The Generalization of the Periodic Orbit Dividing Surface for Hamiltonian Systems with Three or More Degrees of Freedom-III

Abstract

In two previous papers [Katsanikas & Wiggins, 2021a, 2021b], we developed two methods for the construction of periodic orbit dividing surfaces for Hamiltonian systems with three or more degrees of freedom. We applied the first method (see [Katsanikas & Wiggins, 2021a]) in the case of a quadratic Hamiltonian system in normal form with three degrees of freedom, constructing a geometrical object that is the section of a 4D toroidal structure in the 5D energy surface with the space x = 0. We provide a more detailed geometrical description of this object within the family of 4D toratopes. We proved that this object is a dividing surface and it has the no-recrossing property. In this paper, we extend the results for the case of the full 4D toroidal object in the 5D energy surface. Then we compute this toroidal object in the 5D energy surface of a coupled quadratic normal form Hamiltonian system with three degrees of freedom.

Full text

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OPEN ACCESS
International Journal of Bifurcation and Chaos, Vol. 33, No. 7 (2023) 2350088 (13 pages)
c
The Author(s)
DOI: 10.1142/S0218127423500888
The Generalization of the Periodic Orbit Dividing
Surface for Hamiltonian Systems with Three
or More Degrees of Freedom-III
Matthaios Katsanikas∗,†,§and Stephen Wiggins†,‡,¶
∗
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, UK
‡
Department of Mathematics, United States Naval Academy,
Chauvenet Hall, 572C Holloway Road, Annapolis,
MD 21402-5002, USA
§
mkatsan@academyofathens.gr
¶
s.wiggins@bristol.ac.uk
Received April 15, 2023
In two previous papers [Katsanikas & Wiggins,2021a, 2021b], we developed two methods for
the construction of periodic orbit dividing surfaces for Hamiltonian systems with three or more
degrees of freedom. We applied the first method (see [Katsanikas & Wiggins,2021a]) in the case
of a quadratic Hamiltonian system in normal form with three degrees of freedom, constructing a
geometrical object that is the section of a 4D toroidal structure in the 5D energy surface with the
space x= 0. We provide a more detailed geometrical description of this object within the family
of 4D toratopes. We proved that this object is a dividing surface and it has the no-recrossing
property. In this paper, we extend the results for the case of the full 4D toroidal object in the
5D energy surface. Then we compute this toroidal ob ject in the 5D energy surface of a coupled
quadratic normal form Hamiltonian system with three degrees of freedom.
Keywords: Chemical reaction dynamics; phase space; Hamiltonian system, periodic orbits;
dividing surfaces; normally hyperbolic invariant manifold; four-dimensional toratopes; ditorus;
dynamical astronomy.
1. Introduction
This paper is the third paper in the series of the
papers (the previous two papers were [Katsanikas &
Wiggins,2021a, 2021b]) about the construction of
the periodic orbit dividing surfaces for Hamilto-
nian systems with three degrees of freedom. There-
after we will refer to the papers [Katsanikas & Wig-
gins,2021a] and [Katsanikas & Wiggins,2021b] as
papers I and II. The dividing surfaces are phase
space objects that are (2n−2)-dimensional ob jects
in the (2n−1)-dimensional energy surface of a
Hamiltonian system with ndegrees of freedom.
Prior to this series of papers, the basis of the com-
putation of these objects was the periodic orbit
in Hamiltonian system with two degrees of free-
dom (using the classical algorithm of [Pechukas &
McLafferty,1973; Pechukas & Pollak,1977; Pol-
lak & Pechukas,1978; Pechukas,1981; Pollak,1985])
This is an Open Access article published by World Scientific Publishing Company. It is distributed under the terms of the
Creative Commons Attribution 4.0 (CC BY) License which permits use, distribution and reproduction in any medium, provided
the original work is properly cited.
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June 10, 2023 16:40 WSPC/S0218-1274 2350088
M. Katsanikas &S. Wiggins
or the Normally Hyperbolic Invariant manifold
(NHIMs) in Hamiltonian systems with ndegrees of
freedom [Wiggins,1994; Wiggins et al.,2001; Uzer
et al.,2002; Wiggins,2016]. The computation of
NHIMs is very difficult and it requires extensive
computations using normal form theory [Wiggins
et al.,2001; Uzer et al.,2002; Waalkens et al.,2007;
Toda,2003; Komatsuzaki & Berry,2003]. This is the
reason that in this series of papers we proposed new
methods for the construction of dividing surfaces
using periodic orbits. This is very important in order
to detect new dynamical phenomena in 3D chemical
and astronomical systems like the phenomenon of
dynamical matching in a 3D Caldera-type Hamilto-
nian system (see [Katsanikas & Wiggins,2022]). The
2D cases of this type of Hamiltonian systems have
been studied in many papers (see [Katsanikas &
Wiggins,2018,2019; Katsanikas et al.,2022b; Kat-
sanikas et al.,2020; Geng et al.,2021; Katsanikas
et al.,2022a; Katsanikas et al.,2022c]).
As we mentioned above, in papers I and II we
presented two methods for the construction of peri-
odic orbit dividing surfaces for Hamiltonian systems
with three or more degrees of freedom. In paper I,
we presented a method that produces periodic orbit
dividing surfaces with the topology of a toroidal
object. In paper I, we presented an example of this
method computing the section of this toroidal object
with the plane x= 0 for the case of a quadratic
normal form Hamiltonian system with three degrees
of freedom (because the reaction occurs when x
changes sign). In this paper, we will extend this com-
putation in order to compute and visualize the full
toroidal object (a 4D object in the 5D energy sur-
face). Furthermore, we will present our results for
the same object in a coupled quadratic normal form
Hamiltonian system with three degrees of freedom
and we will compare our results.
We describe our results for the case of an uncou-
pled and a coupled Quadratic normal form Hamilto-
nian system with three degrees of freedom in Secs. 2
and 3 respectively. Our conclusions are presented in
the last section (Sec. 4).
2. Periodic Orbit Dividing Surfaces
in a Quadratic Normal Form
Hamiltonian System with Three
Degrees of Freedom
In this section, we will construct the periodic orbit
dividing surfaces associated with the index-1 saddle
for a Quadratic normal form Hamiltonian system
with three degrees of freedom. In the previous
papers, we studied the section of the periodic orbit
dividing surfaces with the plane x=0.Inthissec-
tion, we will investigate the periodic orbit divid-
ing surfaces without this restriction. Our system
is described by the following Hamiltonian (λ>0,
ω2>0,ω
3>0 — see [Katsanikas & Wiggins,
2021a, 2021b]):
H=λ
2(p2
x−x2)+ω2
2(p2
y+y2)+ω3
2(p2
z+z2).(1)
This system is composed of three subsys-
tems that are described from the Hamiltonians
H1,H
2,H
3:
H1=λ
2(p2
x−x2),
H2=ω2
2(p2
y+y2),
H3=ω3
2(p2
z+z2).
(2)
The following equations are the equations of
motion of the system:
˙x=∂H
∂px
=λpx,(3)
˙px=−∂H
∂x =λx, (4)
˙y=∂H
∂py
=ω2py,(5)
˙py=−∂H
∂y =−ω2y, (6)
˙z=∂H
∂pz
=ω3pz,(7)
˙pz=−∂H
∂z =−ω3z. (8)
This system is composed of three subsystems
that are described by the uncoupled Hamiltoni-
ans H1,H2and H3. There is an index-1 sad-
dle (x, px,y,p
y,z,p
z)=(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).
The NHIM is described by the following equa-
tion [Ezra & Wiggins,2018]:
ω2
2(p2
y+y2)+ω3
2(p2
z+z2)=E. NHIM (9)
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Generalization of the Periodic Orbit Dividing Surface for Hamiltonian Systems
The dividing surface (a four-sphere) associated with
the NHIM is (see [Ezra & Wiggins,2018]):
λ
2p2
x+ω2
2(p2
y+y2)+ω3
2(p2
z+z2)=E. (10)
The periodic orbits PO1 and PO2 associated
with the index-1 saddle are described by the fol-
lowing analytical formulas [Katsanikas & Wiggins,
2021a, 2021b]:
ω2
2(p2
y+y2)=E, PO1 (11)
ω3
2(p2
z+z2)=E. PO2 (12)
Now we will construct the periodic orbit divid-
ing surfaces associated with the periodic orbits PO1
and PO2 and we will prove that they have the no-
recrossing property.
2.1. Periodic orbit dividing
surface — PO1
We will apply the first method of constructing peri-
odic orbit dividing surfaces in Hamiltonian systems
with three degrees as presented in [Katsanikas &
Wiggins,2021a].
(1) The PO1 is described by the analytical for-
mula (11).
(2) We will use the first version of the algo-
rithm that was presented in [Katsanikas & Wiggins,
2021a] because the periodic orbit is projected as a
closed curve (a circle) in the space (y, py).
(3) We will construct the product of the projection
of the periodic orbit in the (y, py)spacewithtwocir-
cles (the radius of these circles r>0 is fixed). The
one circle is in the subspace (y,py,x) and the other
in the subspace (y,py,z). The result of this product
is a hypertorus with equation (see Appendix A):
⎛
⎜
⎝


p2
y+y2−2E
ω22
+x2−r⎞
⎟
⎠
2
+z2=r2.
(13)
This is the equation for a ditorus (http://hi.gher.
space/wiki/Ditorus), which is one of the possi-
ble four 4D tori http://hi.gher.space/wiki/Four-
dimensional torii.
(4) We compute the pmax
zand pmin
z.Thenweusea
sampling in the interval [pmin
z,p
max
z] and we compute
for every point the pxthrough the Hamiltonian of
the system.
Now we will prove that this dividing surface has
the no-recrossing property.
The following equation is a consequence of
Eq. (13):
z2=r2−⎛
⎜
⎝


p2
y+y2−2E
ω22
+x2−r⎞
⎟
⎠
2
.
(14)
We have from Eq. (1) (for a fixed value of
energy E):
λ
2p2
x=E+λ
2x2−ω2
2(p2
y+y2)−ω3
2(p2
z+z2).
(15)
Equation (15) through the equation E=H1+
H2+H3with H2=ω2
2(p2
y+y2)[seeEq.(2)]and
substituting the term z2from Eq. (14), we get
λ
2p2
x=H1+λ
2x2+H3−ω3
2p2
z
+ω3
2⎛
⎜
⎝


p2
y+y2−2E
ω22
+x2−r⎞
⎟
⎠
2
−ω3
2r2.(16)
We have that the terms H3−ω3
2p2
z≥0[see
Eq. (2)],
ω3
2⎛
⎜
⎝


p2
y+y2−2E
ω22
+x2−r⎞
⎟
⎠
2
≥0
and λ
2x2≥0. This means that Eq. (16) is equivalent
with the condition (see Appendix B):
H1−ω3
2r2≥0.(17)
Consequently, we have:
r≤2H1
ω3
.(18)
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M. Katsanikas &S. Wiggins
The condition (18) implies that we have to choose a small radius r. Then Eq. (16) becomes:
px=





2
λ⎛
⎜
⎝H1+H3−ω3
2p2
z+ω3
2⎛
⎜
⎝


p2
y+y2−2E
ω22
+x2−r⎞
⎟
⎠
2
−ω3
2r2⎞
⎟
⎠+x2px>0,
px=−





2
λ⎛
⎜
⎝H1+H3−ω3
2p2
z+ω3
2⎛
⎜
⎝


p2
y+y2−2E
ω22
+x2−r⎞
⎟
⎠
2
−ω3
2r2⎞
⎟
⎠+x2px<0.
(19)
This means that the dividing surface that was con-
structed has the no-recrossing property because
˙x=λpx.
2.2. Periodic orbit dividing
surface — PO2
We will apply the first method of constructing
the periodic orbit dividing surfaces in Hamiltonian
systems with three degrees as presented in [Kat-
sanikas & Wiggins,2021a].
(1) The PO2 is described by the analytical for-
mula (12).
(2) We will use the first version of the algo-
rithm that was presented in [Katsanikas & Wiggins,
2021a] because the periodic orbit is projected as a
closed curve (a circle) in the space (z,pz).
(3) We will construct the product of the projection
of the periodic orbit in the (z, pz)spacewithtwocir-
cles (the radius of these circles r1>0 is fixed). The
one circle is in the subspace (z, pz,y) and the other
in the subspace (z,pz,x). The result of this product
is a hypertorus with equation (see Appendix A):
⎛
⎜
⎝


p2
z+z2−2E
ω32
+x2−r1
⎞
⎟
⎠
2
+y2=r2
1.
(20)
(4) We compute the pmax
yand pmin
y.Thenweuse
a sampling in the interval [pmin
y,p
max
y]andwecom-
pute for every point the pxthrough the Hamiltonian
of the system.
Now we will prove that this dividing surface has
the no-recrossing property.
The following equation is a consequence of
Eq. (20):
y2=r2
1−⎛
⎜
⎝


p2
z+z2−2E
ω32
+x2−r1⎞
⎟
⎠
2
.
(21)
We have from Eq. (1) (for a fixed value of
energy E):
λ
2p2
x=E+λ
2x2−ω2
2(p2
y+y2)−ω3
2(p2
z+z2).
(22)
Equation (22) through the equation E=H1+
H2+H3with H3=ω3
2(p2
z+z2)[seeEq.(2)]and
substituting the term y2from Eq. (21), we get
λ
2p2
x=H1+λ
2x2+H2−ω2
2p2
y
+ω2
2⎛
⎜
⎝


p2
z+z2−2E
ω32
+x2−r1
⎞
⎟
⎠
2
−ω2
2r2
1.(23)
We hav e t he term s H2−ω2
2p2
y≥0[seeEq.(2)],
ω2
2⎛
⎜
⎝


p2
z+z2−2E
ω32
+x2−r1⎞
⎟
⎠
2
≥0
and λ
2x2≥0. This means that Eq. (23) is equivalent
to the condition (see Appendix B):
H1−ω2
2r2
1≥0.(24)
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Generalization of the Periodic Orbit Dividing Surface for Hamiltonian Systems
Consequently, we have:
r1≤2H1
ω2
.(25)
The condition (25) implies that we have to choose a small radius r1. Then Eq. (23) becomes:
px=





2
λ⎛
⎜
⎝H1+H2−ω2
2p2
y+ω2
2⎛
⎜
⎝


p2
z+z2−2E
ω32
+x2−r1⎞
⎟
⎠
2
−ω2
2r2
1⎞
⎟
⎠+x2px>0,
px=−





2
λ⎛
⎜
⎝H1+H2−ω2
2p2
y+ω2
2⎛
⎜
⎝


p2
z+z2−2E
ω32
+x2−r1⎞
⎟
⎠
2
−ω2
2r2
1⎞
⎟
⎠+x2px<0.
(26)
This means that the dividing surface that was con-
structed has the no-recrossing property because
˙x=λpx.
2.3. The structure of the periodic
orbit dividing surfaces PO1
and PO2
In this section, we computed the dividing surfaces
that are based on the periodic orbits PO1 and PO2
(see the previous subsection for the algorithm of
their construction). For this reason, we used the
parameters E=14,H
1=4,λ=1,ω
2=√2and
ω3= 1 (as in our previous papers see [Katsanikas &
Wiggins,2021a, 2021b]).
We began by computing the dividing surfaces
from the periodic orbits PO1 and PO2. Firstly,
we computed the dividing surfaces using as a base
the periodic orbit PO1 for different values of the
radius as used for our construction (see Sec. 2.1).
These values are Rmax1/20,Rmax1/10.Rmax1/5,
Rmax1/2,Rmax1whereRmax1 is the upper limit
of the radius for the construction of the periodic
orbit dividing surface of PO1 [see Eq. (18)]. Then
we did the same computation for the dividing sur-
faces of PO2 using again the same ratios of max-
imum radius Rmax2 [for the dividing surface of
PO2 — see Eq. (25)]. These dividing surfaces were
computed in the five-dimensional energy surface
(x, y, z, py,p
z) (we obtain the pxfor every value of
energy from the Hamiltonian).
We investigated the structure of the periodic
orbit dividing surfaces of PO1 and PO2 (we will
refer to these surfaces as PO1-dividing surface and
PO2-dividing surface many times in the text) in
the energy surface. The PO1-dividing surface is pre-
sented either as a toroidal structure in the 3D pro-
jections of the energy surface (x, y, py), (y, z, py)and
(y,py,p
z) (see for example Fig. 1) and as a Box
structure in the 3D projections of the energy sur-
face (x, y, z), (x, y , pz), (x, z, pz), (x, z, py), (y, z, pz),
(x, py,p
z), and (z, py,p
z) (see for example Fig. 2).
The PO2-dividing surface is also presented as a
toroidal structure or a box structure in the 3D pro-
jections of the energy surface. The difference is that
the PO2-dividing surface is a torus in the 3D pro-
jections (x, z, pz), (y,z, pz)and(z,py,p
z) and not in
the 3D projections (x, y, py), (y, z, py)and(y,py,p
z)
(as in the case of the PO1-dividing surface). In the
other 3D projections the PO2-dividing surface has
the morphology of a Box structure.
We compared the two dividing surfaces, using
2D projections, in the energy surface (x, y, z, py,p
z).
For this reason, we selected the 2D projections
(x, y), (x, z)and(py,p
z). These projections include
Fig. 1. The 3D projection (x, y , py) of the dividing surface
that is constructed from the PO1. The radius for this con-
struction was r=Rmax1/20. The color indicates the distri-
bution of the third dimension py.
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June 10, 2023 16:40 WSPC/S0218-1274 2350088
M. Katsanikas &S. Wiggins
Fig. 2. The 3D projection (x, y , z) of the dividing surface
that is constructed from the PO1. The radius for this con-
struction was r=Rmax1/20.
all dimensions of the five-dimensional energy sur-
face and can help us to investigate the extension
of every dividing surface in all dimensions of the
energy surface. We depicted the PO1-dividing sur-
face in these 2D projections varying the radius of
its construction from Rmax1/20 to Rmax1(see
Figs. 3–7). Similarly, we depicted the PO2-dividing
surface in these 2D projections varying the radius
Fig. 3. The 2D projections (x, y ), (x, z)and(py,p
z) of the dividing surfaces that are constructed from the PO1 (first row)
and the PO2 (second row). The radius for this construction was r=Rmax1/20 for the PO1 and r=Rmax2/20 for the PO2.
of its construction from Rmax2/20 to Rmax2(see
Figs. 3–7). We saw in these figures that the (x, y)
and (x, z) projections of the PO1-dividing surface
is represented by an area that is inside a rectan-
gle and an area inside a square respectively. On
the contrary, the PO2-dividing surface was repre-
sented by an area that is inside a square and an area
inside a rectangle in the (x, y)and(x, z )projections
respectively. This means that the PO1-dividing sur-
face is more elongated in the y-direction and the
PO2-dividing surface was more elongated in the z-
direction. As we increased the radius of the con-
struction of the PO1-dividing surface and of the
PO2-dividing surface, the areas that are occupied
by the rectangle and the square became larger. The
representation of these two periodic orbits dividing
surface in the 2D projection (py,p
z) is an ellipti-
cal surface (see for example Fig. 3). The projection
(py,p
z) of the PO1-dividing surface is more elon-
gated in the py-direction and the projection (py,p
z)
of the PO2-dividing surface is more elongated in
the pz-direction (see Figs. 3–5). Now if we increase
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Generalization of the Periodic Orbit Dividing Surface for Hamiltonian Systems
Fig. 4. The 2D projections (x, y ), (x, z)and(py,p
z) of the dividing surfaces that are constructed from the PO1 (first row)
and the PO2 (second row). The radius for this construction was r=Rmax1/10 for the PO1 and r=Rmax2/10 for the PO2.
Fig. 5. The 2D projections (x, y ), (x, z)and(py,p
z) of the dividing surfaces that are constructed from the PO1 (first row)
and the PO2 (second row). The radius for this construction was r=Rmax1/5 for the PO1 and r=Rmax2/5forthePO2.
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M. Katsanikas &S. Wiggins
Fig. 6. The 2D projections (x, y ), (x, z)and(py,p
z) of the dividing surfaces that are constructed from the PO1 (first row)
and the PO2 (second row). The radius for this construction was r=Rmax1/2 for the PO1 and r=Rmax2/2forthePO2.
Fig. 7. The 2D projections (x, y ), (x, z)and(py,p
z) of the dividing surfaces that are constructed from the PO1 (first row)
and the PO2 (second row). The radius for this construction was r=Rmax1 for the PO1 and r=Rmax2forthePO2.
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Generalization of the Periodic Orbit Dividing Surface for Hamiltonian Systems
Fig. 8. The 3D projection (x, y , z) of the dividing surface
that is constructed from the NHIM.
Fig. 9. The 3D projection (y, z, py) of the dividing surface
that is constructed from the NHIM.
the radius of the construction of these two peri-
odic orbit dividing surfaces, the pyand pzvalues of
the PO1-dividing surface approach the respective
pyand pzvalues of the PO2-dividing surface (see
Figs. 6 and 7).
In addition, we computed the dividing sur-
face from the NHIM (for the same set of param-
eters of the system) and we compared it with the
PO1-dividing surface and the PO2-dividing surface.
This surface was computed using the algorithm
and analytical formulas of [Ezra & Wiggins,2018]
(that were also used in the papers I and II [Kat-
sanikas & Wiggins,2021a, 2021b]). This surface is
a 4D ellipsoidal that is presented as an ellipsoidal
in 3D projections (y,z,py), (y,z,pz), (y,py,p
z)and
(z,py,p
z) (see for example Fig. 9) and as a 2D
elliptical or circular surface in the other 3D pro-
jections (see for example Fig. 8 — this happens
because the dividing surface constructed from the
NHIM has x= 0 see [Ezra & Wiggins,2018]).
Comparing Fig. 10 with Figs. 3–7, we see that
the PO1-dividing surface and PO2-dividing surface
are more elongated in the x-direction (for all val-
ues of the radius of their construction) than the
dividing surface associated with the NHIM. On the
other hand, the dividing surface associated with the
NHIM is more elongated in the zand pzdirec-
tions than the PO1-dividing surfaces. Furthermore,
the dividing surface associated with the NHIM is
more elongated in the yand pydirections than the
PO2-dividing surfaces. We observe that by increas-
ing the radius of construction of the PO1-dividing
surface and PO2-dividing surface, these surfaces
approximate the dividing surface associated with
the NHIM in the directions in which initially these
surfaces were less elongated. We see also that the
combination of the PO1-dividing surface and PO2-
dividing surface approximate the dividing surface
from the NHIM in all directions except x-direction
in which they are more elongated. This means that
the combination of these surfaces is a larger set
than that of the dividing surface associated with
the NHIM.
Fig. 10. The 2D projections (x, y ), (x, z)and(py,p
z) of the dividing surfaces that are constructed from the NHIM.
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M. Katsanikas &S. Wiggins
3. Periodic Orbit Dividing Surfaces
in a Coupled Quadratic Normal
Form Hamiltonian System with
Three Degrees of Freedom
In this section, we computed the periodic orbit
dividing surface of the coupled quadratic normal
form Hamiltonian systems with three degrees of
freedom. We compare our results with those of the
previous section for the uncoupled case of this sys-
tem. The Hamiltonian of the systems is (with λ>0,
ω2>0,ω
3>0):
H=λ
2(p2
x−x2)+ω2
2(p2
y+y2)
+ω3
2(p2
z+z2)+cyz2.(27)
This is the Hamiltonian of the previous section
plus a coupling term cyz2.
The following equations are the equations of
motion of the coupled system:
˙x=∂H
∂px
=λpx,(28)
˙px=−∂H
∂x =λx, (29)
˙y=∂H
∂py
=ω2py,(30)
˙py=−∂H
∂y =−ω2y−cz2,(31)
˙z=∂H
∂pz
=ω3pz,(32)
˙pz=−∂H
∂z =−ω3z−2cyz. (33)
There is an index-1 saddle (x, px,y,p
y,z,p
z)=
(0,0,0,0,0,0) for energy (numerical value of the
Hamiltonian) E= 0. We consider that the reac-
tion occurs when xchanges sign [Ezra & Wiggins,
2018] (x=0).
We applied the algorithm in Secs. 2.1 and 2.2
of the previous section to construct periodic orbit
dividing surfaces for the periodic orbits PO1 and
PO2 of the coupled system. This has been done for
E=14,H
1=4,λ=1,ω
2=√2andω3=1(as
in the previous section) and for c=0.1. The initial
conditions of PO1 are (x, px,z,p
z)=(0,0,0,0) in
the Poincar´e section y=0withpy>0[thiscan
be obtained from the Hamiltonian — see Eq. (27)]
Fig. 11. The 3D projection (x, y , py) of the dividing surface
that is constructed from the PO1 in the case of the coupled
case of our system. The radius for this construction was r=
Rmax1/20. The color indicates the distribution of the third
dimension py.
and the initial conditions of PO2 are (x, px,y,p
y)=
(0,0,0,0) in the Poincar´esectionz=0withpz>0
[this can be obtained from the Hamiltonian — see
Eq. (27)]. 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 steps 2
to 4 of the algorithms in Secs. 2.1 and 2.2 of the pre-
vious section and we constructed the corresponding
periodic orbit dividing surfaces from PO1 and PO2.
These periodic orbit dividing surfaces have a simi-
lar morphology to that of the periodic orbit divid-
ing surfaces of the previous section (see for example
Fig. 11 and compare it with that of Fig. 1). This
means that the periodic orbit dividing surfaces of
the coupled case of our system have a similar struc-
ture to those of the uncoupled case.
4. Conclusions
In previous papers (see [Katsanikas & Wiggins,
2021a, 2021b] we presented two methods for the
construction of periodic orbit dividing surfaces in
Hamiltonian systems with three degrees of freedom.
Then we presented these algorithms and we proved
the no-recrossing property of the section of these
surfaces with the space x=0forthecaseofthe
uncoupled quadratic normal form Hamiltonian sys-
tem with three degrees of freedom. In this paper,
we applied the first method of the construction of
periodic orbit dividing surfaces to construct the full
4D periodic orbit dividing surfaces in the uncou-
pled and coupled quadratic normal form Hamilto-
nian system with three degrees of freedom. Our
conclusions are:
(1) We proved the no-recrossing property for the
full 4D periodic orbit dividing in the uncoupled
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Generalization of the Periodic Orbit Dividing Surface for Hamiltonian Systems
case of the quadratic normal form Hamiltonian
systems with three degrees of freedom.
(2) The 4D periodic orbit dividing surfaces have a
morphology of a toroidal object or a box struc-
ture in the 3D projections of the energy surface.
(3) The PO1-dividing surfaces are elongated more
in the y-axis and the PO2-dividing surfaces are
elongated more in the z-axis. This means that
the one periodic orbit dividing surface gives
more information in the plane and the other
gives more information out of the plane. The
combination of these periodic orbit dividing
surfaces gives us all the valuable information.
(4) The radius that we used for the construction
of the periodic orbit dividing surfaces is cru-
cial because by increasing this radius we have
a corresponding increase of the periodic orbit
dividing surfaces into all dimensions.
(5) The periodic orbit dividing surfaces of the cou-
pled case of our system have a similar structure
to those of the uncoupled case.
(6) The full periodic orbit dividing surfaces are
more elongated in the x-direction than the
dividing surface from the NHIM and they
approximate this surface in all other directions
increasing the radius of their construction.
(7) The combination of the PO1-dividing surface
and PO2-dividing surface (increasing the radius
of their construction) is a larger set than that of
the dividing surface associated with the NHIM.
Acknowledgment
We acknowledge the support of EPSRC Grant
No. EP/P021123/1 and ONR Grant No. N00014-
01-1-0769.
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Appendix A
Hypertorus and Dividing Surfaces
In this section, we will describe the equation of the
ditorus and how we applied this to the construction
of the dividing surfaces associated with the peri-
odic orbits PO1 and PO2 (PO1-dividing surface and
PO2-dividing surface respectively — see Eqs. (13)
and (20) for the corresponding ditori). Firstly, we
will present the equations of a hypertorus (a 3D
torus in the 4D space, that is often referred to as a
4D torus). This is the 3-Torus T3=S1×S1×S1,
see http://hi.gher.space/wiki/Ditorus. These equa-
tions are given as (with R1>0,R
2>0andr0>0
with a, b, c ∈[0,2π)):
x=(R1+(R2+r0cos a)cosb)cosc, (A.1)
y=(R1+(R2+r0cos a)cosb)sinc, (A.2)
z=(R2+r0cos a)sinb, (A.3)
w=r0sin a. (A.4)
The following equation represents the above
parametric equations for the ditorus in cartesian
coordinates:
((x2+y2−R1)2+z2−R2)2+w2=r2
0.
(A.5)
If we set as R1=2E
ω2,R2=r0=r,x=py,
y=y, z =xand w=z, we have the hypertorus of
Eq. (13) that is needed for the construction of the
PO1-dividing surface. Now if we set R1=2E
ω3,
R2=r0=r1,x=pz,y=z, z =xand w=y,
we have the hypertorus of Eq. (20) that is needed
for the construction of the PO2-dividing surface.
We remark that the ditorus is an example
of a toratope http://hi.gher.space/wiki/Toratope.
There are four distinct types of 4D tori: the tori-
sphere, the spheritorus, the ditorus, and the tiger
http://hi.gher.space/wiki/Four-dimensional torii. It
would be interesting to understand if the remaining
three of four tori arise naturally as dividing surfaces
in 3 DoF Hamiltonian systems.
Appendix B
Analysis of the Conditions for
the Maximum Radius of
the Construction of the Dividing
Surfaces
Here we will analyze why we have the condi-
tions (18) and (25) for the radius that is needed for
the construction of the dividing surface associated
with the periodic orbits PO1 and PO2 respectively
(PO1-dividing surface and PO2-dividing surface).
Firstly, we will prove the first condition (18)
for the PO1-dividing surface. In Sec. 2.1, the right-
hand side of Eq. (16) has three terms A1,A
2and
A3.A1=H3−ω3
2p2
z≥0[seeEq.(2)],A2=
ω3
2((p2
y+y2−2E
ω2)2+x2−r)2≥0andA3=
λ
2x2≥0. Furthermore, except for these terms
Eq. (16) has two other terms, H1≥0 [from Eq. (2)]
and the term −ω3
2r2. The right-hand side of this
equation has four possible cases:
(1) Firstly, we have that:
H1+A1+A2+A3−ω3
2r2≥0.(B.1)
This is equivalent with this equation:
r≤2(H1+A1+A2+A3)
ω3
.(B.2)
The terms A1,A
2and A3are dependent on
the variables (x, y, z, px,p
y,p
z). This means that
the radius r(the radius of the construction of the
PO1-dividing surface) is dependent and restricted
by the values on one or more of the variables
(x, y, z, px,p
y,p
z). This cannot happen because the
radius of the construction is defined to be fixed
(from the beginning) in our algorithm and it can-
not be dependent or to be restricted by any of the
variables (x, y, z, px,p
y,p
z).
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Generalization of the Periodic Orbit Dividing Surface for Hamiltonian Systems
(2) The second case is to have the term H1plus one
or two of the terms Ai,i={1,2,3}to satisfy the
condition:
H1+Ai−ω3
2r2≥0.(B.3)
This is equivalent to this equation:
r≤


2H1+Ai
ω3
.(B.4)
For the same reason as in the first case, this cannot
happen because Aiis dependent on the variables
(x, y, z, px,p
y,p
z).
(3) The third case is to have only terms Ai,i=
{1,2,3}(one or two or three) to satisfy the next
condition:
Ai−ω3
2r2≥0.(B.5)
This is equivalent to this equation:
r≤


2Ai
ω3
.(B.6)
For the same reason as in the first case, this
cannot happen because Aiis dependent on the
variables (x, y, z, px,p
y,p
z).
(4) The last case, is to have only H1to sat-
isfy Eq. (17) and consequently to have the condi-
tion (18). This condition has only the values of H1
(the value of one of the three integrals of motion
that are fixed for every energy surface). This gives
us the opportunity to fix the radius for the con-
struction of the PO1-dividing surface. This solution
is accepted because the radius is not dependent on
any of the variables (x, y, z, px,p
y,p
z).
In conclusion, only the fourth case is valid
for the construction of the PO1-dividing surface
according to our algorithm that is presented in [Kat-
sanikas & Wiggins,2021a]. This means that only
the condition (18) is needed for the construction of
the PO1-dividing surface.
Similarly, we can prove the same for the con-
dition (25) for the PO2-dividing surface and the
conditions (39) and (46) for the periodic orbit divid-
ing surfaces associated with the periodic orbits PO1
and PO2 in paper I (see [Katsanikas & Wiggins,
2021a]).
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