,2,i,j1,j 2
2m2
=E
(7)
and we find the maximum and minimum values pmax
x,2,i,j1,j 2and pmin
x,2,i,j1,j 2. We choose points px,2,i,j1,j 2
with j2 = 1, ..., n1in the interval pmin
x,2,i,j1,j 2≤px,2,i,j1,j 2≤pmax
x,2,i,j1,j 2. These points can be uniformly
distributed in this interval. Then we obtain the value pz,2,i,j1,j2from the Hamiltonian:
V(x2,i,j1,j 2, y2,i,j1,j 2, z2,i,j1,j 2) + p2
x,2,i,j1,j 2
2m1
+p2
y,2,i,j1,j 2
2m2
+p2
z,2,i,j1,j2
2m3
=E
(8)
This algorithm constructs exactly the same object as the algorithm of subsection 4.1 of [Katsanikas &
Wiggins, 2021a], a 4D torus in the 5D energy surface. The difference is that now we did not have as a base
a periodic orbit but a 2D surface (the 2D generating surface).
The algorithm of the second version is:
(1) We use as a base the 2D generating surface (the 2D torus), that was constructed through the second
version of the method that was presented in the subsection 2.1. For each point x1,i,j1, y1,i,j 1, z1,i,j1on
this torus we must calculate the pmax
x,1,i,j1and pmin
x,1,i,j1by solving the following equation for a fixed value
of energy (Hamiltonian) Ewith py=pz= 0:
1In algorithms of the papers [Katsanikas & Wiggins, 2021a,b], we have θj2=j22kπ
n1with j2=1, ..., n1, but actually we
consider k= 1.

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6M. Katsanikas and S. Wiggins
V(x1,i,j1, y1,i,j 1, z1,i,j1) + p2
x,1,i,j1
2m1
=E
(9)
and we find the maximum and minimum values pmax
x,1,i,j1and pmin
x,1,i,j1. We choose points px,1,i,j1with
j1=1, ..., n1in the interval pmin
x,1,i,j1≤px,1,i,j 1≤pmax
x,1,i,j1. These points can be uniformly distributed in
this interval.
(2) Now for every point x1,i,j1, y1,i,j1, z1,i,j1, px,1,i,j1we must calculate the pmax
y,1,i,j1and pmin
y,1,i,j1by solving
the following equation for a fixed value of energy (Hamiltonian) Ewith pz= 0:
V(x1,i,j1, y1,i,j 1, z1,i,j1) + p2
x,1,i,j1
2m1
+p2
y,1,i,j1
2m2
=E
(10)
We choose points py,1,i,j1with j1 = 1, ..., n1in the interval pmin
y,1,i,j1≤py ,1,i,j1≤pmax
y,1,i,j1. These points
can be uniformly distributed in this interval. Then we obtain the value pz,1,i,j1from the Hamiltonian:
V(x1,i,j1, y1,i,j 1, z1,i,j1) + p2
x,1,i,j1
2m1
+p2
y,1,i,j1
2m2
+p2
z,1,i,j1
2m3
=E
(11)
This algorithm constructs exactly the same object as the algorithm of subsection 4.2 of [Katsanikas &
Wiggins, 2021a], a 4D torus in the 5D energy surface. The difference is that now we did not have as a base
a periodic orbit but a 2D surface (the 2D generating surface).
3.2. The algorithm of the second method for the construction of the dividing
surfaces
In this subsection, we will present the algorithm of the construction of dividing surfaces using the 2D
generating surfaces that are constructed in the subsection 2.2. The algorithm is the following:
(1) We use as a base the 2D generating surface (a 2D torus or a 2D cylindrical surface), that was constructed
through the method that was presented in the subsection 2.2. For each point x1,i,j1, y1,i,j1, z1,i,j1on
this torus or cylinder we must calculate the pmax
x,1,i,j1and pmin
x,1,i,j1by solving the following equation for
a fixed value of energy Ewith py=pz= 0:
V(x1,i,j1, y1,i,j 1, z1,i,j1) + p2
x,1,i,j1
2m1
=E
(12)
and we find the maximum and minimum values pmax
x,1,i,j1and pmin
x,1,i,j1. We choose points px,1,i,j1with
j1=1, ..., n1in the interval pmin
x,1,i,j1≤px,1,i,j 1≤pmax
x,1,i,j1. These points can be uniformly distributed in
this interval.
(2) Now for every point x1,i,j1, y1,i,j1, z1,i,j1, px,1,i,j1we must calculate the pmax
y,1,i,j1and pmin
y,1,i,j1by solving
the following equation for a fixed value of energy Ewith pz= 0:
V(x1,i,j1, y1,i,j 1, z1,i,j1) + p2
x,1,i,j1
2m1
+p2
y,1,i,j1
2m2
=E
(13)
We choose points py,1,i,j1with j1 = 1, ..., n1in the interval pmin
y,1,i,j1≤py ,1,i,j1≤pmax
y,1,i,j1. These points
can be uniformly distributed in this interval. Then we obtain the value pz,1,i,j1from the Hamiltonian:

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2D Generating surfaces and Dividing surfaces in Hamiltonian systems with three degrees of freedom 7
V(x1,i,j1, y1,i,j 1, z1,i,j1) + p2
x,1,i,j1
2m1
+p2
y,1,i,j1
2m2
+p2
z,1,i,j1
2m3
=E
(14)
This algorithm constructs exactly the same object as the algorithm of section 3 of [Katsanikas &
Wiggins, 2021b], a 4D torus or 4D cylinder in the 5D energy surface. The difference is that now we did
not have as a base a periodic orbit but a 2D surface (the 2D generating surface).
4. Application of the algorithms in the Quadratic Normal Form Hamiltonian
System with three degrees of freedom
In this section, we apply the first and the second methods of the construction of 2D generating surfaces in
the quadratic normal form Hamiltonian system. As we described in the previous sections, these generating
surfaces make the same dividing surfaces as the periodic orbits do. The no-recrossing property of the
periodic orbit dividing surfaces for the quadratic normal form Hamiltonian systems has been proved in a
series of previous papers (see [Katsanikas & Wiggins, 2021a,b, 2023a,b]).
In the first subsection, we present the properties of the Hamiltonian system (see subsection 4.1). Then
we construct the 2D generating surfaces through the first (see subsection 4.2) and the second method (see
subsection 4.3) of the construction of these surfaces.
4.1. Basic properties of the Quadratic Normal Form Hamiltonian System with
Three Degrees of Freedom
The quadratic normal form Hamiltonian system [Wiggins, 2016] is described by the following Hamiltonian
(with λ > 0, ω2>0, ω3>0):
H=λ
2(p2
x−x2) + ω2
2(p2
y+y2) + ω3
2(p2
z+z2)
(15)
The three subsystems of our model are described by the Hamiltonians H1,H2and H3:
H1=λ
2(p2
x−x2),
H2=ω2
2(p2
y+y2),
H3=ω3
2(p2
z+z2).
(16)
The following equations are the equations of motion of the system:

February 14, 2024 14:56 output
8M. Katsanikas and S. Wiggins
˙x=∂H
∂px
=λpx,(17)
˙px=−∂H
∂x =λx, (18)
˙y=∂H
∂py
=ω2py,(19)
˙py=−∂H
∂y =−ω2y, (20)
˙z=∂H
∂pz
=ω3pz,(21)
˙pz=−∂H
∂z =−ω3z.
(22)
This system has an index-1 saddle (x, px, y, py, z, pz) = (0,0,0,0,0,0)) for a value of energy (numerical
value of the Hamiltonian) E= 0. A chemical reaction occurs when a trajectory cross the plane x= 0 (see
[Ezra & Wiggins, 2018]).
The NHIM is described by the following equation ([Ezra & Wiggins, 2018]):
ω2
2(p2
y+y2) + ω3
2(p2
z+z2) = E, N HIM
(23)
The dividing surface (a four-sphere) from the NHIM is (see [Ezra & Wiggins, 2018]):
λ
2p2
x+ω2
2(p2
y+y2) + ω3
2(p2
z+z2) = E.
(24)
The periodic orbits PO1 and P02 associated with the index-1 saddle are described by the following
analytical formulas ([Katsanikas & Wiggins, 2021a,b]):
ω2
2(p2
y+y2) = E, P O1
(25)
ω3
2(p2
z+z2) = E. P O2
(26)
The periodic orbits PO1 and PO2 are circles in the planes (y, py) and (z, pz) respectively.
We constructed the 2D generating surfaces using these periodic orbits.
4.2. 2D Generating surfaces that are constructed through the first method
In this subsection, we will construct the 2D generating surfaces of PO1 and PO2 using the first method
that was described in the subsection 2.1.
If we apply the algorithm of the subsection 2.1 in PO1, we have:

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2D Generating surfaces and Dividing surfaces in Hamiltonian systems with three degrees of freedom 9
(1) The PO1 is described by the analytical formula (25) . We will use the first version of the algorithm
that was presented in the subsection 2.1 because the periodic orbit is projected as a closed curve (a
circle) in the space (y, py).
(2) We will construct the product of the projection of the periodic orbit in the (y , py) space with one circle
(the radius of this circles r > 0 is fixed). This circle is in the subspace (y, py, x). The equation of this
object (a torus) is given by the following analytical formula:
(qp2
y+y2−r2E
ω2
)2+x2=r2
(27)
If we apply the algorithm of the subsection 2.1 in PO2, we have:
(1) The PO2 is described by the analytical formula (26) . We will use the first version of the algorithm
that was presented in the subsection 2.1 because the periodic orbit is projected as a closed curve (a
circle) in the space (z, pz).
(2) We will construct the product of the projection of the periodic orbit in the (z , pz) space with one circle
(the radius of this circle r1>0 is fixed). This circle is in the subspace (z, pz, x). The equation of this
object (a torus) is given by the following analytical formula:
(pp2
z+z2−r2E
ω3
)2+x2=r2
1
(28)
These 2D generating surfaces produce the same dividing surfaces as the dividing surfaces that are
produced from the periodic orbits (see the previous section). The no-recrossing property of these surfaces
was proved in [Katsanikas & Wiggins, 2023a] and the conditions for the radius rand the radius r1are the
equations (18) and (25) of [Katsanikas & Wiggins, 2023a]. In this section, we presented the construction
of a 2D generating surface using the first method in the case of the quadratic normal form Hamiltonian
system. In the next section, we will present the structure of the generating surfaces for specific values of
the parameters of the system.
4.3. 2D Generating surfaces that are constructed through the second method
In this subsection, we will construct the 2D generating surfaces of PO1 and PO2 using the second method
that was described in the subsection 2.2.
If we apply the algorithm of the subsection 2.2 in PO1, we have:
(1) The PO1 is described by the analytical formula (25) . We will use the second method that was presented
in the subsection 2.2 and we consider the projection of the periodic orbit PO1 in the (x, y) subspace
of the configuration space. This projection is a line (x= 0) because the periodic orbit lies only in the
(y, py) subspace of the phase space. The ycoordinate takes values from −p2E/ω2to p2E/ω2(see
equation (25)).
(2) In this step, we will construct the product of the projection of the periodic orbit with a circle having
radius rin the 2D subspace (y, z).
y2+z2=r2, x = 0
(29)
If we apply the algorithm of the subsection 2.2 in PO2, we have:

February 14, 2024 14:56 output
10 M. Katsanikas and S. Wiggins
(1) The PO2 is described by the analytical formula (26) . We will use the second method that was presented
in the subsection 2.2 we consider the projection of the periodic orbit PO2 into the (x, z) subspace of
the configuration space. This projection is a line (x= 0) since the periodic orbit lies only in the
(z, pz) subspace of the phase space. The zcoordinate takes values from −p2E/ω3to p2E/ω3(see
the equation (26)).
(2) In this step, we will construct the product of the projection of the periodic orbit with a circle with a
radius r1into the 2D subspace (y, z).
y2+z2=r2
1, x = 0
(30)
In these cases, the algorithms produce an area in the plane (y, z ) (because x= 0). This area is the
result of the section of a cylinder with the plane (y, z) and this is the reason that we call this area (2D
surface) as a cylindrical surface.
These 2D generating surfaces produce the same dividing surfaces as the dividing surfaces that are
produced from the periodic orbits (see the previous section). The no-recrossing property of these surfaces
was proved in [Katsanikas & Wiggins, 2023b] and the conditions for the radius rand the radius r1are the
equations (18) and (25) of [Katsanikas & Wiggins, 2023b]. In this section, we presented the construction
of a 2D generating surface using the second method in the case of the quadratic normal form Hamiltonian
system. In the next section, we will present the structure of the generating surfaces for specific values of
the parameters of the system.
5. The structure of 2D generating surfaces
In this section, we computed the 2D generating surfaces that are based on the periodic orbits PO1 and
PO2 of the quadratic normal form Hamiltonian system with three degrees of freedom using the first and
second method of construction (see subsections 4.2 and 4.3). In particular, we used the same parameters
(E= 14, H1= 4, λ = 1, ω2=√2 and ω3= 1) as in our previous papers [Katsanikas & Wiggins, 2021a,b].
We begin by computing the 2D generating surfaces from the periodic orbits PO1 and PO2 using the
first and the second methods of construction. Firstly, we computed the 2D generating surfaces using as a
base the periodic orbit PO1 for different values of the radius that was used for our construction (see the
subsections 4.2 and 4.3). These values are Rmax/20, Rmax/5, Rmax/2, Rmax where Rmax is the upper
limit of the radius for the construction of the 2D generating surface (that is given by the equation (18)
of [Katsanikas & Wiggins, 2023a] for the first method of construction or the equation (18) of [Katsanikas
& Wiggins, 2023b] for the second method of construction - see the subsections 4.2 and 4.3). Then we did
the same computation for the 2D generating surfaces of PO2 using again the same ratios of maximum
radius Rmax1 (that is given by the equation (25) of [Katsanikas & Wiggins, 2023a] for the first method of
construction or the equation (25) of [Katsanikas & Wiggins, 2023b] for the second method of construction
- see the subsections 4.2 and 4.3). The 2D generating surfaces that are computed using the first method,
they are embedded in the three-dimensional subspaces of the phase space: (x, y, py) for the case of PO1
and (x, z, pz) for the case of PO2. The 2D generating surfaces that are computed, using the second method,
they are embedded in the three-dimensional subspace of the phase space (x, y, z).
The 2D generating surfaces of PO1 and PO2, which are constructed through the first method, are
tori in the 3D subspaces of the phase space (see these surfaces for different values of Rmax and Rmax1
in Figs. 1 and 2). The only difference between the generating surfaces of PO1 and the generating surfaces
of PO2 is that they are embedded in a different 3D subspace of the phase space (see Figs. 1 and 2 and
the construction of these surfaces in subsections 4.2 ans 4.3). We observe in Fig. 1 that as we increase the
radius of the construction of the 2D generating surfaces of PO1 (using the first method), the generating
surfaces are extended more in the y−and x−directions. In addition, we observe in Fig. 2 that as we
increase the radius of the construction of the 2D generating surfaces of PO2 (using the first method), the
generating surfaces are extended more in the z−and x−directions.
The 2D generating surfaces of PO1 and PO2, that are constructed through the second method, are

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2D Generating surfaces and Dividing surfaces in Hamiltonian systems with three degrees of freedom 11
cylindrical surfaces in the (x, y, z) subspace of the phase space. Actually, they have x= 0 because of their
construction (using the second method of construction - see the subsection 4.3). This means that we need
only a 2D projection (y, z) to describe the structure of these generating surfaces of PO1 and PO2 that are
constructed using the second method (see Fig. 3 for the case of PO1 and Fig. 4 for the case of PO2). In
Figs. 3 and 4, we see that the generating surfaces are presented as cylindrical surfaces in the space (y, z).
We observe also that the generating surfaces of PO1 are more elongated in the y−axis and the generating
surfaces of PO2 are more elongated in the z−axis. In addition, we observe that increasing the radius of
the construction of the 2D generating surfaces of PO1 and PO2, the generating surfaces are extended more
and more in the y−and z−directions.
Fig. 1. The 3D projection (y, py, x) of the 2D generating surfaces that are constructed from the PO1, using the first method
of construction, for radius Rmax/20 (upper left panel), Rmax/5(upper right panel), Rmax/2 (lower left panel) and Rmax
(lower right panel).
6. 2D generating surfaces in a coupled case of the quadratic normal form
Hamiltonian system
In this section, we computed the 2D generating surfaces of the coupled quadratic normal form Hamiltonian
systems with three degrees of freedom (this system was used in our previous papers [Katsanikas & Wiggins,
2023a,b]). We compare our results with those of the previous section for the uncoupled case of this system.
We recall the Hamiltonian of the system (with λ > 0, ω2>0, ω3>0):
H=λ
2(p2
x−x2) + ω2
2(p2
y+y2) + ω3
2(p2
z+z2) + cyz2
(31)
The equations of motion are:

February 14, 2024 14:56 output
12 M. Katsanikas and S. Wiggins
Fig. 2. The 3D projection (z, pz, x) of the 2D generating surfaces that are constructed from the PO2, using the first method
of construction, for radius Rmax1/20 (upper left panel), Rmax1/5(upper right panel), Rmax1/2 (lower left panel) and Rmax1
(lower right panel).
˙x=∂H
∂px
=λpx,(32)
˙px=−∂H
∂x =λx, (33)
˙y=∂H
∂py
=ω2py,(34)
˙py=−∂H
∂y =−ω2y−cz2,(35)
˙z=∂H
∂pz
=ω3pz,(36)
˙pz=−∂H
∂z =−ω3z−2cyz.
(37)
There is an index-1 saddle (x, px, y, py, z, pz) = (0,0,0,0,0,0) for energy (numerical value of the Hamil-
tonian) E= 0. We consider that the reaction occurs when xchanges sign [Ezra & Wiggins, 2018] (x= 0).
We applied the algorithms of the subsections 4.2 and 4.3 to construct the 2D generating surfaces
associated with the periodic orbits PO1 and PO2 of the coupled system. This has been done for E=
14, H1= 4, λ = 1, ω2=√2 and ω3= 1 (as in the previous section) and for c= 0.1 (see [Katsanikas &
Wiggins, 2023a,b]). The initial conditions of PO1 are (x, px, z, pz) = (0,0,0,0) in the Poincare section y= 0

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2D Generating surfaces and Dividing surfaces in Hamiltonian systems with three degrees of freedom 13
Fig. 3. The 2D projection (y, z ) of the 2D generating surfaces that are constructed from the PO1, using the second method
of construction, for radius Rmax/20 (upper left panel), Rmax/5(upper right panel), Rmax/2 (lower left panel) and Rmax
(lower right panel).
with py>0 (this can be obtained from the Hamiltonian - see the equation 31) and the initial conditions of
PO2 are (x, px, y, py) = (0,0,0,0) in the Poincare section z= 0 with pz>0 (this can be obtained from the
Hamiltonian - see the equation 31). Then we integrate the initial condition of these periodic orbits for a
period (this means that the computation of these two periodic orbits has been done numerically). Then we
followed the steps of the algorithms of the subsections 4.2 and 4.3 and we constructed the corresponding 2D
generating surfaces from PO1 and PO2. These surfaces have a similar morphology to this of the surfaces
of the previous section (see for example Fig. 5 and compare it with this of Fig. 1). This means that the 2D
generating surfaces of the coupled case of our system have a similar structure to those of the uncoupled
case.
7. Conclusions
In this paper, we constructed 2D geometrical objects (2D generating surfaces) that produce dividing
surfaces in Hamiltonian systems with three degrees of freedom. For this reason, we proposed two methods
of construction of these objects. These geometrical objects are constructed using as a starting point a
periodic orbit. This is the first time that we have constructed 2D objects that can produce dividing
surfaces. Then we applied these methods to the coupled and uncoupled cases of the quadratic normal
form Hamiltonian system using as a basis the periodic orbits PO1 and PO2 (see the section 4). The main
conclusions for this construction are:
(1) There are two methods of construction of 2D generating surfaces. Both of them use as a starting point
a periodic orbit. The first method has as a condition the periodic orbit to be presented as a closed
curve in a 2D projection of the phase space. The second method does not have this restriction.
(2) One method of construction of 2D generating surfaces produces toroidal surfaces and the other method
produces toroidal or cylindrical surfaces.

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14 M. Katsanikas and S. Wiggins
Fig. 4. The 2D projection (y, z ) of the 2D generating surfaces that are constructed from the PO2, using the second method
of construction, for radius Rmax1/20 (upper left panel), Rmax1/5(upper right panel), Rmax1/2 (lower left panel) and Rmax1
(lower right panel).
Fig. 5. The 3D projection (y, py, x) of the 2D generating surface that is constructed from the PO1 in the coupled case of our
Hamiltonian system, using the first method of construction, for radius Rmax.
(3) The 2D generating surfaces produce the same dividing surfaces as the periodic orbit dividing surfaces
and this implies that they have the no-recrossing property.
(4) The 2D generating surfaces of PO1 and PO2 using the first method are embedded in a different

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2D Generating surfaces and Dividing surfaces in Hamiltonian systems with three degrees of freedom 15
subspace of the phase space.
(5) The 2D generating surfaces of PO1 and PO2 using the second method are elongated in different
directions.
(6) The 2D generating surfaces are extended more and more as we increase the radius of the construction.
(7) The 2D generating surfaces of the coupled case of our system have a similar structure to those of the
uncoupled case of our system.
Acknowledgments
We acknowledge the support of EPSRC Grant No. EP/P021123/1. SW acknowledges the support of
the William R. Davis ’68 Chair in the Department of Mathematics at the United States Naval Academy.

February 14, 2024 14:56 output
16 REFERENCES
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