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International Journal of Bifurcation and Chaos
©World Scientific Publishing Company
2D Generating Surfaces in a Quartic Hamiltonian system with
three degrees of freedom - I
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
Francisco Gonzalez Montoya
Faculty of Physical Sciences and Engineering, University of Leeds, Leeds LS2 9JT, United Kingdom.
Instituto de Ciencias F´ısicas, Universidad Nacional Aut´onoma de M´exico, Av. Universidad s/n, Col.
Chamilpa, CP 62210, Cuernavaca, Morelos, M´exico.
Facultad de Ciencias, Universidad Nacional Aut´onoma de M´exico, Av. Universidad 3000, Circuito
Exterior s/n, Coyoac´an, CP 04510, Ciudad Universitaria, Ciudad de M´exico, M´exico.
f.gonzalezmontoya@leeds.ac.uk
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 studies, we developed two techniques aimed at expanding the scope of constructing
a periodic orbit dividing surface within a Hamiltonian system with three or more degrees of
freedom. Our approach involved extending a periodic orbit into a torus or cylinder, thereby
elevating it into a higher-dimensional entity within the energy surface (see [Katsanikas & Wig-
gins, 2021a,b, 2023a,b]). Recently, we introduced two alternative methods for creating dividing
surfaces, distinct from the utilization of periodic orbits, by employing 2D surfaces (geometric
entities) or 3D surfaces within a Hamiltonian system with three degrees of freedom (refer to
[Katsanikas & Wiggins, 2024a,b,c]). In these studies, we applied these surfaces in a quadratic
normal form Hamiltonian system with three degrees of freedom. In this series of two papers, we
extend our results to 2D generating surfaces for quartic Hamiltonian systems with three degrees
of freedom. This paper presents the first method of constructing 2D generating surfaces.
Keywords: Chemical reaction dynamics; phase space; Hamiltonian system; periodic orbit; Di-
viding surface; normally hyperbolic invariant manifold; Dynamical Astronomy;
1
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2M. Katsanikas et al.
1. Introduction
The concept of dividing surfaces plays a pivotal role in analyzing and interpreting the dynamics of Hamil-
tonian systems and finds applications across diverse fields such as chemistry and dynamical astronomy.
Traditionally, constructing dividing surfaces relied on periodic orbits, which posed limitations as a result
of restrictions to Hamiltonian systems featuring two degrees of freedom [Pechukas & McLafferty, 1973;
Pechukas & Pollak, 1977; Pollak & Pechukas, 1978; Pechukas, 1981; Pollak, 1985]). However, this method
has shown promising results across various systems, notably in 2D caldera-type Hamiltonian systems, as
evidenced by the studies conducted by Katsanikas et al. [Katsanikas & Wiggins, 2018, 2019; Katsanikas
et al., 2022b, 2020b; Geng et al., 2021a; Katsanikas et al., 2022a,c; Geng et al., 2021b; Katsanikas et al.,
2023].
However, when dealing with Hamiltonian systems having three or more degrees of freedom, the task of
computing dividing surfaces becomes considerably more complex. One viable approach involves leveraging
Normally Hyperbolic Invariant Manifolds (NHIMs) alongside normal form theory. Nonetheless, the com-
putation of NHIMs presents significant computational challenges and complexity ([Wiggins et al., 2001;
Uzer et al., 2002; Waalkens et al., 2007; Toda, 2003; Komatsuzaki & Berry, 2003]). This challenge has
spurred the advancement of periodic orbit dividing surfaces for Hamiltonian systems featuring three or
more degrees of freedom. Recent progress in this field has been detailed in a sequence of papers (refer to
[Katsanikas & Wiggins, 2021a,b, 2023a,b; Gonzalez Montoya et al., 2024a,b]).
This approach has been practically utilized for identifying dynamical matching within a 3D caldera-
type Hamiltonian system (as investigated in [Katsanikas & Wiggins, 2022; Wiggins & Katsanikas, 2023]).
Future research endeavours will focus on employing PODS to tackle selectivity issues ([Katsanikas et al.,
2020a; Agaoglou et al., 2020; Katsanikas et al., 2021]), extending its scope to address challenges in 3D
astronomical potentials (e.g., [Katsanikas et al., 2011a,b])), and exploring its potential applications in 4D
symplectic maps (e.g., [Zachilas et al., 2013]). Additionally, alternative approaches to constructing dividing
surfaces have been developed that utilize 2D and 3D generating surfaces rather than periodic orbits (as
outlined in [Katsanikas & Wiggins, 2024a,b,c]).
In the paper [Katsanikas & Wiggins, 2024a], we examine two methods for constructing dividing surfaces
from 2D surfaces (2D closed and orientable objects without the requirement of knowing or assuming their
invariance). These objects yield identical dividing surfaces to those generated by periodic orbits. In this
context, we expanded the concept of the initial objects that may be used for constructing dividing surfaces
to include two-dimensional objects. Furthermore, we applied these methods exclusively to quadratic normal
form Hamiltonian systems. In this series of two papers, we also broaden the application of these methods
to a quartic Hamiltonian system. This paper focuses on detailing the application of the first method for
constructing 2D generating surfaces for this system. Additionally, we investigated the structure of these
surfaces.
In Section 2, we describe a comprehensive overview of the quartic Hamiltonian systems under scrutiny,
along with providing analytical expressions for their periodic orbits. Transitioning to Section 3, we conduct
computations for 2D generating surfaces within this system. Subsequently, in Section 4, we delve into an
examination of the structure and morphology of these surfaces. Additionally, we investigate the structure
of these surfaces in a coupled scenario of quartic Hamiltonian systems with three degrees of freedom, as
elaborated in Section 5. Finally, our conclusions are presented in the concluding section.
2. The Quartic Hamiltonian system with three degrees of freedom
The quartic Hamiltonian system is characterized by the following Hamiltonian equation, as exemplified in
literature such as [Lyu & Wiggins, 2020; Gonzalez Montoya et al., 2024a]:
H=1
2(p2
x−αx2+β
2x4) + ω2
2(p2
y+y2) + ω3
2(p2
z+z2)
(1)
with α > 0, β < 0, ω2>0, ω3>0 and
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2D Generating Surfaces in a Quartic Hamiltonian system - I 3
H1=1
2(p2
x−αx2+β
2x4),
H2=ω2
2(p2
y+y2),
H3=ω3
2(p2
z+z2).
(2)
H1,H2, and H3are integrals of motion, rendering this system integrable.
The corresponding equations of motion are:
˙x=∂H
∂px
=px,
˙px=−
∂H
∂x =αx −βx3,
˙y=∂H
∂py
=ω2py,
˙py=−
∂H
∂y =−ω2y,
˙z=∂H
∂pz
=ω3pz,
˙pz=−
∂H
∂z =−ω3z.
(3)
The point (x, px, y, py, z, pz) = (0,0,0,0,0,0) corresponds to E= 0 (where Edenotes the numerical
value of the Hamiltonian, referred to as the energy), serving as an index-1 saddle point of the complete
system. Within this framework, we define the reaction event as the transition characterized by a change in
the xcoordinate (for H1>0). By adopting the condition x= 0, as proposed in [Ezra & Wiggins, 2018],
we construct a five-dimensional surface within the six-dimensional phase space.
The dividing surface, the normally hyperbolic invariant manifold (NHIM), and the periodic orbits
PO1 and PO2 can be obtained from the following analytical expressions (for further details, refer to
[Gonzalez Montoya et al., 2024a]):
1
2p2
x+ω2
2(p2
y+y2) + ω3
2(p2
z+z2) = E,
(4)
ω2
2(p2
y+y2) + ω3
2(p2
z+z2) = E(NHIM),
(5)
ω2
2(p2
y+y2) = E(PO1),
(6)
ω3
2(p2
z+z2) = E(PO2).
(7)
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4M. Katsanikas et al.
3. 2D Generating surfaces in a Quartic Hamiltonian system with three degrees
of freedom
In this section, we applied the algorithm of the first method for the construction of 2D generating surfaces
(see subsection 2.1 of [Katsanikas & Wiggins, 2024a]) in periodic orbits PO1 and PO2 (see the previous
section). The 2D generating surfaces produce the same dividing surfaces as the periodic orbit dividing
surfaces (see [Katsanikas & Wiggins, 2024a]). In the next subsections, we give two different methods of the
construction of 2D generating surfaces of PO1 and PO2.
3.1. First Method
In this subsection, we construct the 2D generating surfaces of PO1 through the cartesian product of the
projection of the periodic orbit PO1 in subspace (y, py) of the phase space and a circle in the subspace (x, y)
of the phase space. Furthermore, we construct the 2D generating surfaces of PO2 through the cartesian
product of the projection of the periodic orbit PO2 in subspace (z , pz) of the phase space and a circle in
the subspace (x, z) of the phase space.
If we apply the algorithm for PO1, we have:
(1) The PO1 is described by the analytical formula (6) . We will use the first version of the algorithm
that was presented in subsection 2.1 of [Katsanikas & Wiggins, 2024a] 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 circle 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.
(8)
If we apply the algorithm for PO2, we have:
(1) The PO2 is described by the analytical formula (7). We will use the first version of the algorithm
that was presented in subsection 2.1 of [Katsanikas & Wiggins, 2024a] 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.
(9)
3.2. Second Method
In this subsection, we construct the 2D generating surfaces of PO1 through the cartesian product of the
projection of the periodic orbit PO1 in subspace (y, py) of the phase space and a circle in the subspace (y, z)
of the phase space. Furthermore, we construct the 2D generating surfaces of PO2 through the cartesian
product of the projection of the periodic orbit PO2 in subspace (z , pz) of the phase space and a circle in
the subspace (y, z) of the phase space.
If we apply the algorithm for PO1, we have:
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2D Generating Surfaces in a Quartic Hamiltonian system - I 5
(1) The PO1 is described by the analytical formula (6). We will use the first version of the algorithm
that was presented in subsection 2.1 of [Katsanikas & Wiggins, 2024a] 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 circle r > 0 is fixed). This circle is in the subspace (y, py, z ). The equation of this
object (a torus) is given by the following analytical formula:
qp2
y+y2
−r2E
ω2!2
+z2=r2.
(10)
If we apply the algorithm for PO2, we have:
(1) The PO2 is described by the analytical formula (7) . We will use the first version of the algorithm
that was presented in subsection 2.1 of [Katsanikas & Wiggins, 2024a] 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, y). The equation of this
object (a torus) is given by the following analytical formula:
pp2
z+z2
−r2E
ω3!2
+y2=r2
1.
(11)
4. The structure of 2D Generating surfaces
In this section, we determined 2D generating surfaces based on the periodic orbits PO1 and PO2 using the
construction algorithm detailed in the previous section. The parameters employed for this computation
were E= 14, H1= 4, α= 2, β=−1, ω2=√2, and ω3= 1.
The 2D generating surfaces produced by our algorithm can serve as the basis for constructing dividing
surfaces (see [Katsanikas & Wiggins, 2024a]). This is achieved by computing the portion of these surfaces
that lies within the energy surface. In [Katsanikas & Wiggins, 2024a], we presented the structure of the
2D generating surfaces directly from our algorithm without computing their intersection with the energy
surface. In this paper (as well as in the second paper of this series), we present the structure of the portions
of these generating surfaces that lie within the energy surface. This is done to provide clarity for readers
who wish to compute the corresponding dividing surfaces and need to understand the structure of the
generating surfaces within the energy surface.1For completeness we are doing the same for the quadratic
normal Hamiltonian systems with three degrees of freedom (see the appendix 1).
The 2D and 3D generating surfaces produce the same dividing surfaces as the periodic orbits (see
[Katsanikas & Wiggins, 2024a,b,c]). This means that the radius constraints required for the no-recrossing
property also apply to the construction of dividing surfaces from the 2D and 3D generating surfaces.
Therefore, these radius constraints also apply to the construction of both 2D and 3D generating surfaces,
a point not mentioned in the papers [Katsanikas & Wiggins, 2024a,b,c]. This is because the radius used
for constructing generating surfaces is the same as the radius used for constructing dividing surfaces from
them.
For the Hamiltonian system discussed in this series of papers (specifically the quartic Hamiltonian
system with three degrees of freedom), there is a maximum allowable radius for the construction of periodic
1We have followed the same approach for the structure of 3D generating surfaces in our recent papers [Katsanikas & Wiggins,
2024b,c].
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6M. Katsanikas et al.
orbit dividing surfaces of PO1 and PO2 (see [Gonzalez Montoya et al., 2024a,b]). These maximum radii are
also the limits for constructing the 2D generating surfaces of PO1 and PO22. For the construction method
discussed in this paper these values are provided in [Gonzalez Montoya et al., 2024a].
The process begins by computing the 2D generating surfaces derived from the periodic orbits PO1 and
PO2 using the first method (see subsection 3.1). Initially, we computed the 2D generating surfaces using
PO1 as a foundation for various values of the radius in our construction (refer to the previous section).
These values included Rmax1/10, Rmax1/5, Rmax1/2 and Rmax1where Rmax1signifies the upper limit of
the radius in the construction of the 2D generating surface for PO1. Subsequently, a similar computation
was conducted for the 2D generating surface of PO2, using the same ratios of the maximum radius Rmax2.
These generating surfaces were computed within the three-dimensional subspace (x, y, py) of the energy
surface for PO1 and the three-dimensional subspace (x, z, pz) of the energy surface for PO2. These surfaces
are presented as toroidal structures, as shown in Figs. 1 and 2. Additionally, we observe that as we increase
the radius in the construction of the 2D generating surfaces of PO1 and PO2, the generating surfaces
extend further in the x-direction.
The process continues by computing the 2D generating surfaces derived from the periodic orbits PO1
and PO2 using the second method (see subsection 3.2). Initially, we calculated the 2D generating surfaces
using PO1 as a basis, varying the radius in our construction (refer to the previous section). These values
included Rmax1/10, Rmax1/5, and Rmax1/2 and Rmax1where Rmax1represents the maximum radius for
the 2D generating surface construction for PO1. Similarly, we performed the computation for the 2D
generating surface of PO2, using the same ratios of the maximum radius Rmax2. These generating surfaces
were computed within the three-dimensional subspace (y, py, z) of the energy surface for PO1 and the three-
dimensional subspace (y, z, pz) of the energy surface for PO2. These surfaces appear as toroidal structures,
as illustrated in Figs. 3 and 4. Additionally, we observe that as the radius increases in the construction of
the 2D generating surfaces of PO1 and PO2, the generating surfaces extend further in the z-direction and
y-direction, respectively.
5. 2D Generating surfaces in a coupled case of a Quartic Hamiltonian system
with three degrees of freedom
In this section, we calculate the 2D generating surface for periodic orbits in a coupled quartic Hamiltonian
system with three degrees of freedom. Our objective is to compare our findings with those presented in the
previous section, which pertained to the uncoupled case of this system. The Hamiltonian governing these
systems is expressed as follows (assuming α > 0, β < 0, ω2>0, ω3>0, and c > 0):
H=H1+H2+H3+cyz2
=1
2(p2
x−αx2+β
2x4) + ω2
2(p2
y+y2) + ω3
2(p2
z+z2) + cyz2.
(12)
The equations of motion are:
2The same applies to the 2D and 3D generating surfaces of the quadratic normal form Hamiltonian system with three degrees
of freedom, as discussed in the papers [Katsanikas & Wiggins, 2024a,b,c]. The only difference is that the maximum values for
the construction of periodic orbit dividing surfaces of PO1 and PO2 are provided in [Katsanikas & Wiggins, 2023a,b].
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2D Generating Surfaces in a Quartic Hamiltonian system - I 7
Fig. 1. The 3D projection (x, y, py) of the 2D generating surfaces that are constructed from the PO1 (using the first method),
for radius Rmax1/10 (upper left panel), Rmax 1/5 (upper right panel), Rmax1/2 (lower left panel) and Rmax1(lower right
panel).
˙x=∂H
∂px
=px,
˙px=−
∂H
∂x =αx −βx3,
˙y=∂H
∂py
=ω2py,
˙py=−
∂H
∂y =−ω2y−cz2,
˙z=∂H
∂pz
=ω3pz,
˙pz=−
∂H
∂z =−ω3z−2cyz.
(13)
We employed the algorithms outlined in section 3 to construct generating surfaces for the coupled
system’s periodic orbits, namely PO1 and PO2. This was carried out with specific parameter values:
E= 14, H1= 4, α= 2, β=−1, ω2=√2, and ω3= 1 (consistent with the previous section), and
additionally with c= 0.1 (as in [Gonzalez Montoya et al., 2024a]). The periodic orbits PO1 and PO2 were
computed numerically in [Gonzalez Montoya et al., 2024a]. These surfaces have a similar morphology to
those in the previous section (see for example Fig. 5). This indicates that the 2D generating surfaces of
the coupled system have a similar structure to those of the uncoupled system.
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8M. Katsanikas et al.
Fig. 2. The 3D projection (x, z, pz) of the 2D generating surfaces that are constructed from the PO2 (using the first method),
for radius Rmax2/10 (upper left panel), Rmax2/5 (upper right panel), Rmax2/2 (lower left panel) and Rmax2(lower right
panel).
6. Conclusions
In our previous work (see [Katsanikas & Wiggins, 2024a]), we introduced two methods for constructing
2D generating surfaces in Hamiltonian systems with three degrees of freedom. We applied these methods
specifically to the quadratic normal form Hamiltonian system with three degrees of freedom. In the current
study, we used the first method to generate 2D generating surfaces for both uncoupled and coupled quartic
Hamiltonian systems with three degrees of freedom. This is the first time we have extended our theory
to other types of Hamiltonian systems with three degrees of freedom. Our findings can be summarized as
follows:
(1) We constructed 2D generating surfaces of PO1 in the subspaces (x, y, py) and (y, py, z) of the energy
surface. These are toroidal structures.
(2) We constructed 2D generating surfaces of PO2 in the subspaces (x, z, pz) and (y, z , pz) of the energy
surface. These are toroidal structures.
(3) The 2D generating surfaces of PO1 and PO2 are embedded in a different subspace of the phase space.
(4) The 2D generating surfaces are extended more and more as we increase the radius of the construction.
(5) The 2D generating surfaces of the coupled case of our system have a similar structure to those of the
uncoupled case of our system.
Appendix A 2D generating surfaces in the quadratic normal form Hamiltonian sys-
tem
In a previous work [Katsanikas & Wiggins, 2024a], we explored the structure of 2D generating surfaces using
the first construction method within a quadratic normal form Hamiltonian system with three degrees of
freedom. While that work established the general framework (see section 4), it did not explicitly calculate
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REFERENCES 9
Fig. 3. The 3D projection (y, py, z) of the 2D generating surfaces that are constructed from the PO1 (using the second
method), for radius Rmax1/10 (upper left panel), Rmax1/5 (upper right panel), Rmax1/2 (lower left panel) and Rmax1(lower
right panel).
the portion of these surfaces confined by the energy surface. This appendix rectifies that omission by
providing the specific structure of this confined portion.
As highlighted in Section 4, the maximum radii utilized for constructing periodic orbit dividing surfaces
are also applied to the 2D generating surfaces. Consistent with the approach described in Section 4,
this appendix explores 2D generating surfaces constructed using various radii. The radius values chosen
were Rmax/20, Rmax/5, Rmax/2, and Rmax itself. Here, Rmax refers to the maximum radius used for
the periodic orbit dividing surface of PO1 in [Katsanikas & Wiggins, 2023a]. We then applied the same
methodology to compute the 2D generating surface for PO2, utilizing corresponding ratios of its maximum
radius, denoted as Rmax1 (also defined in [Katsanikas & Wiggins, 2023a]).
Computing these 2D generating surfaces for PO1 (see Fig. A.1 for the uncoupled case and Fig. A.3 for
the coupled case) and PO2 (see Fig. A.2) in the energy surface yields the same results as those detailed in
Sections 5 and 6 of our previous paper ([Katsanikas & Wiggins, 2024a]).
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. FGM
acknowledges the support of DGAPA UNAM grant number AG–101122, CONAHCyT CF–2023–G–763,
and CONAHCyT fronteras grant number 425854.
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Fig. 4. The 3D projection (y , z, pz) of the 2D generating surfaces that are constructed from the PO2 (using the second
method), for radius Rmax2/10 (upper left panel), Rmax2/5 (upper right panel), Rmax2/2 (lower left panel) and Rmax2(lower
right panel).
Fig. 5. The 3D projection (x, y , py) of the 2D generating surface that is constructed from the PO1 (using the first method)
in the coupled case of our Hamiltonian system for radius Rmax1/10.
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Fig. A.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
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12 REFERENCES
Fig. A.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).
Fig. A.3. 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.
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