3D Generating Surfaces in a Quartic Hamiltonian System with Three Degrees of Freedom – II

Abstract

In our previous studies [Katsanikas & Wiggins, 2021a, 2021b, 2023a, 2023b, 2024a, 2024b, 2024c], we presented two methods for building up dividing surfaces based on either periodic orbits or 2D/3D generating surfaces, specifically for Hamiltonian systems with three or more degrees of freedom. These papers extended these dividing surface constructions to allow for more complex forms, such as tori or cylinders, embedded within the energy surface of the Hamiltonian system. These studies were applied to a quadratic normal form Hamiltonian system with three degrees of freedom. This series of papers extends our findings to 3D generating surfaces for three degrees of freedom quartic Hamiltonian systems. This paper focuses on the second approach for constructing these 3D generating surfaces.

Full text

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International Journal of Bifurcation and Chaos
©World Scientific Publishing Company
3D 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 our previous studies [Katsanikas & Wiggins, 2021a,b, 2023a,b, 2024a,b,c], we presented two
methods for building up dividing surfaces based on either periodic orbits or 2D/3D generating
surfaces, specifically for Hamiltonian systems with three or more degrees of freedom. These
papers extended these dividing surface constructions to allow for more complex forms, such as
tori or cylinders, embedded within the energy surface of the Hamiltonian system. These studies
were applied to a quadratic normal form Hamiltonian system with three degrees of freedom.
This series of papers extends our findings to 3D generating surfaces for three degrees of freedom
quartic Hamiltonian systems. The current paper focuses on the second approach for constructing
these 3D generating surfaces.
Keywords: Chemical reaction dynamics; Dynamical Astronomy; Phase space; Hamiltonian sys-
tem; Periodic orbit; Normally hyperbolic invariant manifold; Dividing surface
1. Introduction
The dividing surfaces are important conceptually and practically for understanding global trajectory be-
haviour in Hamiltonian systems, with important applications in chemical reaction dynamics and dynamical
1

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2M. Katsanikas et al.
astronomy. Traditionally these surfaces are constructed using only unstable periodic orbits, but originally,
the unstable periodic orbit method was limited to only two degrees of freedom Hamiltonian systems, as sup-
ported by numerous studies ([Pechukas & McLafferty, 1973; Pechukas & Pollak, 1977; Pollak & Pechukas,
1978; Pechukas & Pollak, 1979; Pechukas, 1981; Pollak, 1985]).
For Hamiltonian systems with three degrees of freedom dividing surfaces were originally computed
using Normally Hyperbolic Invariant Manifolds (NHIMs) in conjunction with normal form theory. However,
this approach suffers from computational complexity and challenges, as highlighted in previous research
[Wiggins et al., 2001; Uzer et al., 2002; Toda, 2003; Komatsuzaki & Berry, 2003; Waalkens et al., 2007;
Waalkens & Wiggins, 2010; Wiggins, 2016].
In our earlier studies, we introduced two novel methods that circumvent the need to compute NHIMs
by employing periodic orbits instead ([Katsanikas & Wiggins, 2021a,b, 2023a,b; Gonzalez Montoya et al.,
2024a,b]). These methods offer an alternative approach for identifying dividing surfaces in Hamiltonian
systems.
One of these approaches was effectively utilized to find the phenomenon of “dynamical matching”
in a three-dimensional extension of the caldera potential, as described in [Katsanikas & Wiggins, 2022;
Wiggins & Katsanikas, 2023]. The dynamical matching has been extensively studied in two-dimensional
caldera-type Hamiltonian systems, with several studies providing support ([Katsanikas & Wiggins, 2018,
2019; Katsanikas et al., 2022b, 2020b; Geng et al., 2021; Katsanikas et al., 2022a,c]).
Future research will focus on using Periodic Orbit Dividing Surfaces (PODS) to address selectivity
issues ([Katsanikas et al., 2020a; Agaoglou et al., 2020; Katsanikas et al., 2021]), expanding its application
to 3D astronomical potentials ([Katsanikas et al., 2011a,b]), and exploring its potential in 4D symplectic
maps [Zachilas et al., 2013].
In our previous research [Katsanikas & Wiggins, 2024a; Katsanikas et al., 2024a,b], we detailed the
building up of dividing surfaces from two-dimensional (2D) surfaces, specifically closed and orientable
objects, without requiring knowledge or assumptions about their invariance. Further studies ([Katsanikas
& Wiggins, 2024b,c] explored two methods for creating dividing surfaces from three-dimensional (3D)
surfaces (3D generating surfaces), which are also closed and orientable, again without the need for invariance
assumptions. These methods produce dividing surfaces equivalent to those obtained from periodic orbits.
In the current study, we extend the use of 3D generating surfaces for a quartic Hamiltonian system. This
article, the second in a series ([Katsanikas et al., 2024c] and the present paper), focuses on the second
method of this construction.
In Section 2, we present a comprehensive overview of the quartic Hamiltonian systems under analysis,
including the analytical expressions for their periodic orbits. Section 3 details the computations for 3D
generating surfaces within these systems. In Section 4, we examine the structure and morphology of these
surfaces in depth. Section 5 expands this investigation to explore the structure of these surfaces in a
coupled scenario of quartic Hamiltonian systems with three degrees of freedom. Finally, our conclusions
are summarized in the last section.
2. The Quartic Hamiltonian System with three degrees of freedom
The quartic Hamiltonian system is characterized by the following Hamiltonian function [Lyu et al., 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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3D Generating Surfaces-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 Hamilton’s equations of motion are given by:
˙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 origin in the phase space (x, px, y, py, z, pz) = (0,0,0,0,0,0) corresponds to the value E= 0 (where
Eis the value of the Hamiltonian function, representing the total energy of the system), is the index-1
saddle point of the entire system. Within this context, we define the reaction event as the transition
characterized by a change in the xcoordinate (for H1>0). By setting x= 0, as proposed by [Ezra &
Wiggins, 2018], we establish a five-dimensional manifold within the six-dimensional phase space.
The dividing surface, its corresponding normally hyperbolic invariant manifold NHIM, and the unstable
periodic orbits for its construction, 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(Dividing surface).
(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. 3D generating Surfaces in the Quartic Hamiltonian system with three degrees
of freedom
In this section, we will leverage the 3D generating surface construction algorithm to the quartic Hamiltonian
system with three degrees of freedom. We will focus on the specific cases of 3D generating surfaces of the
periodic orbits PO1 and PO2.
3D generating surface of PO1
(1) PO1 is a well-defined periodic orbit within the Hamiltonian system. As previously mentioned, it is
given by the analytical formula given by Equation (6). We are interested in the projection of this
periodic orbit onto the (x, y) plane. This projection results in a straight line confined entirely to the
y-axis. In other words, for all points on this projected line, the x-coordinate remains constant at zero.
The y-coordinate, however, varies within a specific range, as defined in Equation (6), from −p2E/ω2
to p2E/ω2(see equation (6)).
(2) Following the line projection onto the (x, y) plane, we define a circle centred at every point of the
periodic orbit within the (y, z) subspace. This circle, with radius r, represents a specific region in
that subspace. Next, we perform a mathematical operation that combines the properties of both the
projected line and the circle. This operation, often referred to as a product or a Cartesian product,
results in a cylindrical surface. Notably, this surface is characterized by having a constant x= 0 value
across its entire structure.
y2+z2=r2, x = 0 .
(8)
(3) Then, we compute pmax
yand pmin
yand we use a sampling in the interval [pmin
y, pmax
y].
3D generating surface of PO2
(1) Similar to PO1, PO2 is a well-defined periodic orbit within the Hamiltonian system, described by
the analytical formula in Equation (7). We are interested in its projection onto the (x, z) plane. This
projection results in a straight line confined entirely to the z-axis. In other words, for all points on this
projected line, the x-coordinate remains constant at zero. The z-coordinate, however, varies within a
specific range defined in Equation (7), ranging from −p2E/ω3to p2E/ω3(see the equation (7)).
(2) Similar to the analysis for PO1, we define a circle centred at every point of the periodic orbit within
the (y, z) subspace. This circle, with a radius r1, represents a specific region. Next, we perform a
mathematical operation, often referred to as a Cartesian product, that combines the information from
the projected line (obtained in step 1) and the circle. The result of this operation is a cylindrical
surface. Notably, this surface shares the characteristic of having a constant x= 0 value across its entire
structure.
y2+z2=r2
1, x = 0 .
(9)
(3) Then, we compute pmax
yand pmin
yand we use a sampling in the interval [pmin
y, pmax
y].
4. The structure of 3D generating Surfaces
This study successfully computed 3D generating surfaces using the framework established in the previous
section. The analysis focused on periodic orbits PO1 and PO2 within a quartic normal form Hamiltonian
system with three degrees of freedom. Notably, the chosen parameters (E= 14, H1= 4, α= 2, β=−1,
ω2=√2, and ω3= 1) were consistent with those employed in our previous works [Gonzalez Montoya et al.,
2024a,b] for better comparability.
This section investigates the influence of radius on the computed 3D generating surfaces associated
with unstable periodic orbits PO1 and PO2. We begin by leveraging these orbits as the foundation for

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3D Generating Surfaces-II 5
surface generation. For each orbit, we calculate multiple surfaces by varying the radius parameter used
in their construction. The chosen radii for PO1-based surfaces are: Rmax/10, Rmax/5, Rmax /2, and Rmax ,
where Rmax is the upper limit for radius ras defined previously. Similarly, for PO2-based surfaces, the
radii are: Rmax1/10, Rmax1/5, Rmax1/2, and Rmax1, where Rmax1is the upper limit for radius r1. This
systematic variation allows us to analyze the sensitivity of the generating surfaces to changes in radius.
Finally, the resulting 3D surfaces are embedded within a four-dimensional subspace of the phase space
(x, y, z, py). It’s important to note that the maximum radii used to construct the periodic orbit dividing
surfaces are also applied consistently to the 3D generating surfaces (refer to Section 3 of [Katsanikas et al.,
2024c] for the explanation). The specific values of these maximum radii for our system can be found in
[Gonzalez Montoya et al., 2024b].
The 3D generating surfaces are depicted as solid cylinders in the 3D projection (y, z, py) (see Figs. 1
and 2). Other 3D projections do not provide additional information because the generating surfaces have
x= 0. Increasing the radius of the construction for the 3D generating surfaces of PO1 and PO2 leads to
three significant effects.
Firstly, in Fig. 1, the 3D generating surfaces of PO1 expand primarily in the z-direction, indicating a
more pronounced extension in this direction as the radius increases.
Secondly, in Fig. 2, a similar trend is observed for the 3D generating surfaces of PO2, where an increase
in radius results in growth in the y-direction.
Lastly, Fig. 3 shows that while both PO1 and PO2 exhibit elongation, the directions of this elongation
differ.
Fig. 1. The 3D projection (y, z, py) of the 3D generating surfaces that are constructed from the unstable periodic orbit PO1,
for radius Rmax/10 (top left panel), Rmax /5 (top right panel), Rmax/2 (bottom left panel) and Rmax (bottom right panel).

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6M. Katsanikas et al.
Fig. 2. The 3D projection (y, z, py) of the 3D generating surfaces that are constructed from the unstable periodic orbit PO2,
for radius Rmax1/10 (top left panel), Rmax1/5 (top right panel), Rmax1/2 (bottom left panel) and Rmax1(bottom right panel).
Fig. 3. The 3D projection (y, z, py) of the 3D generating surfaces that are constructed from the PO1 (with green colour) and
PO2 (with violet colour) of our Hamiltonian system, for radii Rmax/10 and Rmax1/10 respectively.
5. 3D Generating Surfaces in a coupled case of the Quartic Hamiltonian systems
with three degrees of freedom
This section explores the derivation of 3D generating surfaces for periodic orbits within a coupled quartic
Hamiltonian system with three degrees of freedom. We aim to compare these results with those obtained in

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3D Generating Surfaces-II 7
the previous section, which investigated the uncoupled version of the same system. To begin, let us revisit
the Hamiltonian that governs the dynamics of these systems (with 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.
(10)
The Hamilton’s equations of motion are:
˙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)
Building upon the algorithms outlined in Section 3, we constructed 3D generating surfaces for the
coupled system’s unstable periodic orbits, PO1 and PO2. These surfaces were generated using the same
parameter values (E= 14, H1= 4, α= 2, β=−1, ω2=√2, and ω3= 1), employed in the previous
section to ensure consistency. Additionally, a coupling parameter c= 0.1 was introduced, as detailed in
[Gonzalez Montoya et al., 2024a,b]. Notably, the numerical computations for PO1 and PO2 were presented
in the aforementioned references.
Interestingly, the resulting surfaces exhibit a morphology strikingly similar to those observed in the
uncoupled system (e.g., Fig. 4). This resemblance suggests that the 3D generating surfaces in the coupled
system likely share a comparable structure with their counterparts in the uncoupled system.
6. Conclusions
Our previous work ( [Katsanikas & Wiggins, 2024b,c])) introduced two methods for constructing 3D gener-
ating surfaces in Hamiltonian systems with three degrees of freedom. These methods were initially demon-
strated for the quadratic normal form. This study leverages the second method to construct 3D generating
surfaces for both uncoupled and coupled quartic Hamiltonian systems with three degrees of freedom. This
application represents a significant advancement, as it marks the first time our theoretical framework has
been extended to encompass broader classes of Hamiltonian systems with three degrees of freedom. The
main conclusions are:
(1) The 3D generating surfaces are depicted as solid cylinders within the phase space’s 3D projections.
(2) The extent of PO1’s generating surface increases in the z-direction with increasing construction radius.
Similarly, PO2’s generating surface extends further in the y-direction as the radius grows.
(3) Notably, the 3D generating surfaces of PO1 and PO2 exhibit elongation in distinct directions.
(4) Interestingly, the morphology (overall shape and structure) of the 3D generating surfaces remains
consistent across both the coupled and uncoupled variants of our Hamiltonian system.

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8REFERENCES
Fig. 4. The 3D projection (y, z, py) of the 3D generating surface that is constructed from the periodic orbit PO1 in the
coupled quartic Hamiltonian system, for radius Rmax/10.
Acknowledgments
The authors acknowledge the financial support of EPSRC grant number EP/P021123/1. F. Gonzalez
Montoya acknowledges the financial support of CONAHCyT fronteras grant number 425854, CONAHCyT
grant number CF–2023–G–763, and DGAPA UNAM grant number AG–101122. S. Wiggins acknowledges
the support of the William R. Davis ’68 Chair in the Department of Mathematics at the United States
Naval Academy.
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