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June 1, 2021 13:0 katsanikas˙wiggins
International Journal of Bifurcation and Chaos
c
World Scientific Publishing Company
The Generalization of the Periodic Orbit Dividing Surface for
Hamiltonian Systems with Three or more Degrees of Freedom -II
MATTHAIOS KATSANIKAS AND STEPHEN WIGGINS
School of Mathematics, University of Bristol,
Fry Building, Woodland Road, Bristol, BS8 1UG, United Kingdom.
matthaios.katsanikas@bristol.ac.uk, s.wiggins@bristol.ac.uk
Received (to be inserted by publisher)
We develop a method for the construction of a dividing surfaces using periodic orbits in Hamil-
tonian systems with three or more degrees-of-freedom that is an alternative to the method
presented in [Katsanikas & Wiggins, 2021]. Similar to that method, for an ndegrees-of-freedom
Hamiltonian system we extend a 1-dimensional object (the periodic orbit) to a 2n−2 dimen-
sional geometrical object in the energy surface of a 2n−1 dimensional space that has the desired
properties for a dividing surface. The advantage of this new method is that it avoids the compu-
tation of the normally hyperbolic invariant manifold (NHIM) (as the first method did) and it is
easier to numerically implement than the first method of the constructing periodic orbit dividing
surfaces. Moreover, this method has less strict required conditions than the first method for con-
structing periodic orbit dividing surfaces. We apply the new method to a benchmark example
of a Hamiltonian system with three degrees of freedom for which we are able to investigate the
structure of the dividing surface in detail. We also compare the periodic orbit dividing surfaces
constructed in this way with the dividing surfaces that are constructed starting with a NHIM.
We show that these periodic orbit dividing surfaces are subsets of the dividing surfaces that are
constructed from the NHIM.
Keywords: Chemical reaction dynamics; phase space; Hamiltonian system, periodic orbits; Di-
viding surfaces; normally hyperbolic invariant manifold; Dynamical Astronomy;
1. Introduction
This paper is a consequence of a previous paper ( we will refer to that paper as paper I- [Katsanikas &
Wiggins, 2021]) concerning the construction of dividing surfaces in Hamiltonian systems with three or more
degrees of freedom using as a starting point a periodic orbit.
Dividing surfaces are surfaces of one less dimension than that of the energy surface of a Hamiltonian
system and they are an essential ingredient for the transition state theory [Wigner, 1938; Waalkens et al.,
2007] in chemical reaction dynamics. The computation of these structures can be done through the classical
method of [Pechukas & McLafferty, 1973; Pechukas & Pollak, 1977; Pollak & Pechukas, 1978; Pechukas,
1981; Pollak, 1985], using periodic orbits, in Hamiltonian systems with two degrees of freedom. In Hamil-
tonian systems with three or more degrees of freedom, this method cannot be used since the periodic orbit
(1-dimensional object) does not have enough dimensions to guarantee the construction of a dividing sur-
face. This construction can be done using a higher dimensional object, the Normally Hyperbolic Invariant
Manifold -NHIM ([Wiggins, 1994; Wiggins et al., 2001; Uzer et al., 2002; Wiggins, 2016]). The dimension
1
June 1, 2021 13:0 katsanikas˙wiggins
2M. Katsanikas and S. Wiggins
of this object can guarantee the construction of an dividing surface [Waalkens et al., 2007; Waalkens &
Wiggins, 2010]. The computation of this structure can be carried out using normal form theory ([Wiggins
et al., 2001], [Uzer et al., 2002], [Waalkens et al., 2007],[Toda, 2003],[Komatsuzaki & Berry, 2003]).
The computation of the NHIM is very difficult in many cases and it requires extensive computations.
There is the need for a method of constructing dividing surfaces using as starting point a lower dimension
object, like a periodic orbit, that is more computationally efficient. For this reason in paper I we generalized
the classical method of periodic orbit dividing surfaces of [Pechukas, 1981; Pollak, 1985] in Hamiltonian
systems with three and more degrees of freedom. This was done for the first time using as a starting point
a periodic orbit. The method in paper I has one necessary condition in order to be applied. This condition
is that the projection of the periodic orbit into a 2Dsubspace of the phase space is a closed curve. In this
paper, we present a method that is based on the method that is constructed in paper I but it does not
require this condition. This means that this method can be applied in all cases of Hamiltonian systems
with ndegrees of freedom. The basic concept of this method is that starting from a projection of a periodic
orbit we construct an object from the Cartesian product of the projection of the periodic orbit with n−2
circles in the configuration space. Then we create n−1 additional segments using n−1 momenta and we
obtain the last momentum from the Hamiltonian. The pseudocode in Fig.1 describes this construction in
Hamiltonian systems with ndegrees of freedom.
In this paper, we give an introduction (section 1) to the new method of constructing periodic orbit
dividing surfaces in Hamiltonian systems with three and more degrees of freedom. Then we give the
description of the algorithm for Hamiltonian systems with nand three degrees of freedom (see sections
2 and 3). In section 4 we apply our algorithm to a Hamiltonian system with three degrees of freedom
(quadratic normal form Hamiltonian system with three degrees of freedom) in which we have analytical
expressions for the NHIM and periodic orbits [Ezra & Wiggins, 2018] in order to compare the periodic
orbit dividing surface that is constructed by our method with the dividing surface that are constructed
using the NHIM. In section 5 we show that the algorithm becomes exactly the same with the classical
method of [Pechukas, 1981; Pollak, 1985] in Hamiltonian systems with two degrees of freedom. In the final
section, we present our conclusions.
2. The Algorithm for Hamiltonian systems with ndegrees of freedom
In this section we present our algorithm for the general case of Hamiltonian systems with n degrees of
freedom having a potential energy function V(x1, x2, ..., xn) with n≥2 of the form:
T+V(x1, x2, ..., xn) = E
(1)
where Tis the kinetic energy and Eis the numerical value of the Hamiltonian (we will refer to it as
”energy”).
T=p2
x1/2m1+p2
x2/2m2+... +p2
xn/2mn
(2)
This algorithm requires only the projection of a periodic orbit in a 2D subspace of the configuration
space. This method is not dependent on the morphology of the periodic orbits in the 2D subspaces of the
configuration space as in the algorithm that is presented in paper I [Katsanikas & Wiggins, 2021].
The algorithm is :
(1) Locate an unstable periodic orbit PO for a fixed value of Energy E.
(2) Project the PO into a 2D subspace of the configuration space (for example in the (x1, x2) space).
(3) From the projection of the periodic orbit into the configuration space, we construct a torus or a cylinder
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 (x1, x2) space). If
June 1, 2021 13:0 katsanikas˙wiggins
The Generalization of the Periodic Orbit Dividing Surface 3
the projection of the periodic orbit is a circle in the 2D subspace of the configuration space the above
structure will be topologically equivalent to the Cartesian product of n−1 circles S1×S1×S1×...×S1.
This is a n−1-dimensional torus. If the projection of the periodic orbit is a line or curve the above
structure will be topologically equivalent to the Cartesian product of a line or curve with n−2 circles
R×S1×S1×S1×... ×S1. This is a n−1-dimensional cylinder.This structure (in both of the cases)
can be achieved through the construction of one circle around every point of the periodic orbit in a
2D subspace of the configuration space. For example we compute a circle (with a fixed radius r) in
the plane (x2, x3) around every point of the periodic orbit in the 2D subspace of the nD configuration
space (x1, x2, ..., xn). Then we construct a new circle around every point of the previous structure in
other 2D subspace of the nD configuration space (x1, ..., xn). This can be done by constructing a circle
(with a fixed radius r) in the plane (x2, x4). Then we continue adding circles until we have added n−2
circles to the initial projection of the periodic orbit. The goal of this step is to include all coordinates
of the configuration space in the construction of this object.
x2,1,i,j1=x2,0,i +rcos(θj1)
x3,1,i,j1=x3,0,i +rsin(θj1)
x1,1,i,j1=x1,0,i
x4,1,i,j1=x4,0,i
...
xn,1,i,j1=xn,0,i
(3)
x2,2,i,j1,j 2=x2,1,i,j1+rcos(θj2)
x4,2,i,j1,j 2=x4,1,i,j1+rsin(θj2)
x1,2,i,j1,j 2=x1,1,i,j1
x3,2,i,j1,j 2=x3,1,i,j1
xn,2,i,j1,j 2=xn,1,i,j1
... (4)
...
x2,n−2,i,j1,j 2,...,j(n−2) =x2,n−3,i,j1,j2,...,j(n−3) +rcos(θj(n−2))
xn,n−2,i,j1,j 2...j(n−2) =xn,n−3,i,j1,j2,...,j(n−3) +rsin(θj(n−2))
x1,n−2,i,j1,j 2,...j(n−2) =x1,n−3,i,j1,j2,...,j(n−3)
...
xn−1,n−2,i,j1,j 2,...,j(n−2) =xn−1,n−3,1,i,j1,j2,...,j(n−3)
(5)
(x1,0,i, x2,0,i , ..., xn,0,i), i = 1, ...N are the points of the periodic orbit in the nD configuration space
(x1, x2, ..., xn). We have the angle θj1=j12π
n1with j1 = 1, ..., n1for the first circle and θj2=j22kπ
n1
with j2 = 1, ..., n1for the second circle and so on with θj(n−2) =j(n−2)2π
n1with j(n−2) = 1, ..., n1
for the n−2 circle that we need for the construction of the dividing surface.
x1,1,i,j1, x2,1,i,j 1, ..., xn,1,i,j1with i= 1, ..., N and j1=1, ..., n1are the points of the torus or cylinder that
is constructed from the Cartesian product of the projection of the periodic orbit into the 2D subspace
(x1, x2) and a circle in the (x2, x3) space in the nD space (x1, x2, ...xn). x1,2,i,j1,j 2, x2,2,i,j1,j2, ...xn,2,i,j1,j2
with i= 1, ..., N ,j1=1, ..., n1and j2=1, ..., n1are the points of the torus or cylinder that is
June 1, 2021 13:0 katsanikas˙wiggins
4M. Katsanikas and S. Wiggins
constructed from the Cartesian product of the projection of the periodic orbit in the 2D subspace
(x1, x2), and 2 circles in 2D subspaces of the configuration space in the nD space (x1, x2, ..., xn). And so
on x1,n−2,i,j1,j 2,...j(n−2), x2,n−2,i,j1,j2,...j(n−2), ...xn,n−2,i,j 1,j2,..j (n−2) with i= 1, ..., N and j1, j2, ....j(n−
2) = 1, ..., n1are the points of the torus or cylinder that is constructed from the Cartesian product of
the projection of the periodic orbit in the 2D subspace (x1, x2), and other n−2 circles in 2D subspaces
of the configuration space in the nD space (x1, x2, ..., xn).
(4) For each point x1,n−2,i,j 1,j2,...j(n−2), x2,n−2,i,j1,j2,...j(n−2) , ...xn,n−2,i,j1,j 2,..j(n−2) with i= 1, ..., N and
j1, j2, ....j (n−2) = 1, ..., n1on this torus or cylinder we must calculate the pmax
x1,n−2,i,j1,j 2,,...j(n−2)
and pmin
x1,n−2,i,j1,j 2,,...j(n−2) by solving the following equation for a fixed value of energy Ewith
px2=px3=... =pxn= 0:
V(x1,n−2,i,j1,j 2,...j(n−2), x2,n−2,i,j1,j2,...j(n−2), ...xn,n−2,i,j 1,j2,..j (n−2)) +
p2
x1,n−2,i,j1,j 2,...j(n−2)
2m1
=E
(6)
and we find the maximum and minimum values
pmax
x1,n−2,i,j1,j 2,...j(n−2) and pmin
x1,n−2,i,j1,j 2,...j(n−2). We choose points px1,n−2,i,j1,j 2,...j(n−2) with j1, j 2...j(n−
2) = 1, ..., n1in the interval pmin
x1,n−2,i,j1,j 2,...j(n−2) ≤px1,n−2,i,j1,j2,...j(n−2) ≤pmax
x1,n−2,i,j1,j 2,...j(n−2). These
points can be uniformly distributed in this interval. We will repeat the same procedure to compute
the values px2,n−2,i,j1,j2,...j(n−2), ..., pxn−1,n−1,i,j1,j 2,...j(n−2) . In general for n2≤n−1 for each point
x1,n−2,i,j1,j 2,...j(n−2), x2,n−2,i,j1,j2,...j(n−2), ...xn,n−2,i,j 1,j2,..j (n−2), px1,n−2,i,j 1,j2,...j (n−2), ...
pxn2−1,n−2,i,j1,j 2,...j(n−2) with i= 1, ..., N and j1, j2, ....j(n−2) = 1, ..., n1we must calculate the
pmax
xn2,n−2,i,j1,j 2,,...j(n−2) and pmin
xn2,n−2,i,j1,j 2,,...j(n−2) by solving the following equation for a fixed value
of energy Ewith pxn2+1 =... =pxn= 0:
V(x1,n−2,i,j1,j 2,...j(n−2), x2,n−2,i,j1,j2,...j(n−2), ...xn,n−2,i,j 1,j2,..j (n−2)) +
p2
x1,n−2,i,j1,j 2,...j(n−2)
2m1
+... +p2
xn2,n−2,i,j1,j 2,...j(n−2)
2mn2
=E
(7)
and we find the maximum and minimum values pmax
xn2,n−2,i,j1,j 2,...j(n−2) and pmin
xn2,n−2,i,j1,j 2,...j(n−2).
We choose points pxn2,n−2,i,j1,j2,...j(n−2) with j1, j 2, j(n−2) = 1, ..., n1in the interval
pmin
xn2,n−2,i,j1,j 2,...j(n−2) ≤pxn2,n−2,i,j1,j2,...j(n−2) ≤pmax
xn2,n−2,i,j1,j 2,...j(n−2). These points can be uniformly
distributed in this interval.
Then we obtain the value pxn,n−2,i,j1,j2,...j(n−2) from the Hamiltonian:
V(x1,n−2,i,j1,j 2,...j(n−2), x2,n−2,i,j1,j2,...j(n−2), ...xn,n−2,i,j 1,j2,..j (n−2)) +
p2
x1,n−2,i,j1,j 2,...j(n−2)
2m1
+... +p2
xn,n−2,i,j1,j 2,...j(n−2)
2mn
=E
(8)
Dimensionality and Topology: This algorithm constructs a torus or cylinder as the product of the projection
of the periodic orbit ( a curve or a line or a closed curve) in a 2D subspace of the configuration space with
n−2 circles in the nD configuration space. This structure is a n−1-dimensional torus or cylinder. 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 object, from
n−1 to 2n−2 dimensions, which is embedded in the 2n−1 dimensional energy surface. Then we obtain
the value of the last momenta from the Hamiltonian of the system.
June 1, 2021 13:0 katsanikas˙wiggins
The Generalization of the Periodic Orbit Dividing Surface 5
3. The Algorithm for Hamiltonian systems with three degrees of freedom
In this section, we present our algorithm for a Hamiltonian system with three degrees of freedom with
momenta px, py, pz, potential V(x, y, z) and the corresponding masses m1, m2, m3:
p2
x/2m1+p2
y/2m2+p2
z/2m3+V(x, y, z) = E
(9)
Eis the numerical value of the Hamiltonian (we call it as energy).
The algorithm is:
(1) Locate an unstable periodic orbit PO for a fixed value of energy E.
(2) Project the PO into the configuration space and we consider the projection of the periodic orbit in a
2D subspace of the configuration space (for example in the (x, y) space).
(3) We construct a torus or a cylinder 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 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 (10)
(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
dividing 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).
(4) For each point x1,i,j1, y1,i,j 1, z1,i,j1on this torus or cylinder we must calculate the pmax
x,1,i,j1and pmin
x,1,i,j1
by 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
(11)
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.
(5) 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
(12)
We choose points py,1,i,j 1with 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:
June 1, 2021 13:0 katsanikas˙wiggins
6M. Katsanikas and S. Wiggins
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
(13)
Dimensionality and Topology: This algorithm constructs a torus or a cylinder as the product of the
projection of the periodic orbit (1D object) in a 2D subspace with one circle in the configuration space.
This object is a 2-dimensional torus or cylinder. 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 object, 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 object, 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. Application of the Algorithm in the Quadratic Normal Form Hamiltonian
System with Three Degrees of Freedom
In this section we will use a benchmark example in order to show how to apply our algorithm to Hamiltonian
systems with three degrees of freedom. We choose the quadratic normal form (NF) Hamiltonian system
with three degrees of freedom. We did this because we have analytical formulas for the NHIM and we can
compare our results from the construction of the dividing surfaces from periodic orbits with the dividing
surfaces from the NHIM. This system [Wiggins, 2016] is described by the following Hamiltonian:
H=λ
2(p2
x−x2) + ω2
2(p2
y+y2) + ω3
2(p2
z+z2)
(14)
with λ > 0, ω2>0, ω3>0 and
H1=λ
2(p2
x−x2),
H2=ω2
2(p2
y+y2),
H3=ω3
2(p2
z+z2).
(15)
The following equations are the equations of motion of the system:
June 1, 2021 13:0 katsanikas˙wiggins
The Generalization of the Periodic Orbit Dividing Surface 7
˙x=∂H
∂px
=λpx,(16)
˙px=−∂H
∂x =λx, (17)
˙y=∂H
∂py
=ω2py,(18)
˙py=−∂H
∂y =−ω2y, (19)
˙z=∂H
∂pz
=ω3pz,(20)
˙pz=−∂H
∂z =−ω3z.
(21)
This system is composed of three subsystems that are described by the uncoupled Hamiltonians H1,H2
and H3. There is an index-1 saddle (x, px, y, py, z, pz) = (0,0,0,0,0,0)) for energy (numerical value of the
Hamiltonian) E= 0. We consider that the reaction occurs when xchanges sign [Ezra & Wiggins, 2018]
(x= 0). This condition will be applied in our algorithm.
The analytical formula for the NHIM is ([Ezra & Wiggins, 2018]):
ω2
2(p2
y+y2) + ω3
2(p2
z+z2) = E, N HIM
(22)
and the analytical formula for 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.
(23)
The analytical expressions for the periodic orbits PO1 and P02 are the following ([Katsanikas &
Wiggins, 2021]):
ω2
2(p2
y+y2) = E, P O1
(24)
ω3
2(p2
z+z2) = E. P O2
(25)
The periodic orbits PO1 and PO2 are circles in the planes (y, py) and (z, pz) respectively.
We constructed, using our algorithm, the dividing surfaces from PO1 (see subsection 4.1) and PO2 (see
subsection 4.2). Then we compared our results with the dividing surface from the NHIM (see subsection
4.3).
June 1, 2021 13:0 katsanikas˙wiggins
8M. Katsanikas and S. Wiggins
4.1. PO1-Dividing Surface
First we apply our algorithm to PO1 :
(1) The PO1 is given by 24 for every fixed value E.
(2) The periodic orbit is a circle and it lies in a plane (y, py). The radius of this circle is q2E
ω2. This
means that (with x= 0) the projection of the periodic orbit in the configuration space is a line (with
x=z= 0) and ytakes values from −q2E
ω2to q2E
ω2.
(3) In this step the algorithm is simplified since we have x=z= 0 and we do not have to construct any
torus or cylinder in order to have all coordinates of the configuration space in this structure.
(4) Step 4 is carried out easily in this case (with x= 0). We compute the pmax
yand pmin
y. We sample points
in the interval [pmin
y, pmax
y].
(5) Step 5 is carried out easily in this case (with x= 0). We compute the pmax
zand pmin
z. We sample points
in the interval [pmin
z, pmax
z]. For every point in the previous interval we obtain the pxcoordinate from
the Hamiltonian.
No-recrossing Property:
From the equation (14) (for a fixed value of energy E, the numerical value of the Hamiltonian) we have
(using step 2 of the algorithm (x=z= 0)):
λ
2p2
x=E−ω2
2(p2
y+y2)−ω3
2(p2
z)
(26)
The equation (26) using that E=H1+H2+H3with H2=ω2
2(p2
y+y2) (from equation (15)) we have:
λ
2p2
x=H1+H3−ω3
2p2
z
(27)
From equation (15) we have that H3−ω3
2p2
z>0 and H1>0 (with x= 0). Then we have for the
dividing surface :
px=r2
λ(H1+H3−ω3
2p2
z)px>0F orwar d DS
px=−r2
λ(H1+H3−ω3
2p2
z)px<0Backward DS
(28)
The new DS that is constructed has the no-recrossing property since ˙x=λpx.
4.2. PO2-Dividing Surface
We now apply our algorithm to PO2 :
(1) The PO2 is given by 24 for every fixed value E.
(2) The periodic orbit is a circle and it lies on a plane (z, pz). The radius of this circle is q2E
ω3. This
means that (with x= 0) the projection of the periodic orbit in the configuration space is a line (with
x=y= 0) and ztakes values from −q2E
ω3to q2E
ω3.
(3) In this step the algorithm is simplified since we have x=y= 0 and we do not have to construct any
torus or cylinder in order to have all coordinates of the configuration space in this structure.
June 1, 2021 13:0 katsanikas˙wiggins
The Generalization of the Periodic Orbit Dividing Surface 9
(4) Step 4 is carried out naturally in this case (with x= 0). We compute pmax
yand pmin
y. We sample points
in the interval [pmin
y, pmax
y].
(5) Step 5 is carried out naturally in this case (with x= 0). We compute pmax
zand pmin
z. We sample points
in the interval [pmin
z, pmax
z]. For every point in the previous interval we obtain the pxcoordinate from
the Hamiltonian.
No-recrossing Property:
from the equation (14) (for a fixed value of energy E, the numerical value of the Hamiltonian) we have
(using step 2 of the algorithm (x=y= 0)):
λ
2p2
x=E−ω2
2(p2
y)−ω3
2(p2
z+z2)
(29)
Frome equation (29) using that E=H1+H2+H3with H3=ω3
2(p2
z+z2) (from equation (15)) we
have:
λ
2p2
x=H1+H2−ω2
2p2
y
(30)
From equation 15 we have that H2−ω2
2p2
y>0 and H1>0 (with x= 0). Then the expression for the
dividing surface is:
px=r2
λ(H1+H2−ω2
2p2
y)px>0, F orward DS,
px=−r2
λ(H1+H2−ω2
2p2
y)px<0, Backward DS.
(31)
The DS that is constructed has the no-recrossing property since ˙x=λpx.
4.3. The structure and comparison of dividing surfaces
We have constructed the dividing surfaces from the periodic orbits PO1 and PO2 using our algorithm
and we now compare them with the dividing surface constructed from the normally hyperbolic invariant
manifold (NHIM) of the quadratic normal form Hamiltonian for λ= 1, ω2=√2, ω3= 1. These values
were used in [Ezra & Wiggins, 2018], with the value of the energy E= 14, H1= 4. We computed the
dividing surfaces from the periodic orbits PO1 and PO2 using the algorithms and analytical formulas of
the previous section (see subsections 4.1 and 4.2). The dividing surface from the NHIM is obtained using
the algorithm and analytical formulas from ([Ezra & Wiggins, 2018] - see above in the introduction of this
section).
All dividing surfaces have x= 0 (as we mentioned above in this section and we obtain the pxcoordinate
from the Hamiltonian) and this means that the dividing surfaces are 3-dimensional embedded in the 4-
dimensional space (y, z, py, pz).
The dividing surfaces from PO1 and PO2 have a similar ellipsoidal structure (see for example Fig. 4)
similar to the dividing surface from the NHIM (see [Katsanikas & Wiggins, 2021]). The only difference is
that the dividing surface from PO1 is elongated only in the y-direction and the dividing surface from PO2
is elongated only in the z-direction (see Fig. 2). The range of values of the dividing surfaces from PO1
and PO2 is smaller in the y-direction (for the case of the dividing surface from PO1) or smaller in the
z-direction (for the case of the dividing surface from PO2) than the range of values of the dividing surface
from the NHIM (see Fig. 3). This means that the dividing surface from PO1 and PO2 are subsets of the
dividing surface from the NHIM.
June 1, 2021 13:0 katsanikas˙wiggins
10 M. Katsanikas and S. Wiggins
Fig. 1. The pseudocode of our algorithm.
Fig. 2. The 2D projections of the dividing surfaces that are constructed from the PO1 (first row) and PO2 (second row)
using our algorithm in (y, z) (first column), (y , py) (second column) and (z, pz) (third column) subspaces of the phase space.
June 1, 2021 13:0 katsanikas˙wiggins
The Generalization of the Periodic Orbit Dividing Surface 11
Fig. 3. 2D projections of dividing surface that is constructed from the NHIM in (y , z) (first panel), (y, py) (second panel)
and (z, pz) (third panel) subspaces of the phase space.
Fig. 4. The 3D projection (y , py, pz) of the dividing surface which is constructed from the periodic orbit PO1 using our
algorithm. The color indicates the values of the third dimension.The viewpoint is in spherical coordinates is (45o,30o).
5. The Algorithm for Hamiltonian systems with two degrees of freedom
In this section we show that for two degrees of freedom Hamiltonian systems our algorithm coincides with
the classical periodic orbit dividing surface (PODS) construction given in [Pechukas & McLafferty, 1973;
Pechukas & Pollak, 1977; Pollak & Pechukas, 1978; Pechukas & Pollak, 1979; Pollak, 1985]. In particular,
we consider a Hamiltonian system with two degrees of freedom with momenta px, py, potential V(x, y) and
the corresponding masses m1, m2:
p2
x/2m1+p2
y/2m2+V(x, y) = E,
(32)
where Eis the numerical value of the Hamiltonian (which we refer to as energy).
For this a system in this for our algorithm has the form:
(1) Locate an unstable periodic orbit PO for a fixed value of energy E.
(2) (x0,i, y0,i ), i = 1, ...N are the points of the periodic orbit in the 2D configuration space (x, y). This is
a closed curve or a line or a curve in a 2D subspace. It is an 1-dimensional object.
June 1, 2021 13:0 katsanikas˙wiggins
12 M. Katsanikas and S. Wiggins
(3) For each point x0,i , y1,i of the periodic orbit we must calculate the pmax
x,0,i and pmin
x,0,i by solving the
following equation for a fixed value of energy (numerical value of Hamiltonian) Ewith py= 0:
V(x0,i, y0,i ) + p2
x,0,i
2m1
=E,
(33)
and we find the maximum and minimum values pmax
x,0,i and pmin
x,0,i. We choose points px,0,i in the interval
pmin
x,0,i ≤px,0,i ≤pmax
x,0,i. These points can be uniformly distributed in this interval.
(4) Now for every point x0,i, y0,i, px,0,i we must calculate the py,0,i by solving the following equation for a
fixed value of energy E:
V(x0,i, y0,i ) + p2
x,0,i
2m1
+p2
y,0,i
2m2
=E.
(34)
Dimensionality and Topology: This algorithm give us an 1-dimensional object (a circle or ellipse or a line)
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
or a line), 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.
Hence, our algorithm coincides with the classical algorithm for the construction of the 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]) for Hamiltonian systems with two
degrees of freedom.
6. Conclusions
In this paper we have described a second method (the first method was described in paper I - [Katsanikas
& Wiggins, 2021]) of constructing dividing surfaces from periodic orbits in Hamiltonian systems with
three or more degrees of freedom. This method generalizes the classical method of [Pechukas, 1981; Pollak,
1985] that is valid only for Hamiltonian systems with two degrees of freedom. It also avoids the difficult
computation of the NHIM. Furthermore, this method does not require the necessary conditions of the
first method (see paper I -[Katsanikas & Wiggins, 2021]) and can be used in all cases for periodic orbits
in Hamiltonian systems with three or more degrees of freedom. From this study we have the following
remarks:
(1) The algorithm of this paper does not have the restriction of the first method, that is presented in the
paper I, for the periodic orbits. This means that we don’t have the restriction for the periodic orbits
to be closed curves in a 2D subspace of the phase space.
(2) The dividing surfaces that are constructed from the periodic orbits are subsets of the dividing surfaces
that are constructed from the NHIM.
(3) The algorithm of our method becomes identical to the algorithm of the classical method of [Pechukas,
1981; Pollak, 1985] in Hamiltonian systems with two degrees of freedom.
(4) The periodic orbit dividing surfaces, that are constructed using the method of this paper, have the
same topology with the dividing surfaces that are constructed from the NHIM in Hamiltonian systems
with three degrees of freedom.
(5) The periodic orbit dividing surfaces, that are constructed using the method of this paper, give us
information for the trajectory behavior in different regions of the phase space according to the position
of the periodic orbits. In our example (a normal form Hamiltonian system with three degrees of
June 1, 2021 13:0 katsanikas˙wiggins
REFERENCES 13
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.
Acknowledgments
We acknowledge the support of EPSRC Grant No. EP/P021123/1 and ONR Grant No. N00014-01-1-
0769.
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