Also at ]Physics Department, XYZ University.

# Sample Title:
with Forced Linebreak

A. Author  [  B. Author<EMAIL_ADDRESS>Authors’ institution
and/or address  C. Author http://www.Second.institution.edu/~Charlie.Author.
Second institution and/or address

###### Abstract

An article usually includes an abstract, a concise summary of the work covered
at length in the main body of the article. It is used for secondary
publications and for information retrieval purposes.

††preprint: AIP/123-QED

> The “lead paragraph” is encapsulated with the LaTeX `quotation` environment
> and is formatted as a single paragraph before the first section heading.
> (The `quotation` environment reverts to its usual meaning after the first
> sectioning command.) Note that numbered references are allowed in the lead
> paragraph. The lead paragraph will only be found in an article being
> prepared for the journal Chaos.

##  I First-level heading:
The line break was forced via \\\

This sample document demonstrates proper use of REVTeX 4.1 (and LaTeX 2ε) in
manuscripts prepared for submission to AIP journals. Further information can
be found in the documentation included in the distribution or available at
http://authors.aip.org and in the documentation for REVTeX 4.1 itself.

When commands are referred to in this example file, they are always shown with
their required arguments, using normal TeX format. In this format, `#1`, `#2`,
etc. stand for required author-supplied arguments to commands. For example, in
`\section{#1}` the `#1` stands for the title text of the author’s section
heading, and in `\title{#1}` the `#1` stands for the title text of the paper.

Line breaks in section headings at all levels can be introduced using \\\\. A
blank input line tells TeX that the paragraph has ended.

###  I.1 Second-level heading: Formatting

This file may be formatted in both the preprint (the default) and reprint
styles; the latter format may be used to mimic final journal output. Either
format may be used for submission purposes; however, for peer review and
production, AIP will format the article using the preprint class option.
Hence, it is essential that authors check that their manuscripts format
acceptably under preprint. Manuscripts submitted to AIP that do not format
correctly under the preprint option may be delayed in both the editorial and
production processes.

The widetext environment will make the text the width of the full page, as on
page 9. (Note the use the `\pageref{#1}` to get the page number right
automatically.) The width-changing commands only take effect in twocolumn
formatting. It has no effect if preprint formatting is chosen instead.

####  I.1.1 Third-level heading: Citations and Footnotes

Citations in text refer to entries in the Bibliography; they use the commands
`\cite{#1}` or `\onlinecite{#1}`. Because REVTeX uses the `natbib` package of
Patrick Daly, its entire repertoire of commands are available in your
document; see the `natbib` documentation for further details. The argument of
`\cite` is a comma-separated list of _keys_ ; a key may consist of letters and
numerals.

By default, citations are numerical; feyn54 author-year citations are an
option. To give a textual citation, use `\onlinecite{#1}`: (Refs. witten2001;
epr; Bire82). REVTeX “collapses” lists of consecutive numerical citations when
appropriate. REVTeX provides the ability to properly punctuate textual
citations in author-year style; this facility works correctly with numerical
citations only with natbib’s compress option turned off. To illustrate, we
cite several together feyn54; witten2001; epr; Berman1983, and once again
(Refs. epr; feyn54; Bire82; Berman1983). Note that, when numerical citations
are used, the references were sorted into the same order they appear in the
bibliography.

A reference within the bibliography is specified with a `\bibitem{#1}`
command, where the argument is the citation key mentioned above.
`\bibitem{#1}` commands may be crafted by hand or, preferably, generated by
using BibTeX. The AIP styles for REVTeX 4 include BibTeX style files
`aipnum.bst` and `aipauth.bst`, appropriate for numbered and author-year
bibliographies, respectively. REVTeX 4 will automatically choose the style
appropriate for the document’s selected class options: the default is
numerical, and you obtain the author-year style by specifying a class option
of `author-year`.

This sample file demonstrates a simple use of BibTeX via a `\bibliography`
command referencing the `aipsamp.bib` file. Running BibTeX (in this case
bibtex aipsamp) after the first pass of LaTeX produces the file `aipsamp.bbl`
which contains the automatically formatted `\bibitem` commands (including
extra markup information via `\bibinfo` commands). If not using BibTeX, the
`thebibiliography` environment should be used instead.

##### Fourth-level heading is run in.

Footnotes are produced using the `\footnote{#1}` command. Numerical style
citations put footnotes into the bibliography111Automatically placing
footnotes into the bibliography requires using BibTeX to compile the
bibliography.. Author-year and numerical author-year citation styles (each for
its own reason) cannot use this method. Note: due to the method used to place
footnotes in the bibliography, _you must re-run BibTeX every time you change
any of your document’s footnotes_.

##  II Math and Equations

Inline math may be typeset using the `$` delimiters. Bold math symbols may be
achieved using the `bm` package and the `\bm{#1}` command it supplies. For
instance, a bold $\alpha$ can be typeset as `$\bm{\alpha}$` giving
$\bm{\alpha}$. Fraktur and Blackboard (or open face or double struck)
characters should be typeset using the `\mathfrak{#1}` and `\mathbb{#1}`
commands respectively. Both are supplied by the amssymb package. For example,
`$\mathbb{R}$` gives $\mathbb{R}$ and `$\mathfrak{G}$` gives $\mathfrak{G}$

In LaTeX there are many different ways to display equations, and a few
preferred ways are noted below. Displayed math will center by default. Use the
class option `fleqn` to flush equations left.

Below we have numbered single-line equations, the most common kind:

$\displaystyle\chi_{+}(p)\lesssim{\bf[}2|{\bf p}|(|{\bf
p}|+p_{z}){\bf]}^{-1/2}\left(\begin{array}[]{c}|{\bf p}|+p_{z}\\\
px+ip_{y}\end{array}\right)\;,$ (3) $\displaystyle\left\\{\openone
234567890abc123\alpha\beta\gamma\delta
1234556\alpha\beta\frac{1\sum^{a}_{b}}{A^{2}}\right\\}.$ (4)

Note the open one in Eq. (4).

Not all numbered equations will fit within a narrow column this way. The
equation number will move down automatically if it cannot fit on the same line
with a one-line equation:

$\left\\{ab12345678abc123456abcdef\alpha\beta\gamma\delta
1234556\alpha\beta\frac{1\sum^{a}_{b}}{A^{2}}\right\\}.$ (5)

When the `\label{#1}` command is used [cf. input for Eq. (4)], the equation
can be referred to in text without knowing the equation number that TeX will
assign to it. Just use `\ref{#1}`, where `#1` is the same name that used in
the `\label{#1}` command.

Unnumbered single-line equations can be typeset using the `\[`, `\]` format:

$g^{+}g^{+}\rightarrow
g^{+}g^{+}g^{+}g^{+}\dots~{},~{}~{}q^{+}q^{+}\rightarrow
q^{+}g^{+}g^{+}\dots~{}.$

###  II.1 Multiline equations

Multiline equations are obtained by using the `eqnarray` environment. Use the
`\nonumber` command at the end of each line to avoid assigning a number:

$\displaystyle{\cal M}=$ $\displaystyle
ig_{Z}^{2}(4E_{1}E_{2})^{1/2}(l_{i}^{2})^{-1}\delta_{\sigma_{1},-\sigma_{2}}(g_{\sigma_{2}}^{e})^{2}\chi_{-\sigma_{2}}(p_{2})$
(6)
$\displaystyle\times[\epsilon_{j}l_{i}\epsilon_{i}]_{\sigma_{1}}\chi_{\sigma_{1}}(p_{1}),$
$\displaystyle\sum|M^{\text{viol}}_{g}|^{2}$ $\displaystyle=$ $\displaystyle
g^{2n-4}_{S}(Q^{2})~{}N^{n-2}(N^{2}-1)$ (7)
$\displaystyle\times\left(\sum_{i<j}\right)\sum_{\text{perm}}\frac{1}{S_{12}}\frac{1}{S_{12}}\sum_{\tau}c^{f}_{\tau}~{}.$

Note: Do not use `\label{#1}` on a line of a multiline equation if `\nonumber`
is also used on that line. Incorrect cross-referencing will result. Notice the
use `\text{#1}` for using a Roman font within a math environment.

To set a multiline equation without _any_ equation numbers, use the
`\begin{eqnarray*}`, `\end{eqnarray*}` format:

$\displaystyle\sum|M^{\text{viol}}_{g}|^{2}$ $\displaystyle=$ $\displaystyle
g^{2n-4}_{S}(Q^{2})~{}N^{n-2}(N^{2}-1)$
$\displaystyle\times\left(\sum_{i<j}\right)\left(\sum_{\text{perm}}\frac{1}{S_{12}S_{23}S_{n1}}\right)\frac{1}{S_{12}}~{}.$

To obtain numbers not normally produced by the automatic numbering, use the
`\tag{#1}` command, where `#1` is the desired equation number. For example, to
get an equation number of (2.6′),

$g^{+}g^{+}\rightarrow
g^{+}g^{+}g^{+}g^{+}\dots~{},~{}~{}q^{+}q^{+}\rightarrow
q^{+}g^{+}g^{+}\dots~{}.$ (2.6′)

A few notes on `\tag{#1}`. `\tag{#1}` requires amsmath. The `\tag{#1}` must
come before the `\label{#1}`, if any. The numbering set with `\tag{#1}` is
transparent to the automatic numbering in REVTeX; therefore, the number must
be known ahead of time, and it must be manually adjusted if other equations
are added. `\tag{#1}` works with both single-line and multiline equations.
`\tag{#1}` should only be used in exceptional case - do not use it to number
all equations in a paper.

Enclosing single-line and multiline equations in `\begin{subequations}` and
`\end{subequations}` will produce a set of equations that are “numbered” with
letters, as shown in Eqs. (8a) and (8b) below:

$\left\\{abc123456abcdef\alpha\beta\gamma\delta
1234556\alpha\beta\frac{1\sum^{a}_{b}}{A^{2}}\right\\},$ (8a)
$\displaystyle{\cal M}=$ $\displaystyle
ig_{Z}^{2}(4E_{1}E_{2})^{1/2}(l_{i}^{2})^{-1}(g_{\sigma_{2}}^{e})^{2}\chi_{-\sigma_{2}}(p_{2})$
(8b) $\displaystyle\times[\epsilon_{i}]_{\sigma_{1}}\chi_{\sigma_{1}}(p_{1}).$

Putting a `\label{#1}` command right after the `\begin{subequations}`, allows
one to reference all the equations in a subequations environment. For example,
the equations in the preceding subequations environment were Eqs. (8).

####  II.1.1 Wide equations

The equation that follows is set in a wide format, i.e., it spans across the
full page. The wide format is reserved for long equations that cannot be
easily broken into four lines or less:

${\cal
R}^{(\text{d})}=g_{\sigma_{2}}^{e}\left(\frac{[\Gamma^{Z}(3,21)]_{\sigma_{1}}}{Q_{12}^{2}-M_{W}^{2}}+\frac{[\Gamma^{Z}(13,2)]_{\sigma_{1}}}{Q_{13}^{2}-M_{W}^{2}}\right)+x_{W}Q_{e}\left(\frac{[\Gamma^{\gamma}(3,21)]_{\sigma_{1}}}{Q_{12}^{2}-M_{W}^{2}}+\frac{[\Gamma^{\gamma}(13,2)]_{\sigma_{1}}}{Q_{13}^{2}-M_{W}^{2}}\right)\;.$
(9)

This is typed to show the output is in wide format. (Since there is no input
line between `\equation` and this paragraph, there is no paragraph indent for
this paragraph.)

##  III Cross-referencing

REVTeX will automatically number sections, equations, figure captions, and
tables. In order to reference them in text, use the `\label{#1}` and
`\ref{#1}` commands. To reference a particular page, use the `\pageref{#1}`
command.

The `\label{#1}` should appear in a section heading, within an equation, or in
a table or figure caption. The `\ref{#1}` command is used in the text where
the citation is to be displayed. Some examples: Section I on page I, Table 1,

Table 1: This is a narrow table which fits into a text column when using twocolumn formatting. Note that REVTeX 4 adjusts the intercolumn spacing so that the table fills the entire width of the column. Table captions are numbered automatically. This table illustrates left-aligned, centered, and right-aligned columns.  Left222Note a. | Centered333Note b. | Right
---|---|---
1 | 2 | 3
10 | 20 | 30
100 | 200 | 300

and Fig. 1.

##  IV Figures and Tables

Figures and tables are typically “floats”; LaTeX determines their final
position via placement rules. LaTeX isn’t always successful in automatically
placing floats where you wish them.

Figures are marked up with the figure environment, the content of which
imports the image (`\includegraphics`) followed by the figure caption
(`\caption`). The argument of the latter command should itself contain a
`\label` command if you wish to refer to your figure with `\ref`.

Import your image using either the graphics or graphix packages. These
packages both define the `\includegraphics{#1}` command, but they differ in
the optional arguments for specifying the orientation, scaling, and
translation of the figure. Fig. 1

Figure 1:  A figure caption. The figure captions are automatically numbered.

is small enough to fit in a single column, while Fig. 2

Figure 2: Use the figure* environment to get a wide figure, spanning the page
in twocolumn formatting.

is too wide for a single column, so instead the figure* environment has been
used.

The analog of the figure environment is table, which uses the same `\caption`
command. However, you should type your caption command first within the table,
instead of last as you did for figure.

The heart of any table is the tabular environment, which represents the table
content as a (vertical) sequence of table rows, each containing a (horizontal)
sequence of table cells. Cells are separated by the `&` character; the row
terminates with `\\`. The required argument for the tabular environment
specifies how data are displayed in each of the columns. For instance, a
column may be centered (`c`), left-justified (`l`), right-justified (`r`), or
aligned on a decimal point (`d`). (Table 2

Table 2: Numbers in columns Three–Five have been aligned by using the “d” column specifier (requires the dcolumn package). Non-numeric entries (those entries without a “.”) in a “d” column are aligned on the decimal point. Use the “D” specifier for more complex layouts.  One | Two | Three | Four | Five
---|---|---|---|---
one | two | three | four | five
He | 2 | $2.77234$ | $45672.$ | $0.69$
C444Some tables require footnotes. | C555Some tables need more than one footnote. | $12537.64$ | $37.66345$ | $86.37$

illustrates the use of decimal column alignment.)

Extra column-spacing may be be specified as well, although REVTeX 4 sets this
spacing so that the columns fill the width of the table. Horizontal rules are
typeset using the `\hline` command. The doubled (or Scotch) rules that appear
at the top and bottom of a table can be achieved by enclosing the tabular
environment within a ruledtabular environment. Rows whose columns span
multiple columns can be typeset using LaTeX’s `\multicolumn{#1}{#2}{#3}`
command (for example, see the first row of Table 3).

Table 3: This is a wide table that spans the page width in twocolumn mode. It is formatted using the table* environment. It also demonstrates the use of \multicolumn in rows with entries that span more than one column. | $D_{4h}^{1}$ | $D_{4h}^{5}$
---|---|---
Ion | 1st alternative | 2nd alternative | lst alternative | 2nd alternative
K | $(2e)+(2f)$ | $(4i)$ | $(2c)+(2d)$ | $(4f)$
Mn |  $(2g)$666The $z$ parameter of these positions is $z\sim\frac{1}{4}$. | $(a)+(b)+(c)+(d)$ | $(4e)$ | $(2a)+(2b)$
Cl | $(a)+(b)+(c)+(d)$ |  $(2g)$777This is a footnote in a table that spans the full page width in twocolumn mode. It is supposed to set on the full width of the page, just as the caption does. | $(4e)^{\text{a}}$ |
He | $(8r)^{\text{a}}$ | $(4j)^{\text{a}}$ | $(4g)^{\text{a}}$ |
Ag |  | $(4k)^{\text{a}}$ |  | $(4h)^{\text{a}}$

The tables in this document illustrate various effects. Tables that fit in a
narrow column are contained in a table environment. Table 3 is a wide table,
therefore set with the table* environment. Lengthy tables may need to break
across pages. A simple way to allow this is to specify the `[H]` float
placement on the table or table* environment. Alternatively, using the
standard LaTeX 2ε package longtable gives more control over how tables break
and allows headers and footers to be specified for each page of the table. An
example of the use of longtable can be found in the file summary.tex that is
included with the REVTeX 4 distribution.

There are two methods for setting footnotes within a table (these footnotes
will be displayed directly below the table rather than at the bottom of the
page or in the bibliography). The easiest and preferred method is just to use
the `\footnote{#1}` command. This will automatically enumerate the footnotes
with lowercase roman letters. However, it is sometimes necessary to have
multiple entries in the table share the same footnote. In this case, create
the footnotes using `\footnotemark[#1]` and `\footnotetext[#1]{#2}`. #1 is a
numeric value. Each time the same value for #1 is used, the same mark is
produced in the table. The `\footnotetext[#1]{#2}` commands are placed after
the tabular environment. Examine the LaTeX source and output for Tables 1 and
4

Table 4: A table with more columns still fits properly in a column. Note that several entries share the same footnote. Inspect the LaTeX input for this table to see exactly how it is done. |  $r_{c}$ (Å) |  $r_{0}$ (Å) | $\kappa r_{0}$ |  |  $r_{c}$ (Å) |  $r_{0}$ (Å) | $\kappa r_{0}$
---|---|---|---|---|---|---|---
Cu | 0.800 | 14.10 | 2.550 | Sn111Here’s the first, from Ref. feyn54. | 0.680 | 1.870 | 3.700
Ag | 0.990 | 15.90 | 2.710 | Pb222Here’s the second. | 0.450 | 1.930 | 3.760
Au | 1.150 | 15.90 | 2.710 | Ca333Here’s the third. | 0.750 | 2.170 | 3.560
Mg | 0.490 | 17.60 | 3.200 | Sr444Here’s the fourth. | 0.900 | 2.370 | 3.720
Zn | 0.300 | 15.20 | 2.970 | Li222Here’s the second. | 0.380 | 1.730 | 2.830
Cd | 0.530 | 17.10 | 3.160 | Na555And etc. | 0.760 | 2.110 | 3.120
Hg | 0.550 | 17.80 | 3.220 | K555And etc. | 1.120 | 2.620 | 3.480
Al | 0.230 | 15.80 | 3.240 | Rb333Here’s the third. | 1.330 | 2.800 | 3.590
Ga | 0.310 | 16.70 | 3.330 | Cs444Here’s the fourth. | 1.420 | 3.030 | 3.740
In | 0.460 | 18.40 | 3.500 | Ba555And etc. | 0.960 | 2.460 | 3.780
Tl | 0.480 | 18.90 | 3.550 |  |  |  |

for an illustration.

All AIP journals require that the initial citation of figures or tables be in
numerical order. LaTeX’s automatic numbering of floats is your friend here:
just put each figure environment immediately following its first reference
(`\ref`), as we have done in this example file.

###### Acknowledgements.

We wish to acknowledge the support of the author community in using REVTeX,
offering suggestions and encouragement, testing new versions, ….

## Data Availability Statement

AIP Publishing believes that all datasets underlying the conclusions of the
paper should be available to readers. Authors are encouraged to deposit their
datasets in publicly available repositories or present them in the main
manuscript. All research articles must include a data availability statement
stating where the data can be found. In this section, authors should add the
respective statement from the chart below based on the availability of data in
their paper.

AVAILABILITY OF DATA |  STATEMENT OF DATA AVAILABILITY
---|---
Data available on request from the authors |  The data that support the findings of this study are available from the corresponding author upon reasonable request.
Data available in article or supplementary material |  The data that support the findings of this study are available within the article [and its supplementary material].
Data openly available in a public repository that issues datasets with DOIs |  The data that support the findings of this study are openly available in [repository name] at http://doi.org/[doi], reference number [reference number].
Data openly available in a public repository that does not issue DOIs |  The data that support the findings of this study are openly available in [repository name], reference number [reference number].
Data sharing not applicable – no new data generated |  Data sharing is not applicable to this article as no new data were created or analyzed in this study.
Data generated at a central, large scale facility |  Raw data were generated at the [facility name] large scale facility. Derived data supporting the findings of this study are available from the corresponding author upon reasonable request.
Embargo on data due to commercial restrictions |  The data that support the findings will be available in [repository name] at [DOI link] following an embargo from the date of publication to allow for commercialization of research findings.
Data available on request due to privacy/ethical restrictions |  The data that support the findings of this study are available on request from the corresponding author. The data are not publicly available due [state restrictions such as privacy or ethical restrictions].
Data subject to third party restrictions |  The data that support the findings of this study are available from [third party]. Restrictions apply to the availability of these data, which were used under license for this study. Data are available from the authors upon reasonable request and with the permission of [third party].

##  Appendix A Appendixes

To start the appendixes, use the `\appendix` command. This signals that all
following section commands refer to appendixes instead of regular sections.
Therefore, the `\appendix` command should be used only once—to set up the
section commands to act as appendixes. Thereafter normal section commands are
used. The heading for a section can be left empty. For example,

    \appendix
    \section{}

will produce an appendix heading that says “APPENDIX A” and

    \appendix
    \section{Background}

will produce an appendix heading that says “APPENDIX A: BACKGROUND” (note that
the colon is set automatically).

If there is only one appendix, then the letter “A” should not appear. This is
suppressed by using the star version of the appendix command (`\appendix*` in
the place of `\appendix`).

##  Appendix B A little more on appendixes

Observe that this appendix was started by using

    \section{A little more on appendixes}

Note the equation number in an appendix:

$E=mc^{2}.$ (10)

###  B.1 A subsection in an appendix

You can use a subsection or subsubsection in an appendix. Note the numbering:
we are now in Appendix B.1.

####  B.1.1 A subsubsection in an appendix

Note the equation numbers in this appendix, produced with the subequations
environment:

$\displaystyle E$ $\displaystyle=$ $\displaystyle mc,$ (11a) $\displaystyle E$
$\displaystyle=$ $\displaystyle mc^{2},$ (11b) $\displaystyle E$
$\displaystyle\gtrsim$ $\displaystyle mc^{3}.$ (11c)

They turn out to be Eqs. (11a), (11b), and (11c).


# Bifurcation of dividing surfaces constructed from period-doubling
bifurcations of periodic orbits in a caldera potential energy surface

Matthaios Katsanikas<EMAIL_ADDRESS>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.  Makrina
Agaoglou<EMAIL_ADDRESS>Instituto de Ciencias Matemáticas, CSIC,
C/Nicolás Cabrera 15, Campus Cantoblanco, 28049 Madrid, Spain  Stephen Wiggins
<EMAIL_ADDRESS>School of Mathematics, University of Bristol,
Fry Building, Woodland Road, Bristol, BS8 1UG, United Kingdom.

###### Abstract

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 extent 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 for 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
case of bifurcations of dividing surfaces.

Keywords: Bifurcation, periodic orbit dividing surfaces, Caldera Potential,
Phase space structure, Chemical reaction dynamics, dynamical astronomy.

##  I 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 [1, 2]. 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 [1] or to the infinity [2]. This kind of systems is
encountered in many organic chemical reactions and it has been studied
recently in many papers [2, 3, 4, 5, 6, 7, 8]. Similar type of potentials can
be encountered in four-armed barred galaxies [9].

The dividing surfaces play important role in the transition state theory (TST)
[10, 11] in chemical reaction dynamics [10, 11], in dynamical astronomy [12]
and in fluid dynamics [13]. 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 [14, 15, 16, 17, 18]. 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 [19] 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 [20, 21].

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 [22, 23, 24, 25, 26, 27, 28], 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
[29, 30, 31] and Morse bifurcations [32] (for more details about the
references about the bifurcations of the transition states and dividing
surfaces see the introduction of [30]).

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 II. Then we
analyze one supercritical and one subcritical period-doubling bifurcation of
one of the families of the well (section III). We present our results about
the corresponding bifurcations of dividing surfaces in section IV and we
discuss our conclusions in section V.

##  II Model

The analytical form of the PES that we study was inspired by [2] and has the
form:

$V(x,y)=c_{1}(x^{2}+y^{2})+c_{2}y-c_{3}(x^{4}+y^{4}-6x^{2}y^{2})$ (1)

For the parameters $c_{1}=5,c_{2}=3,c_{3}=-0.3$, 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

$H(x,y,p_{x},p_{y})=\frac{p_{x}^{2}}{2m}+\frac{p_{y}^{2}}{2m}+V(x,y),$ (2)

where we consider $m$ to be a constant equal to $1$. The corresponding
Hamiltonian vector field (i.e. Hamilton’s equations of motion) is:

$\displaystyle\dot{x}=\frac{\partial H}{\partial p_{x}}=\frac{p_{x}}{m}$ (3)
$\displaystyle\dot{y}=\frac{\partial H}{\partial p_{y}}=\frac{p_{y}}{m}$
$\displaystyle\dot{p_{x}}=-\frac{\partial V}{\partial
x}(x,y)=-(2c_{1}x-4c_{3}x^{3}+12c_{3}xy^{2})$
$\displaystyle\dot{p_{y}}=-\frac{\partial V}{\partial
y}(x,y)=-(2c_{1}y-4c_{3}y^{3}+12c_{3}x^{2}y+c_{2})$

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

Figure 1: Plot of the PES given in Eq. (1). Figure 2: The coordinate $x$ (in
the Poincaré section $y=0$ with $p_{y}>0$) 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.

##  III 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 [33, 34, 35]). 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 [5] 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.

A) B)
C) D)
E) F)
G) H)
I) J)

Figure 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.

A) B)
C) D)
E) F)
G)

Figure 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.

A) B)
C)

Figure 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.

##  IV 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 (see VI).
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 $(y,p_{x},p_{y})$ (see panel B of Fig. 6) and as filamentary
structures in the 3D projections $(x,y,p_{x})$, $(x,y,p_{y})$ and
$(x,p_{x},p_{y})$ 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 $y$-, $p_{x}$\- and $p_{y}$\- 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 $(y,p_{x},p_{y})$ 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
[30] were presented as ellipsoids in the 3D subspace $(x,p_{x},p_{y})$ (and
not in the 3D subspace $(y,p_{x},p_{y})$) 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 [30],
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
$(x,y,p_{x})$ and $(x,y,p_{y})$ (see for example the panels A and B of Fig.7).
These parabolic cylinders have two branches that move towards the negative $y$
semi-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 projection $(x,p_{x},p_{y})$ 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 $(y,p_{x},p_{y})$ 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 $y$ semi-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 the periodic orbits of the
first period-doubling bifurcation with two basic differences:

  1. 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 $y$ semi-axis (see panel A of
Fig. 8).

  2. 2.

The ellipsoidal rings of the second period-doubling bifurcation have the hole
in the opposite direction in the $y$ axis (it is found for negative $y$
values) 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 bifurcations. Firstly, we analyzed the relative positions of
the dividing surfaces in the 3D projection $(x,y,p_{x})$ 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 $(x,y,p_{y})$ because the dividing surfaces of the family of the
well and of the bifurcating families have similar morphology with this in the
3D projection $(x,y,p_{x})$.

Furthermore, we analyzed the relative positions of the dividing surfaces in
the 3D subspace $(x,p_{x},p_{y})$ 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 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 $(y,p_{x},p_{y})$ 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.

A) B)
C) D)
E) F)

Figure 6: The 3D projections $(x,y,p_{x})$ (panels A, C and E for values of
energy 15, 24 and 30 respectively - first column) and and $(y,p_{x},p_{y})$
(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.

A) B)
C) D)
E) F)

Figure 7: The 3D projections $(x,y,p_{x})$ (panels A and B - first row),
$(x,p_{x},p_{y})$ (panels C and D - second row) and $(y,p_{x},p_{y})$ (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.

A) B)
C)

Figure 8: The 3D projections $(x,y,p_{x})$ (panel A), $(x,p_{x},p_{y})$ (panel
B) and $(y,p_{x},p_{y})$ (panel C) of dividing surfaces associated of the
second period-doubling bifurcation for value of energy 30.

A) B)
C) D)

Figure 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 $E=30$).

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

A) B)
C) D)
E) F)

Figure 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.

  * •

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, 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 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.

A)
B)
C)

Figure 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.

A) B)
C) D)
E) F)

Figure 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.

##  V 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. 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. 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.

  3. 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.

##  VI appendix

The construction of periodic orbit dividing surfaces for Hamiltonian systems
with two degrees of freedom was introduced in [14, 15, 36, 16]. The algorithm
for this construction was described in [37, 38, 39]. Here we give a brief
outline of the algorithm for completeness.

  1. 1.

Locate a periodic orbit.

  2. 2.

Project the periodic orbit into configuration space.

  3. 3.

Choose points on that curve $(x_{i},y_{i})$ for $i=1,...N$ where $N$ is the
desired number of points. Points are spaced uniformly according to distance
along the periodic orbit.

  4. 4.

For each point $(x_{i},y_{i})$ determine $p_{xmax,i}$ by solving:

$\displaystyle H(x_{i},y_{i},p_{x},0)=\frac{p_{x}^{2}}{2m}+V(x_{i},y_{i})$ (4)

for $p_{x}$. Note that a solution of this equation requires
$E-V(x_{i},y_{i})\geq 0$ and that there will be two solutions, $\pm
p_{xmax,i}$.

  5. 5.

For each point $(x_{i},y_{i})$ choose points for $j=1,...,K$ with
$p_{x_{1}}=-p_{xmax,i}$ and $p_{x_{K}}=p_{xmax,i}$ and solve the equation
$H(x_{i},y_{i},p_{x},p_{y})=E$ to obtain $p_{y}$ (we will obtain two solutions
$p_{y}$, 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.

## References

  * Carpenter [1985] B. K. Carpenter, Trajectories through an intermediate at a fourfold branch point. implications for the stereochemistry of biradical reactions, Journal of the American Chemical Society 107, 5730 (1985).
  * Collins  _et al._ [2013] P. Collins, B. K. Carpenter, G. S. Ezra, and S. Wiggins, Nonstatistical dynamics on potentials exhibiting reaction path bifurcations and valley-ridge inflection points, The Journal of Chemical Physics 139, 154108 (2013).
  * Katsanikas  _et al._ [2020a] M. Katsanikas, V. J. García-Garrido, and S. Wiggins, Detection of dynamical matching in a caldera Hamiltonian system using lagrangian descriptors, Int. J. Bifurcation Chaos 30, 2030026 (2020a).
  * Katsanikas  _et al._ [2020b] M. Katsanikas, V. J. García-Garrido, and S. Wiggins, The dynamical matching mechanism in phase space for caldera-type potential energy surfaces, Chemical Physics Letters 743, 137199 (2020b).
  * Katsanikas and Wiggins [2018] M. Katsanikas and S. Wiggins, Phase space structure and transport in a caldera potential energy surface, International Journal of Bifurcation and Chaos 28, 1830042 (2018).
  * Katsanikas and Wiggins [2019] M. Katsanikas and S. Wiggins, Phase space analysis of the nonexistence of dynamical matching in a stretched caldera potential energy surface, International Journal of Bifurcation and Chaos 29, 1950057 (2019).
  * Geng  _et al._ [2021a] Y. Geng, M. Katsanikas, M. Agaoglou, and S. Wiggins, The bifurcations of the critical points and the role of the depth in a symmetric caldera potential energy surface, International Journal of Bifurcation and Chaos 31, 2130034, Feature article (2021a).
  * Geng  _et al._ [2021b] Y. Geng, M. Katsanikas, M. Agaoglou, and S. Wiggins, The influence of a pitchfork bifurcation of the critical points of a symmetric caldera potential energy surface on dynamical matching, Chemical Physics Letters 768, 138397 (2021b).
  * Athanassoula  _et al._ [2009] E. Athanassoula, M. Romero-Gómez, A. Bosma, and J. Masdemont, Rings and spirals in barred galaxies–ii. ring and spiral morphology, Monthly Notices of the Royal Astronomical Society 400, 1706 (2009).
  * Wigner [1938] E. Wigner, The transition state method, Transactions of the Faraday Society 34, 29 (1938).
  * Waalkens  _et al._ [2007] H. Waalkens, R. Schubert, and S. Wiggins, Wigner’s dynamical transition state theory in phase space: classical and quantum, Nonlinearity 21, R1 (2007).
  * Reiff  _et al._ [2022] J. Reiff, J. Zatsch, J. Main, and R. Hernandez, On the stability of satellites at unstable libration points of sun–planet–moon systems, Communications in Nonlinear Science and Numerical Simulation 104, 106053 (2022).
  * Bottaro [2019] A. Bottaro, Flow over natural or engineered surfaces: An adjoint homogenization perspective, Journal of Fluid Mechanics 877, 10.1017/jfm.2019.607 (2019).
  * Pechukas and McLafferty [1973] P. Pechukas and F. J. McLafferty, On transition-state theory and the classical mechanics of collinear collisions, The Journal of Chemical Physics 58, 1622 (1973).
  * Pechukas and Pollak [1977] P. Pechukas and E. Pollak, Trapped trajectories at the boundary of reactivity bands in molecular collisions, The Journal of Chemical Physics 67, 5976 (1977).
  * Pechukas and Pollak [1979] P. Pechukas and E. Pollak, Classical transition state theory is exact if the transition state is unique, The Journal of Chemical Physics 71, 2062 (1979).
  * Pollak [1985] E. Pollak, Periodic orbits and the theory of reactive scattering, Theory of Chemical Reaction Dynamics 3, 123 (1985).
  * Pechukas [1981] P. Pechukas, Transition state theory, Annual Review of Physical Chemistry 32, 159 (1981).
  * Wiggins [2016] S. Wiggins, The role of normally hyperbolic invariant manifolds (nhims) in the context of the phase space setting for chemical reaction dynamics, Regular and Chaotic Dynamics 21, 621 (2016).
  * Katsanikas and Wiggins [2021a] M. Katsanikas and S. Wiggins, The generalization of the periodic orbit dividing surface for Hamiltonian systems with three or more degrees of freedom in chemical reaction dynamics - I, International Journal of Bifurcation and Chaos 31, 2130028 (2021a).
  * Katsanikas and Wiggins [2021b] M. Katsanikas and S. Wiggins, The generalization of the periodic orbit dividing surface for Hamiltonian systems with three or more degrees of freedom in chemical reaction dynamics - II, International Journal of Bifurcation and Chaos 31, 2150188 (2021b).
  * Burghardt and Gaspard [1995] I. Burghardt and P. Gaspard, The molecular transition state: From regular to chaotic dynamics, The Journal of Physical Chemistry 99, 2732 (1995).
  * Li  _et al._ [2009] C. B. Li, M. Toda, and T. Komatsuzaki, Bifurcation of no-return transition states in many-body chemical reactions, J. Chem. Phys. 130, 124116 (2009).
  * Inarrea  _et al._ [2011] M. Inarrea, J. F. Palacian, A. I. Pascual, and J. P. Salas, Bifurcations of dividing surfaces in chemical reactions, J. Chem. Phys. 135, 014110 (2011).
  * Founargiotakis  _et al._ [1997] M. Founargiotakis, S. Farantos, C. Skokos, and G. Contopoulos, Bifurcation diagrams of periodic orbits for unbound molecular systems: Fh2, Chemical Physics Letters 277, 456 (1997).
  * Katsanikas  _et al._ [2020c] M. Katsanikas, V. J. García-Garrido, M. Agaoglou, and S. Wiggins, Phase space analysis of the dynamics on a potential energy surface with an entrance channel and two potential wells, Physical Review E 102, 012215 (2020c).
  * Agaoglou  _et al._ [2020] M. Agaoglou, V. J. García-Garrido, M. Katsanikas, and S. Wiggins, The phase space mechanism for selectivity in a symmetric potential energy surface with a post-transition-state bifurcation, Chemical Physics Letters 754, 137610 (2020).
  * Farantos  _et al._ [2009] S. C. Farantos, R. Schinke, H. Guo, and M. Joyeux, Energy localization in molecules, bifurcation phenomena, and their spectroscopic signatures: The global view, Chemical Reviews 109, 4248 (2009).
  * Mauguiere  _et al._ [2013] F. A. Mauguiere, P. Collins, G. S. Ezra, and S. Wiggins, Bifurcations of normally hyperbolic invariant manifolds in analytically tractable models and consequences for reaction dynamics, International Journal of Bifurcation and Chaos 23, 1330043 (2013).
  * Katsanikas  _et al._ [2021] M. Katsanikas, M. Agaoglou, and S. Wiggins, Bifurcation of dividing surfaces constructed from a pitchfork bifurcation of periodic orbits in a symmetric potential energy surface with a post-transition-state bifurcation, International Journal of Bifurcation and Chaos 31, 2130041 (2021).
  * Lyu  _et al._ [2021] W. Lyu, S. Naik, and S. Wiggins, Hamiltonian pitchfork bifurcation in transition across index-1 saddles, Communications in Nonlinear Science and Numerical Simulation 103, 105949 (2021).
  * MacKay and Strub [2014] R. S. MacKay and D. C. Strub, Bifurcations of transition states: Morse bifurcations, Nonlinearity 27, 859 (2014).
  * Rabinowitz [1982] P. H. Rabinowitz, Periodic solutions of Hamiltonian systems: a survey, SIAM Journal on Mathematical Analysis 13, 343 (1982).
  * Weinstein [1973] A. Weinstein, Normal modes for nonlinear Hamiltonian systems, Inventiones Mathematicae 20, 47 (1973).
  * Moser [1976] J. Moser, Periodic orbits near an equilibrium and a theorem by alan weinstein, CPAM 29, 727 (1976).
  * Pollak and Pechukas [1978] E. Pollak and P. Pechukas, Transition states, trapped trajectories, and classical bound states embedded in the continuum, The Journal of Chemical Physics 69, 1218 (1978).
  * Waalkens and Wiggins [2004] H. Waalkens and S. Wiggins, Direct construction of a dividing surface of minimal flux for multi-degree-of-freedom systems that cannot be recrossed, J. Phys. A: Math. Gen. 37, L435 (2004).
  * Ezra and Wiggins [2018] G. S. Ezra and S. Wiggins, Sampling phase space dividing surfaces constructed from normally hyperbolic invariant manifolds (NHIMs), The Journal of Physical Chemistry A 122, 8354 (2018).
  * Haigh  _et al._ [2021] D. Haigh, M. Katsanikas, M. Agaoglou, and S. Wiggins, The time evolution of the trajectories after the selectivity in a symmetric potential energy surface with a post-transition-state bifurcation, Regular and Chaotic Dynamics 26, 763 (2021).