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International Journal of Bifurcation and Chaos
©World Scientific Publishing Company
Periodic Orbit Dividing Surfaces in Rotating Hamiltonian Systems
with Two Degrees of Freedom
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
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 this paper, we extend the notion of periodic orbit dividing surfaces to rotating Hamiltonian
systems with two degrees of freedom. Firstly, we present a method that enables us to apply the
classical algorithm of the construction of periodic orbit dividing surfaces ([Pechukas & McLaf-
ferty, 1973; Pechukas, 1981; Pollak & Pechukas, 1978; Pollak, 1985]) in rotating Hamiltonian
systems with two degrees of freedom. Then we study the structure of these surfaces in a rotating
quadratic normal-form Hamiltonian system with two degrees of freedom.
Keywords: Chemical reaction dynamics; phase space; Hamiltonian system; periodic orbit; Divid-
ing surface; normally hyperbolic invariant manifold; Dynamical Astronomy; rotational dynamics
1. Introduction
Periodic orbit dividing surfaces (see [Pechukas & McLafferty, 1973; Pechukas & Pollak, 1979; Pechukas,
1981; Pollak & Pechukas, 1978; Pollak, 1985]) play a central role in transition state theory (see [Wigner,
1938; Pelzer & Wigner, 1932; Evans & Polanyi, 1935; Eyring, 1935]) with many applications in chemical
reaction dynamics (see for example [Agaoglou et al., 2019]) and in dynamical astronomy (see for exam-
ple [Reiff et al., 2022]). Bifurcations of these surfaces are also very important (see [Katsanikas et al.,
2021, 2022a]). Periodic orbit dividing surfaces (PODS) were useful for investigating and detecting the phe-
nomenon of dynamical matching in caldera-type Hamiltonian systems (see [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]). Future works will be focused on the applications of the PODS to the problem of
chemical selectivity ([Katsanikas et al., 2020a; Agaoglou et al., 2020]), to 3D astronomical potentials (see
for example [Katsanikas et al., 2011a,b; Lukes-Gerakopoulos et al., 2016]) or to dynamics in 4D symplectic
maps (see for example [Zachilas et al., 2013]).
1
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2Katsanikas &Wiggins
The classical algorithm for the computation of the periodic orbit dividing surfaces for many years
could be applied only in non-rotating Hamiltonian systems with two degrees of freedom. For Hamiltonian
systems with three degrees of freedom the dividing surfaces could be constructed from normally hyperbolic
invariant manifolds (NHIM) (see [Wiggins, 1994; Wiggins et al., 2001; Uzer et al., 2002; Wiggins, 2016]).
However, the computation of the NHIM is much more 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]) or the use of alternative numerical methods originating from studies of the
control of chaos ([Gonzalez Montoya & Jung, 2022; Gonzalez & Jung, 2015; Gonzalez et al., 2014]). The
difficulty of these computations was the motivation behind constructing periodic orbit dividing surfaces in
Hamiltonian systems with three or more degrees of freedom. This motivation led to a recent generalization
detailed in a series of papers (refer to [Katsanikas & Wiggins, 2021a,b, 2023a,b]). This innovative approach
found practical application in detecting dynamical matching within a 3D caldera-type Hamiltonian system
(refer to [Katsanikas & Wiggins, 2022; Wiggins & Katsanikas, 2023]). Additionally, alternative methods for
creating dividing surfaces have been introduced utilizing 2D and 3D generating surfaces instead of periodic
orbits. These methods are discussed in recent works such as [Katsanikas & Wiggins, 2024a,b,c].
In this paper, we show how we can apply the classical algorithm for the construction of periodic orbit
dividing surfaces (PODS) [Pechukas & McLafferty, 1973; Pechukas, 1981; Pollak, 1985] to rotating Hamil-
tonian systems with two degrees of freedom (see section 2). We computed the PODS in a rotating quadratic
normal form Hamiltonian system with two degrees of freedom (see section 3). Finally, we presented our
conclusions in the last section. While there has been very little work on phase space dividing surfaces in
rotating systems we refer the reader to [Jaff´e et al., 1999; Salas et al., 2022].
2. PODS in rotating Hamiltonian systems with two degrees of freedom
In this section we will outline the application of the traditional algorithm from [Pechukas & McLafferty,
1973; Pechukas & Pollak, 1979; Pechukas, 1981; Pollak & Pechukas, 1978; Pollak, 1985] within rotating
Hamiltonian systems featuring two degrees of freedom. We consider a 3D system rotating around its z-axis
at an angular speed Ωb, characterized by the potential V(x, y, z). The Hamiltonian governing the motion
of a test-particle in the plane (considering z=pz= 0) can be written in the form:
H=1
2(p2
x+p2
y) + V(x, y)−Ωb(xpy−ypx)
(1)
where V(x, y) is the potential corresponding to the plane (V(x, y ) = V(x, y, z = 0)). This Hamiltonian
describes a general case of a rotating Hamiltonian system with two degrees of freedom. The presence of
the term −Ωb(xpy−ypx) in the Hamiltonian prevents the direct application of the classical algorithm for
the construction of periodic orbit dividing surfaces described in [Pechukas & McLafferty, 1973; Pechukas,
1981; Pollak & Pechukas, 1978; Pollak, 1985]. In this section we reformulate this Hamiltonian into a new
form where this term is eliminated. The new Hamiltonian will be:
H=1
2( ˙x2+ ˙y2) + V(x, y )−1
2Ω2
b(x2+y2)
(2)
This is a Hamiltonian expressed in the coordinates (x, y, ˙x, ˙y), for more information on the nature of
this coordinate transformation see appendix 1. The corresponding equations of motion are:
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Periodic Orbit Dividing Surfaces in Rotating Hamiltonian Systems 3
˙x= ˙x
¨x=−∂V (x, y)
∂x + 2Ωb˙y+ Ω2
bx
˙y= ˙y
¨y=−∂V (x, y)
∂y −2Ωb˙x+ Ω2
by
(3)
Now with the introduction of the new Hamiltonian H(x, y, ˙x, ˙y) and its corresponding equations of
motion, the obstacle posed by the problematic term in the original Hamiltonian has been successfully
eliminated. The hindrance to applying the classical algorithm, as outlined in [Pechukas & McLafferty,
1973; Pechukas & Pollak, 1979; Pechukas, 1981; Pollak & Pechukas, 1978; Pollak, 1985], for constructing
periodic orbit dividing surfaces has been addressed. With this modification we are now able to employ the
classical algorithm to the transformed Hamiltonian, similar to its application in non-rotating Hamiltonian
systems, considering the effective potential as the potential in this case. The effective potential is defined
as:
Vef f (x, y) = V(x, y)−1
2Ω2
b(x2+y2)
(4)
and the new Hamiltonian becomes:
H(x, y, ˙x, ˙y) = 1
2( ˙x2+ ˙y2) + Vef f (x, y)
(5)
The classical algorithm is now applicable to rotating Hamiltonian systems with two degrees of freedom.
The classical algorithm for the case of the effective potential is as follows:
(1) Locate a periodic orbit.
(2) Project the periodic orbit into configuration space.
(3) Choose points on that curve (xi, yi) for i= 1, ...N where Nis the desired number of points. Points are
spaced uniformly according to distance along the periodic orbit.
(4) For each point (xi, yi) determine ˙xmax,i by solving:
H(xi, yi,˙x, 0) = ˙x2
2+Vef f (xi, yi) (6)
for ˙x. Note that a solution of this equation requires E−Vef f (xi, yi)≥0 and that there will be two
solutions, ±˙xmax,i.
(5) For each point (xi, yi) choose points for j= 1, ..., K with ˙x1=−˙xmax,i and ˙xK= ˙xmax, i and solve the
equation H(xi, yi,˙x, ˙y) = Eto obtain ˙y(we will obtain two solutions ˙y, one negative and one positive).
3. PODS in the rotating quadratic normal form Hamiltonian system with two
degrees of freedom
In this section we will apply the classical algorithm of the previous section to the rotating case of the
quadratic normal form Hamiltonian system (considering ω=λ= 1 for the quadratic normal form Hamil-
tonian system with two degrees of freedom in [Katsanikas & Wiggins, 2021a; Ezra & Wiggins, 2018]). This
rotating system belongs to the class of the rotating Hamiltonian systems with two degrees of freedom of
section 2. This system is described by the following Hamiltonian:
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4Katsanikas &Wiggins
H=1
2(p2
x+p2
y) + V(x, y)−Ωb(xpy−ypx)
(7)
where V(x, y) = 1
2(y2−x2) is the potential. If we consider the case with Ω2
b<1, the system has an index-1
saddle (x, px, y, py) = (0,0,0,0)) for E=0 (see the appendix B). If we apply the transformation of the
previous section and we repeat the same procedure, we have the new Hamiltonian:
H=1
2( ˙x2+ ˙y2) + V(x, y )−1
2Ω2
b(x2+y2) = 1
2( ˙x2+ ˙y2) + Vef f (x, y)
(8)
The corresponding equations of motion can be found as an application of the equation 3 of the previous
section:
˙x= ˙x
¨x=x+ 2Ωb˙y+ Ω2
bx
˙y= ˙y
¨y=−y−2Ωb˙x+ Ω2
by
(9)
From the previous equations the equilibrium point in variables (x, y, ˙x, ˙y) is (0,0,0,0) with the corre-
sponding numerical value of Jacobi constant (energy) to be E= 0.
In this system the reaction occurs when the xcoordinate changes sign. The energy surface is:
1
2( ˙x2−x2) + 1
2( ˙y2+y2)−1
2Ω2
b(x2+y2) = E
(10)
The application of the classical algorithm (see the previous section) to the periodic orbits is carried
out naturally with the only difference being that the periodic orbits can be computed numerically. We
compute a family of periodic orbits that are associated with the index-1 saddle using numerical methods
(an iterative Newton-Raphson method - see for example in [Katsanikas & Wiggins, 2023a]). For this reason,
we used the Poincar´e section (y= 0 with ˙y > 0) to compute the periodic orbits. We computed the periodic
orbits, for different values of Ωb= 0.8,0.6,0.4 and 0.2 (see Fig. 1) for a value of energy E= 0.1. We see
that if we increase the value of Ωb, the periodic orbits tend to be extended more in the x-direction. Then
we computed the associated dividing surfaces and we depicted them in the 3D energy surface (x, y, ˙x) in
Fig. 2 (computing ˙yusing the Hamiltonian). We observe that the periodic orbit dividing surfaces have a
toroidal structure and they extend their size as the rotational velocity Ωbincreases (see Fig.3).
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Periodic Orbit Dividing Surfaces in Rotating Hamiltonian Systems 5
Fig. 1. The periodic orbits associated with the index-1 saddle for Ωb= 0.8,0.6,0.4 and 0.2 (that are depicted by purple,green,
cyan and yellow color respectively) in the configuration space.
Fig. 2. The periodic orbit dividing surfaces for Ωb= 0.8,0.6,0.4 and 0.2 (that are depicted in the upper left panel, the upper
right panel, the lower left panel and the lower right panel respectively) in the energy surface.
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6Katsanikas &Wiggins
Fig. 3. The periodic orbit dividing surfaces for Ωb= 0.8,0.6,0.4 and 0.2 (that are depicted by purple,green cyan and yellow
color respectively) in the energy surface.
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Periodic Orbit Dividing Surfaces in Rotating Hamiltonian Systems 7
4. Conclusions
The main conclusions of our paper are:
(1) We demonstrated how to transform a rotating Hamiltonian with two degrees of freedom from
(x, px, y, py) to (x, y, ˙x, ˙y) in order to eliminate problematic terms of the initial Hamiltonian for the
application of the classical algorithm of [Pechukas & McLafferty, 1973; Pechukas & Pollak, 1979; Pollak,
1985; Pollak & Pechukas, 1978].
(2) We presented the reformulation of the classical algorithm for a rotating Hamiltonian system with two
degrees of freedom (see section 2).
(3) We applied the reformulation of the classical algorithm for the rotating Hamiltonian systems with two
degrees of freedom to the rotating case of the quadratic normal form Hamiltonian system with two
degrees of freedom.
(4) The periodic orbit dividing surfaces are toroidal surfaces in the energy surface of the rotating Hamil-
tonian system with two degrees of freedom that we studied.
(5) The periodic orbits and the associated dividing surfaces increase their size as we increase the rotational
velocity of the Hamiltonian system that we studied.
Appendix A Equations of motion for a rotating Hamiltonian system with two degrees
of freedom
We consider a 3D system rotating around its z-axis at an angular speed Ωb, characterized by the
potential V(x, y, z).The Hamiltonian governing the motion of a test-particle in the plane (considering
z=pz= 0) can be written in the form:
H=1
2(p2
x+p2
y) + V(x, y)−Ωb(xpy−ypx)
(A.1)
where V(x,y) is the potential corresponding to the plane (V(x, y) = V(x, y, z = 0)). This Hamiltonian
describes a general case of a rotating Hamiltonian system with two degrees of freedom. We have the
presence of the term −Ωb(xpy−ypx) in the Hamiltonian. In this section, we aim to reformulate this
Hamiltonian into a new form where this term is eliminated. The equations of motion are:
˙x=∂H
∂px
=px+ Ωby
˙px=−∂H
∂x =−∂V (x, y)
∂x + Ωbpy
˙y=∂ H
∂py
=py−Ωbx
˙py=−∂H
∂y =−∂V (x, y)
∂y −Ωbpx
(A.2)
From the above equations we have:
px= ˙x−Ωby
(A.3)
and
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8Katsanikas &Wiggins
py= ˙y+ Ωbx
(A.4)
In fact we will use the following transformation (that is not canonical):
x=x
px= ˙x−Ωby
y=y
py= ˙y+ Ωbx
(A.5)
If we substitute the equations A.3 and A.4 in the initial Hamiltonian (see the equation A.1) we have
the following Hamiltonian:
H=1
2( ˙x2+ ˙y2) + V(x, y )−1
2Ω2
b(x2+y2)
(A.6)
This is a Hamiltonian with (x, y, ˙x, ˙y). We will use the equations A.3 and A.4 to find the Hamilton
equations for ¨xand ¨y. We compute the derivative with respect time of all terms in the first and third
equation of the equations of motion for the initial Hamiltonian system (see the system of equations A.2).
For the first equation we have:
¨x= ˙px+ Ωb˙y
(A.7)
Then we substitute ˙pxusing the second equation of the equations of motion A.2:
¨x=−∂V (x, y)
∂x + Ωbpy+ Ωb˙y
(A.8)
If we replace pywith the expression from equation A.4 and perform the computations, the equation
transforms into:
¨x=−∂V (x, y)
∂x + 2Ωb˙y+ Ω2
bx
(A.9)
We are now employing a comparable approach to determine the equation for ¨y. Specifically, when differ-
entiating all terms in the third equation of motion (see the system of equations A.2) with respect to time
we obtain:
¨y= ˙py−Ωb˙x
(A.10)
Then we substitute ˙pyusing the fourth equation of the equations of motion A.2:
¨y=−∂V (x, y)
∂y −Ωbpx−Ωb˙x
(A.11)
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Periodic Orbit Dividing Surfaces in Rotating Hamiltonian Systems 9
If we replace pxwith the expression from equation A.3 and perform the computations, the equation
transforms into:
¨y=−∂V (x, y)
∂y −2Ωb˙x+ Ω2
by
(A.12)
From the equations A.9 and A.12 we obtain the equations of motion for the new Hamiltonian
H(x, y, ˙x, ˙y). The equations of motion are:
˙x= ˙x
¨x=−∂V (x, y)
∂x + 2Ωb˙y+ Ω2
bx
˙y= ˙y
¨y=−∂V (x, y)
∂y −2Ωb˙x+ Ω2
by
(A.13)
Appendix B The Equilibrium points of the rotating quadratic normal form Hamil-
tonian system
In this appendix we will investigate the stability of the equilibrium point for the general case of the
rotating quadratic normal form Hamiltonian system with two degrees of freedom. We emphasize the fact
that we consider in this paper a special case of this system. We consider a 3D system rotating around its
z-axis at an angular speed Ωb, characterized by the potential V(x, y, z). The Hamiltonian governing the
motion of a test-particle in the plane (considering z=pz= 0) can be written in the form (with ω > 0 and
λ > 0):
H=λ
2(p2
x−x2) + ω
2(p2
y+y2)−Ωb(xpy−ypx)
(B.1)
The equations of motion are:
˙x=∂H
∂px
=λpx+ Ωby
˙px=−∂H
∂x =−∂V (x, y)
∂x + Ωbpy=λx + Ωbpy
˙y=∂ H
∂py
=ωpy−Ωbx
˙py=−∂H
∂y =−∂V (x, y)
∂y −Ωbpx=−ωy −Ωbpx
(B.2)
where V(x, y) = −λ
2x2+ω
2y2The monodromy matrix corresponding to the previous Hamiltonian system
at the equilibrium point (x, px, y, py) = (0,0,0,0) is:
A=
0λΩb0
λ0 0 Ωb
−Ωb0 0 ω
0−Ωb−ω0
(B.3)
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10 Katsanikas &Wiggins
The characteristic polynomial with respect ψis:
p(ψ) = ψ4+ (ω2+ 2Ω2
b−λ2)ψ2+ (Ω4
b−λ2ω2) (B.4)
We solve the corresponding characteristic equation p(ψ) = 0 (ψare the eigenvalues of the monodromy
matrix) and we have the analytical expression:
D=ω4+ 2λ2ω2+λ4+ 4Ω2
bω2−4Ω2
bλ2= (λ2+ω2)2+ 4Ω2
b(ω2−λ2) (B.5)
This analytical expression must be positive D≥0 to have real roots for ψ2. This is a necessary
condition for the equilibrium point to be a saddle. This means that we have two cases:
(1) ω≥λ:
In this case we have two roots for ψ2, These roots are:
ψ2=−(ω2+ 2Ω2
b−λ2)−√D
2
ψ2=−(ω2+ 2Ω2
b−λ2) + √D
2
(B.6)
The first is always negative (the proof is trivial) and the other is positive if the above condition is
satisfied:
Ω2
b<λ2−ω2+√D
2(B.7)
This means that we have two complex roots and two real roots for ψand the equilibrium point is an
index-1 saddle. These roots are:
ψ=±is|−(ω2+ 2Ω2
b−λ2)−√D
2|
ψ=±s−(ω2+ 2Ω2
b−λ2) + √D
2
(B.8)
Now if the above condition B.7 is not satisfied, we have four complex roots or two complex roots and
two zero roots for ψ. The equilibrium point is a stable equilibrium point or a degenerate equilibrium
point (parabolic nature).
(2) ω < λ. In this case, the following condition must be satisfied:
Ω2
b<(ω2+λ2)2
4(λ2−ω2)(B.9)
In this case we have two roots for ψ2, These roots are:
ψ2=−(ω2+ 2Ω2
b−λ2)−√D
2
ψ2=−(ω2+ 2Ω2
b−λ2) + √D
2
(B.10)
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Periodic Orbit Dividing Surfaces in Rotating Hamiltonian Systems 11
The first is positive if the following condition is satisfied:
Ω2
b<λ2−ω2−√D
2(B.11)
The second is positive if the following condition is satisfied:
Ω2
b<λ2−ω2+√D
2(B.12)
If both of the conditions B.11 and B.12 are not satisfied, we have four complex roots or four zero roots
or two zero roots and two complex roots for ψ. This means that the equilibrium point is stable or
degenerated.
If both of the conditions B.11 and B.12 are satisfied, we have four real roots for ψ. This means that
the equilibrium point is an index-2 saddle.
If one of the conditions B.11 and B.12 is satisfied and the other is not satisfied (this means that the
condition B.12 is satisfied) we have two cases. In the first case we have two complex roots and two real
roots for ψand the equilibrium point is an index-1 saddle.
In this case we have:
Ω2
b>λ2−ω2−√D
2(B.13)
The roots are:
ψ=±is|−(ω2+ 2Ω2
b−λ2)−√D
2|
ψ=±s−(ω2+ 2Ω2
b−λ2) + √D
2
(B.14)
In the second case, we will have two real roots and two zero roots for ψ. The equilibrium point is
degenerated.
Conditions for the existence of an index-1 saddle
From the above analysis, we have the presence of an index-1 saddle if we have one of the following
conditions to be satisfied:
First condition:
ω≥λ
(B.15)
and
Ω2
b<λ2−ω2+√D
2(B.16)
Second condition:
ω < λ,
Ω2
b<(ω2+λ2)2
4(λ2−ω2)
(B.17)
and
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12 REFERENCES
Ω2
b<λ2−ω2+√D
2
Ω2
b>λ2−ω2−√D
2
(B.18)
Special case ω=λ:
If we set ω=λ, then the first condition will become Ω2
b< λ2. Under these circumstances, it is observed
that when we assign ω=λ= 1 (as described in this paper), it is imperative to ensure that the condition
Ω2
b<1 is met to establish the equilibrium point as an index-1 saddle.
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.
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