2D-Generating Surfaces in a Quartic Hamiltonian System with Three Degrees of Freedom – II

Abstract

In earlier research, we developed two techniques designed to expand the construction of a periodic orbit dividing surface for Hamiltonian systems with three or more degrees of freedom. Our methodology involved transforming a periodic orbit into a torus or cylinder, thereby elevating it to a higher-dimensional structure within the energy surface (refer to [Katsanikas & Wiggins, 2021a, 2021b, 2023a, 2023b]). Recently, we introduced two new methods for creating dividing surfaces, which do not rely on periodic orbits. Instead, we used 2D surfaces (geometric entities) or 3D surfaces in a Hamiltonian system with three degrees of freedom (see [Katsanikas & Wiggins, 2024a, 2024b, 2024c]). In these studies, we applied these surfaces within a quadratic normal-form Hamiltonian system with three degrees of freedom. This series of two papers (this paper and [Katsanikas et al., 2024]) extends our results to 2D-generating surfaces for quartic Hamiltonian systems with three degrees of freedom. This paper focuses on presenting the second method of constructing 2D-generating surfaces.

Full text

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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 - II
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 earlier research, we developed two techniques designed to expand the construction of a peri-
odic orbit dividing surface for Hamiltonian systems with three or more degrees of freedom. Our
methodology involved transforming a periodic orbit into a torus or cylinder, thereby elevating
it to a higher-dimensional structure within the energy surface (refer to [Katsanikas & Wiggins,
2021a,b, 2023a,b]). Recently we introduced two new methods for creating dividing surfaces,
which do not rely on periodic orbits. Instead, we used 2D surfaces (geometric entities) or 3D
surfaces in a Hamiltonian system with three degrees of freedom (see [Katsanikas & Wiggins,
2024a,b,c]). In these studies, we applied these surfaces within a quadratic normal form Hamilto-
nian system with three degrees of freedom. This series of two papers (this paper and [Katsanikas
et al., 2024]) extends our results to 2D generating surfaces for quartic Hamiltonian systems with
three degrees of freedom. This paper focuses on presenting the second 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 is crucial for understanding the dynamics of Hamiltonian systems and has
applications in various fields, such as chemistry and dynamical astronomy. Traditionally, constructing di-
viding surfaces depended on periodic orbits, which limited their use primarily to Hamiltonian systems with
two degrees of freedom [Pechukas & McLafferty, 1973; Pechukas & Pollak, 1977; Pollak & Pechukas, 1978;
Pechukas, 1981; Pollak, 1985]. However, this approach has yielded promising results in various systems,
notably in 2D caldera-type Hamiltonian systems, as demonstrated by 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 with three or more degrees of freedom calculating
dividing surfaces becomes significantly more complex. One effective strategy involves using Normally Hy-
perbolic Invariant Manifolds (NHIMs) in conjunction with normal form theory([Wiggins et al., 2001; Uzer
et al., 2002; Waalkens et al., 2007; Toda, 2003; Komatsuzaki & Berry, 2003]). Despite this, computing
NHIMs is highly challenging and computationally intensive. This difficulty has led to the development of
periodic orbit dividing surfaces for Hamiltonian systems with three or more degrees of freedom. Recent
advancements in this area have been documented in a series of papers [Katsanikas & Wiggins, 2021a,b,
2023a,b; Gonzalez Montoya et al., 2024a,b]. This method has been practically applied to identify dynam-
ical matching within a 3D caldera-type Hamiltonian system [Katsanikas & Wiggins, 2022; Wiggins &
Katsanikas, 2023]. Future research efforts will focus on utilizing periodic orbit dividing surfaces (PODS)
to address selectivity issues, expanding their application to 3D astronomical potentials and 4D symplectic
maps [Katsanikas et al., 2020a; Agaoglou et al., 2020; Katsanikas et al., 2021, 2011a,b; Zachilas et al.,
2013]). Additionally, alternative methods for constructing dividing surfaces have emerged, using 2D and
3D generating surfaces instead of periodic orbits [Katsanikas & Wiggins, 2024a,b,c].
In the paper [Katsanikas & Wiggins, 2024a], we investigated two approaches for creating dividing sur-
faces using 2D surfaces (closed and orientable objects) without needing to know or assume their invariance.
These objects result in the same dividing surfaces as those generated by periodic orbits. In this work, we
expanded the initial concept for constructing dividing surfaces to include two-dimensional objects. We
applied these methods specifically to quadratic normal form Hamiltonian systems. In this series of two
papers, we further extend the use of these methods to a quartic Hamiltonian system. This paper focuses on
the second method for constructing 2D generating surfaces within this system and examines the structure
of these surfaces.
In Section 2, we present a detailed overview of the quartic Hamiltonian systems under investigation and
provide analytical expressions for their periodic orbits. In Section 3, we calculate the 2D generating surfaces
within this system. Following this, Section 4 explores the structure and morphology of these surfaces. We
further examine the structure of these surfaces in a coupled scenario involving quartic Hamiltonian systems
with three degrees of freedom in Section 5. Finally, we summarize our findings in the last section.
2. The Quartic Hamiltonian system with three degrees of freedom
The quartic Hamiltonian system is the following Hamiltonian equation [Lyu & Wiggins, 2020; Gonza-
lez Montoya et al., 2024a; Katsanikas et al., 2024]:
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 - II 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 Erepresents the Hamilto-
nian’s numerical value, known as the energy. This point serves as an index-1 saddle point for the entire
system. In this context, we define the reaction event as a transition indicated by a change in the xcoordi-
nate (for H1>0). By setting x= 0, as suggested in [Ezra & Wiggins, 2018], we establish 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 derived from the following analytical expressions (for more details, see [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 will construct the 2D generating surfaces of PO1 and PO2 using the second method
described in [Katsanikas & Wiggins, 2024a].
Applying the algorithm of the second method in [Katsanikas & Wiggins, 2024a] to PO1 involves the
following steps:
(1) The PO1 is defined by the analytical formula (6). We use the second method from [Katsanikas &
Wiggins, 2024a] and project the periodic orbit PO1 onto the (x, y) subspace of the configuration space.
This projection results in a line (x= 0) because the periodic orbit is confined to the (y, py) subspace
of the phase space. The ycoordinate ranges from −p2E/ω2to p2E/ω2(as seen in equation (6)).
(2) Next, we construct the product of the projection of the periodic orbit with a circle of radius rin the
2D subspace (y, z):
y2+z2=r2, x = 0 .
(8)
Applying the same algorithm to PO2 involves the following steps:
(1) The PO2 is defined by the analytical formula (7). We use the second method from [Katsanikas &
Wiggins, 2024a] and project the periodic orbit PO2 onto the (x, z) subspace of the configuration space.
This projection results in a line (x= 0) because the periodic orbit is confined to the (z, pz) subspace of
the phase space. The zcoordinate ranges from −p2E/ω3to p2E/ω3(as indicated by equation (7)).
(2) Next, we construct the product of the projection of the periodic orbit with a circle of radius r1in the
2D subspace (y, z):
y2+z2=r2
1, x = 0 .
(9)
4. The structure of 2D Generating surfaces
In this section, we computed the 2D generating surfaces of PO1 and PO2 using the algorithms of the
previous section. We used the same parameters as in [Katsanikas et al., 2024] ( E= 14, H1= 4, α= 2,
β=−1, ω2=√2, and ω3= 1). In accordance with our discussion in the initial paper [Katsanikas
et al., 2024], we have calculated the portion of 2D generating surfaces within the energy surface. For
completeness, we are also performing the same analysis for quadratic normal Hamiltonian systems with
three degrees of freedom (refer to Appendix 1). As highlighted in Section 4 of the paper [Katsanikas et al.,
2024], the maximum radii utilized for constructing periodic orbit dividing surfaces are also applied to the
2D generating surfaces (the values of the maximum radii are given in [Gonzalez Montoya et al., 2024b]).
Initially, we compute the 2D generating surfaces utilizing PO1 as a basis, varying the radius values for
our constructions, as detailed in [Katsanikas & Wiggins, 2024a]. These radius values encompass Rmax /10,
Rmax/5, Rmax /2 and Rmax where Rmax represents the maximum radius limit in the construction of the
2D generating surface for PO1 (see equation (8)). Following this, a similar computation is performed for
the 2D generating surface of PO2, employing the same ratios of the maximum radius Rmax1(as specified
for the generating surface of PO2—see equation (9)).
Due to their construction (see the previous section), the 2D generating surfaces have x= 0. This
means that only a 2D projection in the (y, z) plane is needed to describe the structure of these generating
surfaces for PO1 and PO2. Refer to Fig. 1 for the case of PO1 and Fig. 2 for the case of PO2. In Figs. 1
and 2, the generating surfaces are depicted as cylindrical surfaces in the (y, z) space. We also observe that
the generating surfaces of PO1 are more elongated along the y-axis, while those of PO2 are more elongated
along the z-axis. Additionally, as the radius of the 2D generating surfaces for PO1 and PO2 increases, the
generating surfaces extend further.

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2D Generating Surfaces in a Quartic Hamiltonian system - II 5
Fig. 1. 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/10 (upper left panel), Rmax /5 (upper right panel), Rmax/2 (lower left panel) and Rmax
(lower right panel).
5. 2D Generating surfaces in a coupled case of a Quartic Hamiltonian system
with three degrees of freedom
In this section, we computed the 2D generating surfaces of a coupled Quartic Hamiltonian system. This
system is described by the following Hamiltonian (see [Katsanikas et al., 2024]):
H=H1+H2+H3+cyz2
=1
2(p2
x−αx2+β
2x4) + ω2
2(p2
y+y2) + ω3
2(p2
z+z2) + cyz2.
(10)
The equations of motion are:

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6M. Katsanikas et al.
Fig. 2. The 2D projection (y , z) of the 2D generating surfaces that are constructed from the PO2, using the second method
of construction, for radius Rmax1/10 (upper left panel), Rmax1/5 (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.
(11)
We utilized the methodologies delineated in section 3 to generate surfaces for the periodic orbits of
the coupled system, specifically PO1 and PO2. This process was conducted with precise parameter values:
E= 14, H1= 4, α= 2, β=−1, ω2=√2, and ω3= 1, as consistent with the previous section, and
additionally with c= 0.1 as per [Gonzalez Montoya et al., 2024a]. The numerical computation of the
periodic orbits PO1 and PO2 was previously detailed in [Gonzalez Montoya et al., 2024a]. Notably, these
surfaces exhibit a comparable morphology to those discussed in the preceding section (e.g., see Fig.3 ). This
similarity suggests that the 2D generating surfaces of the coupled system share a comparable structural
composition with those of the uncoupled system.

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2D Generating Surfaces in a Quartic Hamiltonian system - II 7
Fig. 3. The 2D projection (y , z) of the 2D generating surface that is constructed from the PO1 in the coupled case of our
Hamiltonian system, for radius Rmax/10.
6. Conclusions
In our earlier research (refer to [Katsanikas & Wiggins, 2024a]), we presented two techniques for building
2D generating surfaces within Hamiltonian systems featuring three degrees of freedom. These approaches
were specifically applied to the quadratic normal form Hamiltonian system with three degrees of freedom.
In the present investigation, we employed the second method to produce 2D generating surfaces for both
uncoupled and coupled quartic Hamiltonian systems with three degrees of freedom. This marks the initial
instance of extending our theory to alternative types of Hamiltonian systems with three degrees of freedom.
To summarize our findings:
(1) The 2D generating surfaces can also be computed using the second method of construction in a quartic
Hamiltonian system with three degrees of freedom.
(2) Due to their construction, the 2D generating surfaces of PO1 and PO2 are presented as cylindrical
surfaces.
(3) The 2D generating surfaces of PO1 extend more in the ydirection, while the 2D generating surfaces
of PO2 extend more in the zdirection.
(4) The 2D generating surfaces extend further as we increase the radius of the construction.
(5) In the coupled case of our system, the 2D generating surfaces have a structure similar to those in the
uncoupled case.
Appendix A 2D generating surfaces in the quadratic normal form Hamiltonian sys-
tem
In Section 4 of the paper [Katsanikas et al., 2024], we referenced our prior work in [Katsanikas & Wiggins,
2024a], where we delineated the structure of 2D generating surfaces using the second construction method
within a quadratic normal form Hamiltonian system with three degrees of freedom. However, in that work,
we did not compute the portion of these surfaces contained within the energy surface. This appendix
addresses that omission by providing the structure of this portion.
As highlighted in Section 4 of the paper [Katsanikas et al., 2024], the maximum radii utilized for
constructing periodic orbit dividing surfaces are also applied to the 2D generating surfaces. Within this
appendix, we computed the 2D generating surfaces for various construction values. These values encom-
passed Rmax/20, Rmax /5, Rmax/2, and Rmax , where Rmax represents the upper limit of the radius in

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8REFERENCES
constructing the periodic orbit dividing surface for PO1 (refer to [Katsanikas & Wiggins, 2023b]). Subse-
quently, a similar computation was conducted for the 2D generating surface of PO2, employing the same
ratios of the maximum radius Rmax1(refer to [Katsanikas & Wiggins, 2023b]).
Calculating these 2D generating surfaces for PO1 (refer to Fig. A.1) and PO2 (see Fig. A.2) on the
energy surface produces results similar to those described in our previous paper ([Katsanikas & Wiggins,
2024a]).
Fig. A.1. 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).
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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