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International Journal of Bifurcation and Chaos

©World Scientiﬁc Publishing Company

Bifurcation of dividing surfaces constructed from period-doubling

bifurcations of periodic orbits in a caldera potential energy surface

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

Makrina Agaoglou

Instituto de Ciencias Matem´aticas, CSIC, C/Nicol´as Cabrera 15, Campus Cantoblanco,28049 Madrid,

Spain

makrina.agaoglou@icmat.es

Stephen Wiggins

School of Mathematics, University of Bristol,

Fry Building, Woodland Road, Bristol, BS8 1UG, United Kingdom.

s.wiggins@bristol.ac.uk

Received (to be inserted by publisher)

In this work we analyze the bifurcation of dividing surfaces that occurs as a result of two

period-doubling bifurcations in a 2D caldera-type potential. We study the structure, the range,

the minimum and maximum extents of the periodic orbit dividing surfaces before and after a

subcritical period-doubling bifurcation of the family of the central minimum of the potential

energy surface. Furthermore, we repeat the same study for the case of a supercritical period-

doubling bifurcation of the family of the central minimum of the potential energy surface. We

will discuss and compare the results for the two cases of bifurcations of dividing surfaces.

Keywords: Bifurcation, periodic orbit dividing surfaces, Caldera Potential, Phase space structure, Chem-

ical reaction dynamics, dynamical astronomy.

1. Introduction

The aim of this paper is to study a bifurcation of periodic orbit dividing surfaces that occurs as a result of a

period-doubling bifurcation of periodic orbits. We studied the structure, the range, and the minimum and

maximum of the dividing surfaces. We study this kind of bifurcation in a caldera type system [Carpenter,

1985; Collins et al., 2013]. The original form of this potential is characterized by one central minimum and

four index-1 saddles around it that control the exit and entrance to four wells [Carpenter, 1985] or to the

inﬁnity [Collins et al., 2013]. This kind of systems is encountered in many organic chemical reactions and

it has been studied recently in many papers [Collins et al., 2013; Katsanikas et al., 2020b,c; Katsanikas &

Wiggins, 2018, 2019; Geng et al., 2021a,b]. Similar type of potentials can be encountered in four-armed

barred galaxies [Athanassoula et al., 2009].

1

February 26, 2022 12:44 katsanikasetal1

2M. Katsanikas et al.

The dividing surfaces play important role in the transition state theory (TST) [Wigner, 1938; Waalkens

et al., 2007] in chemical reaction dynamics [Wigner, 1938; Waalkens et al., 2007], in dynamical astronomy

[Reiﬀ et al., 2022] and in ﬂuid dynamics [Bottaro, 2019]. These surfaces are one less dimension than that of

the potential energy surface. This means that in Hamiltonian systems with two degrees the dividing surfaces

are 2-dimensional objects. We constructed these objects using the classical algorithm of [Pechukas &

McLaﬀerty, 1973; Pechukas & Pollak, 1977, 1979; Pechukas, 1981]. This method is valid only in Hamiltonian

systems with two degrees of freedom. The construction of these objects can be done in Hamiltonian systems

with three or more degrees of freedom, using a higher dimensional object, the Normally Hyperbolic Invariant

Manifold -NHIM (see for example [Wiggins, 2016] and references therein). The construction of these objects,

using periodic orbits, in Hamiltonian systems with three or more degrees of freedom has been done recently

by [Katsanikas & Wiggins, 2021a,b].

In the past, many researchers studied the bifurcations of the transition states (periodic orbits and

NHIMs) that are the basic elements for the construction of dividing surfaces [Burghardt & Gaspard, 1995;

Li et al., 2009; Inarrea et al., 2011; Founargiotakis et al., 1997; Katsanikas et al., 2020a; Agaoglou et al.,

2020; Farantos et al., 2009], but not the bifurcations of the dividing surfaces. The investigation of the

bifurcations of dividing surfaces has been restricted to the case of pitchfork bifurcations [Mauguiere et al.,

2013; Katsanikas et al., 2021; Lyu et al., 2021] and Morse bifurcations [MacKay & Strub, 2014] (for more

details about the references about the bifurcations of the transition states and dividing surfaces see the

introduction of [Katsanikas et al., 2021]).

In this paper, we study for the ﬁrst time the structure of dividing surfaces before and after a period-

doubling bifurcation. Furthermore, we will investigate the range, the minimum, and maximum of the

dividing surfaces before and after a period-doubling bifurcation. We will study all the cases of a period-

doubling bifurcation, the supercritical and subcritical cases. This is the ﬁrst time that it is studied the

structure of the dividing surfaces that are constructed from periodic orbits with high order multiplicity.

The description of the potential energy surface is at section 2. Then we analyze one supercritical and

one subcritical period-doubling bifurcation of one of the families of the well (section 3). We present our

results about the corresponding bifurcations of dividing surfaces in section 4 and we discuss our conclusions

in section 5.

2. Model

The analytical form of the PES that we study was inspired by [Collins et al., 2013] and has the form:

V(x, y) = c1(x2+y2) + c2y−c3(x4+y4−6x2y2) (1)

For the parameters c1= 5, c2= 3, c3=−0.3, the PES has a minimum (well) which is surrounded by 4

index-1 saddles. The energy of the two upper index-1 saddles is higher than the energy of two lower index-1

saddles. These four index-1 saddles control the entrance and the exit from the central area of the caldera.

A three dimensional plot of the PES is given in Fig. 1. The two dimensional (2D) Hamiltonian is given by

H(x, y, px, py) = p2

x

2m+p2

y

2m+V(x, y),(2)

where we consider mto be a constant equal to 1. The corresponding Hamiltonian vector ﬁeld (i.e. Hamilton’s

equations of motion) is:

February 26, 2022 12:44 katsanikasetal1

Bifurcation of Dividing surfaces 3

˙x=∂ H

∂px

=px

m

˙y=∂H

∂py

=py

m

˙px=−∂V

∂x (x, y) = −(2c1x−4c3x3+ 12c3xy2)

˙py=−∂V

∂y (x, y) = −(2c1y−4c3y3+ 12c3x2y+c2)

(3)

Hamilton’s equations conserve the total energy, which we will denote as Ethroughout the paper.

Fig. 1. Plot of the PES given in Eq. (1).

Fig. 2. The coordinate x(in the Poincar´e section y= 0 with py>0) of the periodic orbits of the family of the well and its

bifurcations. The stable and unstable parts of the families are depicted by red and cyan colors, respectively. P1 and P2 are

the points of bifurcation for the supercritical and subcritical cases, respectively.

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4M. Katsanikas et al.

3. Periodic Orbits

As we described in the introduction, we have a central minimum on the potential energy surface of our

model. According to the Lyapunov subcenter theorem we have two families of periodic orbits associated

with this minimum, or ”well”, (see [Rabinowitz, 1982; Weinstein, 1973; Moser, 1976]). In this section, we

describe the period-doubling bifurcations of one of these families of periodic orbits of the well (the family of

periodic orbits with period 1 -see [Katsanikas & Wiggins, 2018] for more details). This family was initially

stable and then it has two period-doubling bifurcations, one supercritical at the point P1 (see Fig. 2 -

at this point the basic family becomes unstable) and one subcritical at the point P2 (see Fig.2 - at this

point the basic family becomes stable). The supercritical and subcritical period-doubling bifurcation will

be referred to in the paper as the ﬁrst and second period-doubling bifurcation of the family of the well,

respectively, this is because of their order of appearance. The periodic orbits of the family of the well are

vertical lines in the conﬁguration space (see Fig. 3), The length of these lines is increased as the energy

increases (see Fig. 3). The periodic orbits of the ﬁrst and second period-doubling bifurcations have a similar

geometry in the conﬁguration space with that of horseshoe curves (see Figs.4, 5). These curves are directed

downward and upward in the case of the ﬁrst and second period-doubling bifurcations, respectively. The

extension of these curves in the x−and y−directions increases as the energy increases.

4. Results

Here we focus our study initially on the evolution of the structure of the dividing surfaces before and

after a supercritical and a subcritical period-doubling bifurcation of one family of the well of our model.

The algorithm for the construction of the dividing surfaces was given in the appendix. First we study the

structure of the dividing surfaces that correspond to the periodic orbits of the family of the well before

the ﬁrst period-doubling bifurcation (supercritical - before the point P1 in Fig. 2). Second, we study the

structure of the dividing surfaces associated with the periodic orbits of the ﬁrst period-doubling bifurcation

and their position with respect to the position of the dividing surfaces associated with the periodic orbits of

the family of the well (after the point P1 and before the point P2 in Fig. 2). Then, we study the structure

of dividing surfaces of the second period-doubling bifurcation(subcritical - after the point P2 in Fig. 2) and

their position with respect to the dividing surfaces with respect to the position of the dividing surfaces

associated with the periodic orbits of the family of the well and the ﬁrst period-doubling bifurcation.

Moreover, we study the range, the minimum, and maximum of the dividing surfaces as the energy varies

in all cases.

First, we study the structure of dividing surfaces of the periodic orbits of the well before the two

period-doubling bifurcations. We see that the dividing surfaces (for value of energy 15) are presented as

ellipsoids in the 3D projection (y, px, py) (see panel B of Fig. 6) and as ﬁlamentary structures in the 3D

projections (x, y, px), (x, y, py) and (x, px, py) of the phase space (see for example panel A of Fig. 6). Then

as we increase the energy we see that the ﬁlamentary structures extend more and more in the y-, px- and

py- directions as we increase the energy (see panels C and E of Fig. 6 for values of energy above the two

period-doubling bifurcations). We observe the same extension in these directions for the ellipsoids in the

3D subspace (y, px, py) of the phase space (see the panels D and F of Fig. 6). We note that the dividing

surfaces of the period-1 family in the Hamiltonian model that we studied in [Katsanikas et al., 2021]

were presented as ellipsoids in the 3D subspace (x, px, py) (and not in the 3D subspace (y, px, py)) and as

ﬁlamentary structures in the other 3D subspaces of the phase space. This happens because the periodic

orbits of the family of the well (that we study in this paper) lie on the y-axis in the conﬁguration space

and the periodic orbits of the basic family (family of the lower saddles) of the system, that we studied in

[Katsanikas et al., 2021], lie on the x- axis in the conﬁguration space.

Then, we study the structure of the dividing surfaces of the periodic orbits of the ﬁrst period-doubling

bifurcation of the family of the well. These dividing surfaces are presented as parabolic cylinders in 3D

projections (x, y, px) and (x, y, py) (see for example the panels A and B of Fig.7). These parabolic cylinders

have two branches that move towards the negative ysemi-axis. As we increase the energy we see that

these branches become more open (they increase the distance between each other in the x-direction).

Furthermore, the dividing surfaces have similar morphology with this of cylinders without holes in the 3D

February 26, 2022 12:44 katsanikasetal1

Bifurcation of Dividing surfaces 5

A) B)

C) D)

E) F)

G) H)

I) J)

Fig. 3. 2D projections of the periodic orbits of the family of the well in the conﬁguration space for energies A) 0.1 B) 5 C)

10 D) 15 E) 20 F) 25 G) 28 H) 30 I) 32 J) 34.

February 26, 2022 12:44 katsanikasetal1

6M. Katsanikas et al.

A) B)

C) D)

E) F)

G)

Fig. 4. 2D projections of the periodic orbits of the ﬁrst period-doubling bifurcation of the family of the well in the conﬁguration

space for energies A) 22 B) 24 C) 26 D) 28 E) 30 F) 32 G) 34.

projection (x, px, py) of the phase space (see panels C and D of Fig. 7). These cylinders are distorted at the

central area close to the plane x= 0. This distortion becomes smaller as we increase the energy (compare

the panels C and D of Fig. 7). The length of these cylinders in the x-direction becomes larger as the energy

increases (compare the panels C and D of Fig. 7). The dividing surfaces of the periodic orbits of the ﬁrst

period-doubling bifurcation are represented as ellipsoidal rings in the 3D subspace (y, px, py) of the phase

space (see panels E and F of Fig. 7). These ellipsoidal rings are characterized as ellipsoids with a hole (that

is found in the positive ysemi-axis) that becomes larger as the energy increases (compare the panels E

and F of Fig. 7).

The next step was to study the structure of dividing surfaces of the periodic orbits of the second

period-doubling bifurcation. These objects (see Fig. 8) have the same topology as the dividing surfaces of

February 26, 2022 12:44 katsanikasetal1

Bifurcation of Dividing surfaces 7

A) B)

C)

Fig. 5. 2D projections of the periodic orbits of the second period-doubling bifurcation of the family of the well in the

conﬁguration space for energies A) 30 B) 32 C) 34.

the periodic orbits of the ﬁrst period-doubling bifurcation with two basic diﬀerences:

(1) The branches of the parabolic cylinders have diﬀerent directions from those of the parabolic cylinders

that we described at the ﬁrst period-doubling bifurcation. They are extended in the positive ysemi-axis

(see panel A of Fig. 8).

(2) The ellipsoidal rings of the second period-doubling bifurcation have the hole in the opposite direction

in the yaxis (it is found for negative yvalues) to the direction of the ellipsoidal rings of the ﬁrst

period-doubling bifurcation (see the panel C of Fig. 8 and compare it with the panels E and F of Fig.

7).

We investigated the relative positions of the dividing surfaces after the two period-doubling bifurca-

tions. Firstly, we analyzed the relative positions of the dividing surfaces in the 3D projection (x, y, px)

after the ﬁrst and second period-doubling bifurcations. After the ﬁrst period-doubling bifurcation, we see

the branches of the parabolic cylinder of the periodic orbits of the ﬁrst period-doubling bifurcation to be

extended on both sides of the ﬁlamentary structure of the family of periodic orbits of the family of the

well (see panel A of Fig. 9). After the second period-doubling bifurcation, except the parabolic cylinder of

the ﬁrst period-doubling bifurcation and the ﬁlamentary structure of the family of the well, we see another

parabolic cylinder, associated with the periodic orbits of the second period-doubling bifurcation, whose

branches are extended on both sides of the ﬁlamentary structure (see panel B of Fig. 9). These branches

lie in the opposite direction to that of the branches of the parabolic cylinder of the ﬁrst period-doubling

bifurcation. The same situation occurs in the 3D projection (x, y, py) because the dividing surfaces of the

family of the well and of the bifurcating families have similar morphology with this in the 3D projection

(x, y, px).

Furthermore, we analyzed the relative positions of the dividing surfaces in the 3D subspace (x, px, py)

of the phase space. In this 3D subspace, we see the ﬁlamentary structure (associated with the family of

the well), that lies in the plane x= 0, to be at the central area of the distorted cylinder without holes,

associated with the periodic orbits of the ﬁrst period-doubling bifurcation (see panel C of Fig. 9). After the

second period-doubling bifurcation, we see a second distorted cylinder without holes (associated with the

periodic orbits of the second period-doubling bifurcation - see panel D Fig. 9). We see that the ﬁlamentary

structure (associated with the family of the well) is at the central area of both distorted cylinders (without

February 26, 2022 12:44 katsanikasetal1

8M. Katsanikas et al.

A) B)

C) D)

E) F)

Fig. 6. The 3D projections (x, y, px) (panels A, C and E for values of energy 15, 24 and 30 respectively - ﬁrst column) and

and (y, px, py) (panels B, D and F for values of energy 15, 24 and 30 respectively - second column) of the dividing surfaces

associated with the periodic orbits of the family of the well.

holes). This can explain why the cylinders, associated with the bifurcating families, are distorted at their

central area in the plane x= 0. The dividing surfaces associated with the family of the well and the

dividing surfaces of the bifurcating families are represented in the 3D subspace (y, px, py) by an ellipsoid

and ellipsoidal rings, respectively (as we explained above). These objects occupy the same volume in this

3D subspace of the phase space.

Finally, we also studied other features of dividing surfaces like their minimum, maximum and the

range:

•Maximum and Minimum of the dividing surfaces: In Figure 12 we visualize the minimum and

maximum of the dividing surfaces with respect to the X coordinate in the panels A and B respectively,

February 26, 2022 12:44 katsanikasetal1

Bifurcation of Dividing surfaces 9

A) B)

C) D)

E) F)

Fig. 7. The 3D projections (x, y, px) (panels A and B - ﬁrst row), (x, px, py) (panels C and D - second row) and (y, px, py)

(panels E and F - third row) of the dividing surfaces associated with the periodic orbits of the ﬁrst period-doubling bifurcation.

The ﬁrst column (panels A, C and E) and the second column (panels B, D and F) are for values of energy 24 and 30 respectively.

to the Y coordinate in the panels C and D and ﬁnally the minimum and maximum to the PX coordinate

in the panels E and F. Notice that the graphs for the minimum and maximum with respect to the PY

coordinate are missing since they can be determined from the other three coordinates (X,Y,PX) of the

Hamiltonian of the system. All families are depicted in green (the family of the well), in red (the ﬁrst

period-doubling bifurcation) and in black ( the second), versus the energy. In the panels A, C and E we

notice that the minimum of the X, Y and PX coordinates respectively of all families is decreasing as we

increase the energy. Analogously, in panels B, D and F the maximum of the X, Y and PX coordinates

respectively of all families is increasing as we increase the energy. Finally it is important to observe that

the minimum and maximum with respect to the X coordinate of the green family (central family) is zero

for all energies. In all other cases the two bifurcating families are represented as branches of the initial

February 26, 2022 12:44 katsanikasetal1

10 M. Katsanikas et al.

A) B)

C)

Fig. 8. The 3D projections (x, y, px) (panel A), (x, px, py) (panel B) and (y, px, py) (panel C) of dividing surfaces associated

of the second period-doubling bifurcation for value of energy 30.

curve that corresponds to the family of the well.

•The range of the dividing surfaces: In Figure 11 we present the range of the dividing surfaces with

respect to the X, Y and PX coordinate respectively of all families in green color (family of the well), red

color (ﬁrst period-doubling bifurcation) and black color (second period-doubling bifurcation). We notice

that the range with respect to all coordinates of all families is increasing as we increase the energy. The

only exception is in the panel A where the range of the central family (green family) of the X coordinate

is zero for all energies. In all other cases the two bifurcating families are represented as branches of the

initial curve that corresponds to the family of the well.

5. Conclusions

We study the geometrical structure of the dividing surfaces, i.e., their minimum, maximum and range

before and after the supercritical and subcritical period-doubling bifurcation of the family of the well

of our model (a caldera-type potential). The structure and the relative position of the dividing surfaces

associated with the period-doubling bifurcations in the 3D subspaces of the phase space are:

(1) Parabolic cylinders that have two branches to be extended on both sides of the dividing surfaces

associated with the periodic orbits of the initial family. The diﬀerence between the two period-doubling

bifurcations for the ﬁrst (supercritical) and the second (subcritical) is that the branches of the parabolic

cylinders are in opposite directions.

(2) Ellipsoidal rings that have a hole. The diﬀerence between the two period-doubling bifurcations, the

ﬁrst (supercritical) and the second (subcritical), is that the hole of the ellipsoidal rings is on opposite

sides of the ellipsoidal ring.

February 26, 2022 12:44 katsanikasetal1

Bifurcation of Dividing surfaces 11

A) B)

C) D)

Fig. 9. The relative positions of the dividing surfaces associated with the periodic orbits of the well (with green color) and the

periodic orbits of the ﬁrst period-doubling bifurcation of the family of the well (with red color) after the ﬁrst period-doubling

bifurcation (for value of energy E= 24). The relative positions of the dividing surfaces associated with the periodic orbits of

the well (with green color), the periodic orbits of the ﬁrst period-doubling bifurcation of the family of the well (with red color)

and the periodic orbits of the second period-doubling bifurcation of the family of the well (with black color) after the second

period-doubling bifurcation (for value of energy E= 30).

(3) Distorted cylinders without holes. The dividing surfaces of the initial family are at the central region

of these cylinders. In this region, we have a distortion of the surface of these cylinders.

The range, minimum, and maximum of the dividing surfaces of bifurcating families versus energy are

represented as branches of the range, minimum and maximum of the dividing surfaces associated with the

initial family.

Appendix A

The construction of periodic orbit dividing surfaces for Hamiltonian systems with two degrees of freedom

was introduced in [Pechukas & McLaﬀerty, 1973; Pechukas & Pollak, 1977; Pollak & Pechukas, 1978;

Pechukas & Pollak, 1979]. The algorithm for this construction was described in [Waalkens & Wiggins, 2004;

Ezra & Wiggins, 2018; Haigh et al., 2021]. Here we give a brief outline of the algorithm for completeness.

(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 pxmax,i by solving:

February 26, 2022 12:44 katsanikasetal1

12 M. Katsanikas et al.

A) B)

C) D)

E) F)

Fig. 10. Diagrams for the A) minimum and B) maximum of the X coordinate of the family of the well (green), ﬁrst period

doubling bifurcation (red) and second period doubling bifurcation (black) families versus the energy, C) minimum and D)

maximum of the Y coordinate of all families versus the energy and E) minimum and F) maximum of the PX coordinate of all

families versus the energy.

H(xi, yi, px,0) = p2

x

2m+V(xi, yi) (A.1)

for px. Note that a solution of this equation requires E−V(xi, yi)≥0 and that there will be two

solutions, ±pxmax,i.

(5) For each point (xi, yi) choose points for j= 1, ..., K with px1=−pxmax,i and pxK=pxmax,i and solve

the equation H(xi, yi, px, py) = Eto obtain py(we will obtain two solutions py, one negative and one

positive).

This algorithm gives produces a phase space dividing surface with the desired properties. It satisﬁes

the no-recrossing property and the phase space ﬂux across the dividing surface is a minimum with respect

to other possible dividing surfaces.

Acknowledgments

The authors acknowledge the ﬁnancial support provided by the EPSRC Grant No. EP/P021123/1

and MA acknowledges support from the grant CEX2019-000904-S and IJC2019-040168-I funded by:

MCIN/AEI/ 10.13039/501100011033.

February 26, 2022 12:44 katsanikasetal1

Bifurcation of Dividing surfaces 13

A)

B)

C)

Fig. 11. A) Range of the X coordinate of the family of the well (green line), the ﬁrst period doubling family (red line) and

second period-doubling bifurcation family (black line) versus the energy, B) Range of the Y coordinate of all families versus

the energy and C) Range of the PX coordinate of all families, versus the energy.

February 26, 2022 12:44 katsanikasetal1

14 M. Katsanikas et al.

A) B)

C) D)

E) F)

Fig. 12. Diagrams for the A) minimum and B) maximum of the X coordinate of the family of the well (green), ﬁrst period

doubling bifurcation (red) and second period doubling bifurcation (black) families versus the energy, C) minimum and D)

maximum of the Y coordinate of all families versus the energy and E) minimum and F) maximum of the PX coordinate of all

families versus the energy.

February 26, 2022 12:44 katsanikasetal1

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