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Error code: DatasetGenerationError
Exception: ArrowInvalid
Message: JSON parse error: Missing a closing quotation mark in string. in row 0
Traceback: Traceback (most recent call last):
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 145, in _generate_tables
dataset = json.load(f)
File "/usr/local/lib/python3.9/json/__init__.py", line 293, in load
return loads(fp.read(),
File "/usr/local/lib/python3.9/json/__init__.py", line 346, in loads
return _default_decoder.decode(s)
File "/usr/local/lib/python3.9/json/decoder.py", line 340, in decode
raise JSONDecodeError("Extra data", s, end)
json.decoder.JSONDecodeError: Extra data: line 2 column 1 (char 31587)
During handling of the above exception, another exception occurred:
Traceback (most recent call last):
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1995, in _prepare_split_single
for _, table in generator:
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 148, in _generate_tables
raise e
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/packaged_modules/json/json.py", line 122, in _generate_tables
pa_table = paj.read_json(
File "pyarrow/_json.pyx", line 308, in pyarrow._json.read_json
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The above exception was the direct cause of the following exception:
Traceback (most recent call last):
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1529, in compute_config_parquet_and_info_response
parquet_operations = convert_to_parquet(builder)
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1154, in convert_to_parquet
builder.download_and_prepare(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1027, in download_and_prepare
self._download_and_prepare(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1122, in _download_and_prepare
self._prepare_split(split_generator, **prepare_split_kwargs)
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1882, in _prepare_split
for job_id, done, content in self._prepare_split_single(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 2038, in _prepare_split_single
raise DatasetGenerationError("An error occurred while generating the dataset") from e
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text
string | meta
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\section{Introduction}
Partial gapping of spectral weight in absence of any metal instability appears in
many strongly correlated electron systems~\cite{mannellaNAT2005,borisenkoPRL2008,uchidaPRL2011,ChandPRB2012}. This so-called pseudogap phenomenon
is, for example, found in the normal state of charge-density-wave (CDW) systems, above
the CDW onset temperature~\cite{borisenkoPRL2009}. A pseudogap phase has also been reported in
the normal state of high-temperature cuprate superconductors.
The nature of these pseudogaps is still being debated~\cite{normanAP2005,chatterjeeNATPHYS2010,jleeSCIENCE2009,rdaouNAT2010,mhashimotoNATPHYS2010,tkondoNAT2009,tkondoNATPHYS2011,tkondoPRL2013,skawasakiPRL2010,yheSCIENCE2014}.
Recently, it has become
clear that charge ordering is a universal property of hole doped
cuprates~\cite{twuNAT2011,twuNATCOMM2013,jchangNATPHYS2012,ghiringhelliSCIENCE2012,achkarPRL2012,
huckerPRB2014,blancocanosaPRB2014,doironleyraudPRX2013,tabisNATCOMM2014,dasilvanetoSCIENCE2014,cominSCIENCE2014,fujitaSCIENCE2014,christensenARXIV2014,thampyPRB2014,croftPRB2014}.
Around the so-called 1/8-doping, the CDW onset temperature appears
much before the superconducting transition temperature.
The normal state of cuprates should hence be revisited to
identify a single particle gap from CDW order and
to investigate the spectral gapping in absence of both superconductivity and CDW order.
We therefore present an angle-resolved photoemission spectroscopy (ARPES) study of
the well-known charge stripe ordered system La$_{1.6-x}$Nd$_{0.4}$Sr$_x$CuO$_4$ (Nd-LSCO),
in which charge and spin orders are coupled~\cite{tranquadaNAT1995,christensenPRL2007}.
As shown in the phase diagram (Fig.~1), this material has a strongly suppressed superconducting transition temperature, which
allows
a low temperature study of the normal state.
We have studied the spectral lineshape evolution as a function
of momentum, temperature and doping.
On the overdoped side, Nd-LSCO $p=0.20$, an antinodal spectral gap is observed.
This gap can be closed by either increasing doping to $p=0.24$, increasing temperature
to $T\sim 80$~K or moving in momentum towards the zone diagonal.
The normal state gap $\Delta$ redistributes spectral weight up to
$\sim 2.5\Delta$, but the total weight remains conserved.
Analysis of the spectral lineshape suggests a correlation between the
gap amplitude and electron scattering.
In the underdoped regime $p<0.15$, the antinodal lineshape changes.
Compared to the overdoped side of the phase diagram, a significant
suppression of spectral weight is observed. This effect is
discussed in terms of quasiparticle decoherence and competing orders.
In particular, the idea that charge stripe order can contribute to the
suppression of antinodal spectral weight is discussed.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.42\textwidth]{FIG1.eps}
\end{center}
\caption{(Color online) (a) Temperature-doping phase diagram of La$_{1.6-x}$Nd$_{0.4}$Sr$_x$CuO$_4$ (Nd-LSCO),
established by diffraction and resistivity experiments~\cite{rdaouNAT2009,cyrchoinierePHYSC2010,tranquadaNAT1995,swakimotoPRB2003,finkPRB2011}.
The temperature scale $T_\rho$ is determined by
the deviation from high-temperature linear
resistivity~\cite{rdaouNAT2009}. The charge ordering temperature ($T_{ch}$) is
obtained from x-ray diffraction~\cite{tranquadaNAT1995,swakimotoPRB2003,finkPRB2011}.
All lines are guides to the eye.
(b)
Charge stripe order parameter $\Delta_{ch}$, derived from hard x-ray diffraction
experiments on La$_{2-x}$Ba$_x$CuO$_4$ (LBCO)~\cite{huckerPRB2013}.
(c) Leading edge gap of LBCO versus doping, from Ref.~\onlinecite{VallaSCIENCE2006}. }
\label{fig:fig1}
\end{figure}
\begin{figure*}[!]
\begin{center}
\includegraphics[width=0.75\textwidth]{FIG2.eps}
\end{center}
\caption{(Color online) (a)-(d) Anti-nodal angle-resolved photoemission spectra, taken in the normal state of
La$_{1.6-x}$Nd$_{0.4}$Sr$_x$CuO$_4$ for different dopings $p=x$ as indicated. Solid white points are momentum disctribution
curves at the Fermi level, indicated by horizontal dashed lines. Top panels schematically show the Fermi
surface topology for each of the doping concentrations.
The red lines indicate the trajectory along which the anti-nodal spectra
were recorded. Solid black points indicate the underlying Fermi momenta
at which symmetrized EDCs are shown in Fig. 3(c-d). }
\label{fig:fig2}
\end{figure*}
\section{Methods}
Our ARPES experiments were carried out at the Swiss Light Source (SLS)
on the Surface and Interface Spectroscopy (SIS) beam line,~\cite{FlechsigAIPCP2004} using 55 eV circular polarized
photons. Single crystals of Nd-LSCO with $x=p=0.12$, 0.15, 0.20 and 0.24 -- grown by the traveling zone
method -- were cleaved \textit{in-situ} under ultra-high vacuum (UHV) conditions ($\sim0.5 \times 10^{-10}$ mbar)
using a top-post technique or a specially designed cleaving tool~\cite{manssonREVSI2007}.
Photo-emitted electrons were analyzed using
a SCIENTA 2002 or a R4000 analyzer. A total energy resolution of $\sim15$ meV was achieved
with this setup.
Due to matrix element effects, all data were recorded in the second Brillouin zone
but represented by the equivalent points in the first zone.
The Fermi level was measured on poly-crystalline copper in thermal and electric contact with the sample.
Copper spectra were also used to normalize detector efficiencies.\\
\section{Results}
Normal state ($T\gtrsim T_c$) energy distribution maps taken in the anti-nodal
($\pi$,0)-region of Nd-LSCO $x=p=0.12$, 0.15, 0.20, and 0.24
are shown in Fig.~\ref{fig:fig2}.
As doping $p$ is reduced, the
"quasiparticle" excitations are gradually broadened.
Finite spectral weight at the Fermi level $E_F$ ($\omega=0$) is, however, found for all compositions even
deep inside the charge stripe ordered phase~\cite{jchangNEWJP2008}.
It is thus possible to define the
underlying Fermi momenta $k_F$
from the maximum intensity of the momentum distribution curves (MDC) at $\omega=0$.
The Nd-LSCO Fermi surface topology~\cite{ClaessonPRB2009}, shown schematically in Fig.~\ref{fig:fig2},
is similar to that of La$_{2-x}$Sr$_x$CuO$_4$ (LSCO)~\cite{tyoshidaPRB2006,razzoliNEWJP2010} and Bi2212~\cite{KaminskiPRB06,BenhabibPRL15}.
A van-Hove singularity crosses $E_F$ at a doping concentration slightly larger than $x = p = 0.20$,
separating electron- from hole-like Fermi surfaces.
\subsection{Spectral lineshapes}
Analysis of symmetrized energy distribution
curves (EDCs) at $k = k_F$ is a standard method to visualize the existence of
a spectral gap near the Fermi level~\cite{normanNAT1998}.
A single-particle gap shifts the spectral weight away
from the Fermi level and hence produces a double peak structure
in the symmetrized curves. In absence of a spectral gap,
the symmetrized EDC at $k_F$ is on the contrary characterized
by a lineshape peaked at the Fermi level.
For overdoped LSCO and Nd-LSCO $p \sim 0.24$, the anti-nodal spectra
have a Voigt-like profile (see top spectrum of Fig.~\ref{fig:fig3}a,b) just above $T_c$, suggesting
resolution limited gapless excitations.
At slightly lower doping in Nd-LSCO $p=0.20$, a clear spectral gap $\Delta\sim 25-30$ meV is found in the anti-nodal region for $T \sim T_c$ (Fig.~\ref{fig:fig3}b).
Similar line-shapes of the ARPES spectra were obtained on
Nd-LSCO $p\sim0.15$ and LSCO with $p=0.105, 0.12$ and 0.15, see Fig.~\ref{fig:fig3}a,b.
As in Bi2212 and Bi2201~\cite{mhashimotoNATPHYS2014,chatterjeePNAS2011,ktanakaSCIENCE2006}, a dramatic change of anti-nodal line shape appears for underdoped Nd-LSCO (Fig.~\ref{fig:fig3}b.).
The peaked lineshape structure -- found for
Nd-LSCO $p=0.15$ and 0.20 -- is strongly depleted.
\begin{figure*}
\begin{center}
\includegraphics[width=0.87\textwidth]{FIG3v6.eps}
\end{center}
\caption{(Color online) Symmetrized normal state energy distribution curves (EDCs) recorded on La$_{2-x}$Sr$_x$CuO$_4$ (LSCO) and La$_{1.6-x}$Nd$_{0.4}$Sr$_x$CuO$_4$ (Nd-LSCO).
All spectra were taken just above $T_c$. In top panels (a)-(f) are raw symmetrized spectra while in bottom panels (g)-(l) are
background subtracted spectra. (a-b) Symmetrized EDCs taken in the anti-nodal region, for doping concentrations of LSCO and Nd-LSCO as indicated. ARPES data on LSCO $x=0.105$ and 0.145
were previously presented in Ref.~\onlinecite{mshiPRL2008,mshiEPL2009,ChangPRB2008ARPES} and all LSCO samples were characterized by neutron scattering experiments~\cite{ChangPRL2007,ChangPRB2008,ChangPRB2012}.
(c-d) Momentum dependence of symmetrized energy distribution curves (EDCs) taken at $k_F$ moving from anti-nodal (bottom) to nodal
(top) region for Nd-LSCO $p=0.12$ and 0.20.
(e-f) Temperature dependence of anti-nodal symmetrized EDCs recorded on
Nd-LSCO $p=0.12$ and 0.20.
For clarity, each spectrum has been given an arbitrary vertical shift. Solid lines in bottom panels
are fits, see text for an explanation. }
\label{fig:fig3}
\end{figure*}
A similar evolution of the line-shape is found when moving
from the anti-nodal to the nodal region in Nd-LSCO at $p=0.12$ (Fig.~\ref{fig:fig3}c).
It resembles the doping dependence (Fig.~\ref{fig:fig3}b):
first the double-peaked structure is recovered and second, upon entering the Fermi arc,
gapless excitations are found~\cite{jchangNEWJP2008}.
For comparison, the momentum dependence
of the EDC lineshapes in Nd-LSCO $p=0.20$ is shown in Fig.~\ref{fig:fig3}d.
At this doping, a peaked structure is found
for all underlying Fermi momenta.
(see Fig.~\ref{fig:fig3}d). The temperature dependence of antinodal spectra
are also very different in Nd-LSCO $p=0.12$ and 0.20 -- see Fig.~\ref{fig:fig3}(e,f) and \ref{fig:fig5}.
For $p=0.20$, the normal state gap closes at $T\approx80$~K, while it persists in
the stripe order $p=0.12$ compound. Furthermore, the peaked structure
in the symmetrized EDC lineshape becomes more pronounced in $p=0.20$ upon cooling (Fig.~\ref{fig:fig3}f).
The opposite trend is observed at 0.12 doping. In fact, as in Bi2201~\cite{mhashimotoNATPHYS2014},
a much sharper anti-nodal line-shape is found at 75~K compared to 17~K.
Finally, the spectral gap in $p=0.20$ seems to conserve
but redistribute the spectral weight (Fig.~\ref{fig:fig5}) as it opens
upon cooling. In contrast, for underdoped Nd-LSCO $p=0.12$, spectral weight is
either lost or redistributed in a non-trivial fashion upon cooling.
The anti-nodal spectra at the anomalous 1/8 doping are thus
behaving very differently from what is found in more overdoped
samples of Nd-LSCO. The 1/8 anti-nodal spectra are also very
different from what is observed in LSCO at similar doping (Fig.~3).
\subsection{Background subtraction}
The raw spectra, described above, are composed of an intrinsic signal on
top of an extrinsic background. Importantly,
the extrinsic background has essentially the same profile for all measured
compounds. It is therefore possible to normalize spectral
intensities relatively to the extrinsic background - see Appendix.
Anti-nodal spectra were recorded on several cleaved surfaces
of Nd-LSCO $p=0.12$ and different ratios between signal and
extrinsic backgrounds were found. As a consequence,
slightly different raw anti-nodal line-shapes were extracted.
However, once background is subtracted, consistent lineshapes
were reproduced (shown in the Appendix).
As shown in Fig.~3(g-l), only the antinodal
lineshape of Nd-LSCO with $p=0.12$ is significantly influenced
by the background subtraction. For all other spectra,
the background subtraction has little impact on the
overall lineshape. In fact, for Nd-LSCO $p=0.12$
the signal is comparable to the background, whereas for
compounds with $p>0.15$ the signal-to-background ratio
is much larger (see Fig.~\ref{fig:fig5}).
Again, this is an indication that the 1/8 anti-nodal
spectra are anomalous.
\section{Discussion}
\subsection{Lineshape modelling}
Lets start by discussing the spectra on the overdoped side of
the phase diagram.
Neglecting matrix
element effects, the symmetrized intensity $I(k_F,\omega)$ is given by the
spectral function~\cite{normanNAT1998}
\begin{equation}
A(k_F,\omega)\sim -\Im\Sigma / [(\omega-\Re\Sigma)^2+\Im\Sigma^2].
\end{equation}
In absence of a spectral gap, $\Re\Sigma =0$ at $k=k_F$ and the spectral function
is nothing else than a Lorentzian function, when approximating $\Im\Sigma$ by a constant
$\Gamma$. If $\Im\Sigma=\Gamma$ is comparable to
the applied energy resolution, a Voigt lineshape is effectively observed.
This is the case for anti-nodal spectra of
Nd-LSCO $p=0.24$ (Fig.~\ref{fig:fig3}h). The intrinsic linewidth $\Gamma$ is a measure of the
"quasiparticle" scattering. With increasing scattering, the linewidth
broadens ($\Gamma$ increases) and the peak amplitude -- sometimes referred to as
the "quasiparticle residue $Z$" -- is lowered.
In this fashion, a metal can loose its coherence.
In presence of a spectral gap, Eliashberg theory
applied to the normal state finds
the Green's function $G(k_F,\omega)=[(\omega+i\Gamma)-\Delta^2/(\omega+i\Gamma)]^{-1}$
to be given by two parameters: the gap $\Delta$ and the scattering rate $\Gamma$~\cite{ChubukovPRB2007}.
This functional form mimics roughly the
observed lineshape, but does not provide a fulfilling description of
the experimental spectra.
We, therefore, adopted a simpler phenomenological Green's function,
$G(k_F,\omega)=[(\omega+i\Gamma)-\Delta^2/\omega]^{-1}$, that contains the same two parameters
and has previously been used to analyze symmetrized energy distribution
curves~\cite{AKanigelNATPHYS2006,jleeSCIENCE2009,normanPRB1998,FranzPRB1998,mshiPRL2008,mshiEPL2009}.
The spectral function $A(k_F,\omega)=\pi^{-1}\Im G(k_F,\omega)$ can now be expressed by two dimensionless quantities,
\begin{equation}
A(x)\sim \frac{1}{\Delta}\frac{\gamma}{(x-1/x)^2+\gamma^2}
\end{equation}
where $x=\omega/\Delta$ and $\gamma=\Gamma/\Delta$.
This phenomenological spectral function preserves the Lorentzian lineshape and total spectral weight, but shifts the
peaks to $x=\pm 1$ ($\omega=\pm\Delta$) while the linewidth $\Gamma/\Delta$
is renormalized by the spectral gap.
For a fixed gap $\Delta$, increasing quasiparticle scattering still leads
to a broader line and weaker peak amplitude.
Absence of a peaked structure may therefore be a signature of
strong quasiparticle scattering.
\begin{figure}
\begin{center}
\includegraphics[width=0.45\textwidth]{FIG5v4.eps}
\end{center}
\caption{(Color online) Comparison of anti-nodal spectra at $T\sim 20$~K (blue) and $75$~K (red).
(a) and (b) show raw energy distribution curves recorded at $k_F$ on Nd-LSCO $p=0.12$ and 0.20
with the respective background intensities, measured at momenta far from $k_F$.
In (c) and (d), the respective background subtracted curves are compared. }
\label{fig:fig5}
\end{figure}
\subsection{Spectral gap and scattering}
Using Eq.~2, analysis of background subtracted spectra~\cite{GweonPRL2011,FatuzzoPRB2014} was carried out.
Resolution effects
are modelled by Gaussian convolution
of the model function $A(k_F,\omega)$ (Eq.~1 and 2).
In this fashion, $\Gamma$ and $\Delta$
were extracted along the underlying Fermi surface
of Nd-LSCO $p=0.20$.
As shown in Fig.~\ref{fig:fig6}, a correlation
between the gap $\Delta$ and the scattering rate $\Gamma$ is found.
A similar trend is observed when the gap $\Delta$ is weakened
by increasing temperature in Nd-LSCO $p=0.20$.
This relation between the antinodal gap (usually referred to as the pseudogap) and electron scattering is
consistent with previous observations.
It is, for example, established
that the pseudogap is largest near the zone boundary~\cite{normanNAT1998,chatterjeeNATPHYS2010,tkondoNAT2009}.
At the same time, the scattering rate $\Gamma$ has been shown to increase
when moving from nodal to antinodal regions~\cite{VallaPRL2000,ChangNatCom2013}.
Furthermore, the photoemission lineshape broadens and the pseudogap increases when doping is reduced
from the overdoped side of the phase diagram~\cite{chatterjeePNAS2011}. The same trend has
been reported by STM studies of the density-of-states~\cite{Alldredge2008,KatoJPSJ2008}.
The exact experimental relation between scattering and pseudogap (normal state gap) has, however,
not been discussed much~\cite{KaminskiPRB2005}. A correlation between scattering and the spectral gap has previously been
predicted by dynamical mean-field theory (DMFT) calculations for the Hubbard model~\cite{SenechalPRL2004}.
Within the DMFT approach~\cite{GullPRL2013,SordiPRB2013,FerreroEPL2009,Alloul2014}, the pseudogap emerges
from electron correlations as a primary effect that, in turn, enhances
the tendency for the system to undergo superconducting and
charge-density-wave instabilities, at lower temperatures. Notice however that, as opposed to superconductivity, charge order has not yet been found directly in DMFT calculations.
From a different point of view,
the pseudogap (normal state gap) emerges as a precursor to superconductivity~\cite{chatterjeeNATPHYS2010,jleeSCIENCE2009,PLeePRX2014}, or
as a precursor to an order competing with superconductivity~\cite{cominSCIENCE2014,VishikPNAS2012,EGMoonPRB2010,mhashimotoNATPHYS2014,MHashimotoNMAT2015}.
In Bi2201, for example, the charge ordering onset temperature is comparable to the pseudogap temperature scale $T^*$~\cite{cominSCIENCE2014}.
Furthermore, a connection between the charge ordering vector and the vector
nesting the Fermi arc tips was found~\cite{cominSCIENCE2014}. It is therefore a possibility that the
pseudogap is related to fluctuating CDW order. In two-dimensional CDW systems, spectral gaps
are indeed observed above the CDW onset temperature~\cite{MonneyPRB2012,InosovPRB2009}.
In cuprates, however, the single particle gap originating from CDW order has
not been clearly elucidated by ARPES experiments.
\begin{figure*}
\begin{center}
\includegraphics[width=0.65\textwidth]{Fig6v6.eps}
\end{center}
\caption{(Color online) Normal state gap $\Delta$ versus the scattering rate $\Gamma$. Both quantities were extracted by fitting
background subtracted symmetrized energy distribution curves along the underlying Fermi surface of Nd-LSCO $p=0.12$, 0.20 and 0.24, as well as antinodal
spectra versus temperature. The fitting
procedure is explained in the text. Gray shaded area indicates schematically the correlation between
the normal state gap and the electron scattering.}
\label{fig:fig6}
\end{figure*}
\subsection{Spectra gaps at 1/8 doping}
It is therefore interesting to discuss the spectral lineshapes at the 1/8-doping,
where the charge order parameter has its maximum (Fig. 1). Charge order -- in principle -- should
open a single-particle gap somewhere on the Fermi surface~\cite{kissNP2007,ChatterjeeNATCOM2015}.
It is commonly assumed that the
stripe ordered ground state found in Nd-LSCO
is identical to that of La$_{2-x}$Ba$_x$CuO$_4$ (LBCO) and La$_{1.8-x}$Eu$_{0.2}$Sr$_x$CuO$_4$ (Eu-LSCO)
with $p=x\simeq 1/8$~\cite{VojtaAP2009}. All three systems have the same
low-temperature tetragonal crystal structure, similar thermopower~\cite{JchangPRL2010,LiPRL2007}, and the
same spin/charge stripe structure~\cite{WilkinsPRB2011,NachumiPRB1998,TranquadaPRB1997,FujitaPRB2004}.
At the particular 1/8 doping -- due to phase competition -- charge stripe order suppresses
almost completely superconductivity.
ARPES studies on these stripe ordered systems commonly report
anti-nodal spectra with little low-energy spectral weight~\cite{VallaSCIENCE2006,rhheNATPHYS2009,ZabalotnyyEPL2009,jchangNEWJP2008,VallaPhysicaC2012}.
Different interpretations have been put forward~\cite{VallaSCIENCE2006,rhheNATPHYS2009}.
In LBCO it was suggested that the pseudogap (normal state gap) has $d$-wave character and
that the gap amplitude $\Delta$ is maximized at 1/8-doping~\cite{VallaSCIENCE2006} (this result is reproduced in Fig.~1c).
Subsequent experiments reported a correction to the $d$-wave
symmetry~\cite{rhheNATPHYS2009}. This led to the proposal of a two-gap scenario~\cite{KondoPRL2007,HuefnerRPP2008,MaPRL2008},
with an additional spectral gap (of unknown origin)
in the anti-nodal region~\cite{rhheNATPHYS2009}.
In Nd-LSCO $p=0.12$, Fermi arcs with finite length were
found even at the lowest measured temperatures~\cite{jchangNEWJP2008}.
To access the intrinsic spectral evolution as a function of
momentum in Nd-LSCO $p=0.12$, background subtracted data
should be considered. In Fig.~\ref{fig:fig3}(i), spectra near the anti-nodal region and
close to the tip of the Fermi arc are compared.
Near to the tip, the spectrum resembles that observed
in overdoped Nd-LSCO. Fitting to Eq.~(2) yields $\Delta=20\pm2$~meV and a scattering constant
$\Gamma=39\pm8$~meV. This is consistent with the approximate
constant ratio of $\Delta/\Gamma$ (see Fig.~\ref{fig:fig6}) found for
Nd-LSCO $p=0.20$.
The lineshape of the anti-nodal spectra is, however,
dramatically modified. A similar evolution was found in
LBCO~\cite{rhheNATPHYS2009}. It seems that the system
has lost coherence.
Fitting using Eq.~2, indeed yields much smaller ratios of $\Delta/\Gamma$ -- see Fig.~\ref{fig:fig6}.
A sudden quasiparticle decoherence effect is therefore one possible
explanation for the different anti-nodal lineshape observed in the underdoped
regime.
\subsection{Effects of competing orders}
Next, we discuss the possible influence of static long-range charge density-wave order.
For conventional CDW systems,
the order parameter is identical to the single-particle
gap~\cite{GruenerRMP1988}, and $\Delta_{ch}$ scales with the lattice distortion
$u$~\cite{GruenerRMP1988}. By measuring this distortion using hard x-ray diffraction, it was found that
$\Delta_{ch}$ has a strong doping dependence~\cite{huckerPRB2013} (reproduced in Fig.~\ref{fig:fig1}b)
-- peaking sharply at the 1/8-doping.
~Just a slight increase of doping, to say $p=0.15$,
results in a single-particle gap $\Delta_{ch}$
renormalized by a factor of five~\cite{huckerPRB2013} (compared to 1/8-doping).
Notice that the charge stripe onset temperature $T_{ch}$ -- observed by x-ray diffraction --
varies more smoothly with doping. Hence, the coupling constant $\alpha=\Delta_{ch}/k_BT_{ch}$
has a strong doping dependence -- being largest at 1/8 doping. It is also around this doping
that quantum oscillation~\cite{DoironLeyraudNAT2007,SESebastianRPP2012,VignolleCRP2011,barisicNP2013} and transport~\cite{LaliberteNATCOMM2011,LeboeufPRB2011,JchangPRL2010,doironleyraudPRX2013} experiments have revealed the Fermi surface reconstruction
in YBCO and Hg1201. Charge ordering has been proposed as the mechanism responsible for this reconstruction~\cite{LaliberteNATCOMM2011,tabisNATCOMM2014}.
Strongly coupled charge order is therefore not necessarily in contradiction with the observation
of quasiparticles with light masses. Interestingly, neither
the Fermi surface reconstruction nor the effect of charge order have
been convincingly probed by photoemission spectroscopy.
The observation of an electronic Fermi surface reconstruction is complicated by orthorhombic distortions, that
fold the bands similarly to what is expected from density-wave orders~\cite{HeNJP2011,MengNAT2009,KingPRL2011}.
Moreover, identification of charge density wave order effects on
the antinodal lineshape in very underdoped compounds is complicated by superconductivity, pseudogaps and possibly also
spin-freezing phenomena~\cite{fujitaPRB2002,WakimotoPRB2000}. The choice of Nd-LSCO ensures, due to it's low $T_c$, that superconductivity
is not influencing the problem.
Furthermore, in this system spin and charge density wave orderings are coupled~\cite{tranquadaNAT1995},
and hence part of the same phenomenon.
When a spectral gap $\Delta$ opens, low-energy spectral weight is either suppressed or
redistributed in $(k,\omega)$-space. It has, for example, been shown that in Bi2212, pronounced
redistribution of spectral weight -- extending beyond 200 meV -- appears inside the pseudogap~\cite{MHashimotoNMAT2015}.
In Fig.~\ref{fig:fig5}b, antinodal spectra
of Nd-LSCO $p=0.20$ display how the normal state gap opens upon cooling. As
the gap opens, spectral weight is transferred to larger energies, while
the total amount of spectral weight remains approximately constant. This
rearrangement of spectral weight manifests itself within an energy scale
$(2-3)\Delta<100$~meV. In the anti-nodal regime of stripe ordered Nd-LSCO $p=0.12$, within
the same temperature and energy window,
the behaviour is very different (see Fig.~\ref{fig:fig5}a).
Upon cooling, low-energy ($\omega<100$~meV) spectral weight is removed with
an apparent net loss of total weight.
The $k-$dependence in Fig.~\ref{fig:fig3}(c,i), does
not suggest any pile up of spectral weight at other locations in momentum space.
Thus either spectral weight is transferred to $\omega>5\Delta$, or
it is simply not conserved.
A system that undergoes a phase transition may not display
spectral weight conservation.
Appearance of charge stripe order in the low-temperature tetragonal
crystal structure may therefore lead to effective loss of
spectral weight.
In that case, stripe order seems to influence mainly the anti-nodal region
and, remarkably, suppression of spectral weight extends up to energies
as large as 100~meV.
\section{Conclusions}
In summary, we have presented a systematic
angle resolved photoemission spectroscopy, normal state study of the charge stripe ordered cuprate compound La$_{1.6-x}$Nd$_{0.4}$Sr$_x$CuO$_4$ (Nd-LSCO).
By varying the doping concentration, antinodal spectra were recorded
from the overdoped metallic phase to the 1/8-doping --
where static charge stripe order is stabilized.
The metallic phase is characterized by gapless excitations even in the
antinodal region. At $x=0.20$, a spectral
gap $\Delta\approx 30$~meV opens in the antinodal region but spectral weight
remains conserved, although shifted to slightly larger energies.
Analysis of the line shape suggests a
correlation between electron scattering and the gap amplitude.
Finally, for underdoped compounds the anti-nodal lineshape is quite
different. Upon cooling into the stripe ordered phase,
spectral weight appears to be lost.
An additional source for spectral weight suppression
is therefore proposed, and charge stripe order is discussed
as an underlying mechanism.\\
\textit{Acknowlegdements.--} This work was supported by the Swiss National Science Foundation
(through grant Nr 200020-105151, 200021-137783 and its NCCR - MaNEP and Sinergia network Mott Physics Beyond the Heisenberg (HPBH)
model), the Ministry
of Education and Science of Japan, and the Swedish Research Council.
Work at ORNL was supported by US-DOE, BES, Materials Sciences and Engineering Division.
JSZ and JBG were supported by the US NSF (DMR 1122603).
The photoemission experiments were performed at SLS of the Paul Scherrer Institut,
Villigen PSI, Switzerland. We thank the X09LA beamline~\cite{FlechsigAIPCP2004} staff
and Xiaoping Wang for technical support. We wish to thank Nicolas Doiron-Leyraud, Paul Freemann, Markus H\"{u}cker, Claude Monney, Henrik R\o{}nnow, Louis Taillefer
and Andr\'{e}-Marie Tremblay for enlightening
discussions.
\clearpage
\section{Appendix A}
All measured ARPES spectra contain background that typically vary slowly with momentum
and excitation energy $\omega$. The background can be evaluated at momenta far away from
$k_F$. We found that across all dopings studied, the background has a very similar
intensity profile as a function of $\omega$. It is thus possible to scale ARPES intensities
using this background. In Fig.~\ref{fig:fig4}, the background of two Nd-LSCO $p=0.12$
anti-nodal spectra recorded under comparable conditions but on different surfaces.
The background can be scaled / normalized to give an essentially perfect match.
Energy-distribution curves recorded at $k_F$ are, however, displaying different
intensities and lineshapes. This demonstrates that from experiment to experiment,
different signal-to-background ratios are observed. We stress that this effect
is most visible at $p=0.12$, where anti-nodal spectral weight appears strongly
suppressed or redistributed. Once the background intensities are
subtracted, the intrinsic lineshape is essentially identical, irrespectively
of the signal-to-background ratio - see Fig.~\ref{fig:fig4}b. Throughout this
work, detailed analysis of lineshapes were carried out on the
background-subtracted data.
\begin{figure
\begin{center}
\includegraphics[width=0.35\textwidth]{FIG4.eps}
\end{center}
\caption{(Color online) Comparison of anti-nodal spectra recorded on different surfaces of Nd-LSCO $p=0.12$ at $T=80$~K.
(a) Raw spectra at $k_F$ and at momentum $k_{BG}$, representing the extrinsic background. Intensities
have been normalized so that the background intensities match across different experiments.
In this fashion, it shown how the same spectral lineshape can appear different due to
a different signal-to-background ratio. Spectra, at $T\sim 80$~K, were taken after
cleaving at $T=20$~K (black) and at $80$~K (red).
(b) Background subtracted spectra, scaled by an arbitrary constant. }
\label{fig:fig4}
\end{figure}
\clearpage
|
{
"timestamp": "2015-09-29T02:20:42",
"yymm": "1509",
"arxiv_id": "1509.08294",
"language": "en",
"url": "https://arxiv.org/abs/1509.08294"
}
|
\section{Introduction}
Defects in the DNA molecule play a crucial role in biological processes such as
replication. This is a known fact that DNA is a long polymeric chain that contains four
different kinds of nitrogenous bases. The allowed pairing in the two complementary
strands follow a simple rule, that is, Adenine (A) can form a hydrogen bond with
Thymine (T) while Guanine (G) can form a hydrogen bond with Cytosine (C)
\cite{watson,stryer}. The hydrogen bonding strength for these two base pairs is not same
as the AT base pair has two hydrogen bonds while GC base pair has three hydrogen bonds.
The approximate ratio of GC and AT bond strengths varies from 1.2 to 1.5 as mentioned
by various research groups \cite{pb,campa,chen,saul,voul,erp,weber,pbd2009,alex,zoli}.
In the absence of the complementary base on the opposite strand, the pairing between
the two bases is absent. This site is called a \textit{defect site} because of
an absence of a stable (or non-existing) bonding between these two bases
on the opposite strands \cite{ns01,ns03}. The presence of defects in DNA is related to
interesting physics and biochemistry of the molecule. The dynamics of these defects
may delay the replication process and hence lead to the breathing dynamics of opening of
the chain \cite{gauth}. It is predicted that in embryonic cells, these delays may
cause the \textit{cell death} while in mature cells like somatic cells,
this damage (defect) may be an initiation step in the development of cancer
\cite{paivi,hensey,vilen,branzei,kaufmann}. These defects are present
in the DNA based actuators. The role of the defects in the designing
of molecular motor has been discussed by McCullagh {\it et al}
\cite{mccullagh}. How the defects affect the melting, elastic and other
properties are problem of scientific interest.
There are many paths to explore the role of the defects and the damage
repair mechanism in the living cells. Theoretical approach to investigate
the problem routes via molecular dynamics or model based calculations
\cite{kafri,amb,joyuex,dudu,kumar,frank,ffalo,macedo,nik}.
Our approach is a model based calculation. We use Peyrard Bishop Dauxois (PBD)
model \cite{pb} to investigate the thermal and mechanical denaturation of DNA molecules
in presence of defects. The main objective of the current study is to investigate
the effect of density and location of defects on the denaturation of
DNA molecule.
In experiments, researchers synthesis and/or characterize the samples
to decipher the information stored by that sample. Accordingly, we choose
four samples of DNA molecules each containing 16 base pairs.
All these samples have different numbers
and distribution of \textit{AT-GC} pairs. We identify all these molecules
according to the distribution of base pairs and named them as follows:
{\it Chain 1: 3'-AAAAAAAAAAAAAAAA-5' } (homogeneous),
{\it Chain 2: 3'-AGAGAGAGAGAGAGAG-5' } (alternating $AT-GC$ pairs),
{\it Chain 3: 3'-AAAAAAAAGGGGGGGG-5' }(50\%$AT$+50\%$GC$),
{\it Chain 4: 3'-TCCCTAGACTTAGGGA-5' }(random sequence).
The prime motivation behind the selection of different kinds of
sequences is to predict the role of defect(s) in the melting or unzipping
of different kinds of DNA molecules. The next task is to introduce the defect in the model.
We have continued from our previous approach \cite{ns01} where the defect
in the model was introduced via Morse potential that represent the hydrogen bonding.
If a pair has a defect that means there is an absence of hydrogen bond and
this feature is reflected from the absence of potential depth while retaining
the repulsive part of the potential in order to avoid the crossing of two
bases in a pair (see Fig. \ref{fig01}).
\begin{figure}[t]
\begin{center}
\includegraphics[height=2.25in,width=2.5in]{fig01.eps}
\caption{\label{fig01}(Color online)
The on-site potential for the defect in a pair is shown by square
symbol and dashed (red) line. While for the bases in a pair is
represented by the depth of the potential (solid line with black circle),
there is no minimum of potential for the defect pair \cite{ns01}.}
\end{center}
\end{figure}
We present the work in different sections. In Section \ref{model},
we provide a brief description of the model and methodology used in this work.
The effect of defect(s) on thermal denaturation of dsDNA molecule
is discussed in Section \ref{melt} while in Section \ref{force},
role of the defect(s) on the mechanical unzipping is discussed.
We finally conclude our results in Section \ref{result}.
\section{The model}
\label{model}
For the current investigation we use a statistical model that was proposed by
Peyrard and Bishop \cite{pb}. The model considers the stretching between the
corresponding bases only. Although the model ignores the helicoidal structure
of the dsDNA molecule, it has enough details to analyze mechanical behavior
at the few angstrom scale relevant to molecular-biological events \cite{pbd2009}.
The Hamiltonian for the considered system of $N$ base pairs unit is
written as,
\begin{equation}
\label{eqn1}
H = \sum_{i=1}^N\left[\frac{p_i^2}{2m}+ V_M(y_i) \right] +
\sum_{i=1}^{N-1}\left[W_S(y_i,y_{i+1})\right]
\end{equation}
where $y_i$ represents the stretching from the equilibrium position of the hydrogen bonds.
the first term in the Hamiltonian represents the momentum ($p_i = m${\it\.{y}}).
The $m$ represents the reduced mass of a base pair which is taken to be same for both
$AT$ and $GC$ base pairs. The stacking interaction between two consecutive base pairs along the
chain is represented by,
\begin{equation}
\label{eqn2}
W_S(y_i,y_{i+1}) = \frac{k}{2}(y_i - y_{i+1})^2[1 + \rho e^{-b(y_i + y_{i+1})}],
\end{equation}
where $k$ represents the single strand elasticity. The anharmonicity in the strand
elasticity is represented by $\rho$ while $b$ represents its range. The stacking interaction
$W_S(y_i,y_{i+1})$ is independent of the nature of the bases at site $i$ and $i+1$ as
these parameters are assumed to be independent of sequence heterogeneity. The sequence
heterogeneity has effect on the stacking interaction along the strand. This can be taken
care of through the single strand elasticity parameter $k$. One can take the
variable $k$ according to the distribution of bases along the strand
\cite{as_phy02}. A defect in a pair will modify the electronic distribution around the
bases hence the stacking parameters.
However, for the current investigation we settled on the average of this
parameter.
The hydrogen bonding between the two bases in the $i^{th}$ pair is represented by
the Morse potential.
\begin{equation}
\label{eqn3}
V_M(y_i) = D_i(e^{-a_iy_i} - 1)^2
\end{equation}
where $D_i$ represents the potential depth which basically represents the
bond strength of that pair. The parameter, $a_i$, represents the inverse of
the width of the potential well. The heterogeneity in the sequence is taken
care of by the values of $D_i$ and $a_i$. These model parameters should be tuned
in order to get physical picture of DNA molecule. For the current investigations,
we choose: $D_{\rm AT} = 0.1 \; {\rm eV}, \; a_{\rm AT} = 4.2 \; {\rm \AA^{-1}}, \;
D_{\rm GC} = 0.15 \; {\rm eV}, \; a_{\rm GC} = 6.3 \; {\rm \AA^{-1}},\;\; \rho = 5.0,
b = 0.35 \; {\rm \AA^{-1}},\; {\rm and} k = 0.021 \; {\rm eV/\AA^{-2}}$.
Thermodynamics of the transition can be investigated by evaluating the expression for
the partition function. For a sequence of $N$ base pairs with periodic boundary conditions,
the partition function can be written as:
\begin{equation}
\label{eqn4}
Z = \int_{-\infty}^{\infty} \prod_{i=1}^{N}\left\{dy_idp_i\exp[-\beta H]\right\} =
Z_pZ_c,
\end{equation}
where $Z_p$ corresponds to the momentum part of the partition function while the $Z_c$
contributes as the configurational part of the partition function. Since
the momentum part is decoupled in the integration, it can be integrated out as a
simple Gaussian integral. This will contribute a factor of $(2\pi mk_BT)^{N/2}$ in the
partition function, where $N$ is the number of base pairs in the chain.
The calculations of the configurational partition function, $Z_c$, is not straight forward.
This is defined as,
\begin{equation}
\label{eqn5}
Z_c = \int_{-\infty}^{\infty} [\prod_{i=1}^{N-1} dy_i K(y_i,y_{i+1})]dy_NK(y_N)
\end{equation}
where $K(y_i,y_{i+1}) = \exp\left[-\beta H(y_i,y_{i+1})\right].$
For the homogeneous chain, one can evaluate the partition function by transfer integral (TI)
method by applying the periodic boundary condition \cite{chen}.
In case of a heterogeneous chain, with open boundary, the configurational part of the
partition function can be integrated numerically with
the help of matrix multiplication method \cite{chen,ns03,erp}. The important part of this
integration is the selection of proper cut-offs for the integral appearing in Eq.5
to avoid the divergence of the partition function.
The method to identify the proper cut-off has been discussed by several
groups \cite{chen,pbd95,erp}.
The calculations done by T.S. van Erp {\it et al}
show that the upper cut-off will be $\approx$ 144 \AA\ with the
our model parameters at $T = 600$ K while the lower cut-off is -0.4 \AA.
In the earlier work by Dauxois and Peyrard it was shown that the $T_m$ converges
rapidly with the upper limit of integration \cite{pbd95}. In that work they
considered an infinite homogeneous chain and solved the partition function using
TI method. For short chains, we
calculate $T_m$ for different values of upper cut-offs which are shown in
Fig. \ref{fig02}. From the plot it is clear that the choice of 200 \AA\ is sufficient
to avoid the divergence of partition function. Thus the configurational space for our
calculations extends from -5 \AA\ to 200 \AA.
\begin{figure}[t]
\begin{center}
\includegraphics[height=2in, width=2in]{fig02.eps}
\caption{\label{fig02}
The melting temperature $T_m$ calculated for different values of upper
cut-off, ($\delta$) for homogeneous chain.
The best straight line fit for this plot is found for $1/\delta$. The
different cut-offs are 8, 10, 20, 50, 100, 200 and 300 \AA.
The model parameters are: $D = 0.1 \; {\rm eV}, \; a = 4.2 \; {\rm \AA^{-1}},
\; \rho = 5.0, \; b = 0.35 \; {\rm \AA^{-1}},\; {\rm and} k = 0.021 \; {\rm eV/\AA^{-2}}$.
}
\end{center}
\end{figure}
Once the limit of integration has been chosen, the
task is reduced to discretizing the space to evaluate the integral numerically.
The space is discretized using the Gaussian quadrature formula.
In our previous studies \cite{ns03}, we observed that in order to get
precise value of melting temperature ($T_m$) one has to choose the large
grid points. We found that 900 is quite sufficient number for this
purpose. As all matrices in Eq.\ref{eqn6} are identical in nature
the multiplication is done very efficiently. The thermodynamic
quantities of interest can be calculated by evaluating the Helmholtz
free energy of the system. The free energy per base pair is,
\begin{equation}
\label{eqn6}
f(T) = -\frac{1}{2\beta}\ln\left(\frac{2\pi m}{\beta}\right) -
\frac{1}{N\beta}\ln Z_c; \qquad\qquad \beta = \frac{1}{k_BT}.
\end{equation}
The thermodynamic quantities like specific heat ($C_v$) as a function of temperature
or the applied force can be evaluated by taking the second derivative of the free energy.
The peak in the specific heat corresponds to the melting temperature or the critical
force of the system.
Other quantities such as the average fraction $\theta(= 1 - \phi)$ of bonded
(or open) base pairs can be calculated by introducing the dsDNA ensemble(dsDNAE)
\cite{erp} or using the phenomenological approach \cite{campa,ns01}.
In general, the $\theta$ is defined as,
\begin{equation}
\label{eq7}
\theta = \theta_{\rm ext}\theta_{\rm int}
\end{equation}
$\theta_{\rm ext}$ is the average fraction of strands forming duplexes, while $\theta_{\rm int}$
is the average fraction of unbroken bonds in the duplexes.
The opening of long and short chains are completely different. For long chains, when the
fraction of open base pairs, $\phi(=1-\theta)$, goes practically from 0 to 1 at
the melting transition, the two strands are not yet completely separated.
At this point, the majority of the bonds are disrupted and the dsDNA is denaturated,
but the few bonds still remaining intact, preventing the two strands parting from each other.
Only at high temperatures will there be a real separation. Therefore for very long chains
the double strand is always a single macromolecule through the transition, thus one can
calculate the fraction of intact or broken base pairs only. For short chains,
the process of single bond disruption and strand dissociation tend to happen
in the same temperature range. Thus, the computation of both $\theta_{\rm int}$ and
$\theta_{\rm ext}$ is essential \cite{campa}. The problem of computation of
$\theta_{\rm ext}$ can be handled efficiently by working in
dsDNA ensemble (dsDNAE) \cite{erp}.
\section{Thermal melting of the DNA molecule}
\label{melt}
We consider the defects (1-4 in number) and their effect on the melting temperature
of the DNA molecule. Since the nature of each chain is different, the number
and location of these defects may modify the melting profile of the chain
in different manner.
\vspace{1cm}
\begin{figure}[h]
\begin{center}
\includegraphics[height=2.5in, width=3.2in]{fig03.eps}
\caption{\label{fig03}(Color online)
The melting temperature, $T_m$, calculated by specific heat and fraction
of open pair $\phi$ for the {\it chain 1} (homogeneous) and {\it chain 4}
(random). The parameters $p$ and $q$ are adjusted in order to get precise
match with peak in specific heat. The values are $p = 12.0$ and $q = 10.0$.
The value of $C_v$ is scaled to show that the peak position and 50\%
of the open pairs meet at the same point (temperature).}
\end{center}
\end{figure}
The melting temperature, $T_m$ is calculated by the peak
in the specific heat as well as from $\theta$ as given
in \cite{campa,ns01}. For pure chain, we show the melting profile of the chain
in Fig. \ref{fig03}.
The melting temperatures for \textit{ chain 1, 2, 3, \& 4} without any defect are 447.5,
508.8, 511.0 and 509.8 K, respectively.
\begin{figure*}[t]
\begin{center}
\includegraphics[height=5.0in, width=6.5in]{fig04.eps}
\caption{\label{fig04} (Color online)
The melting temperature, $T_m$, for all the four chains with different
numbers as well as the locations of these defects along the DNA molecule.
Figures in a row are for a chain with different numbers of defects.
Figures in a column are for different chains with same number of defect(s)
displayed by $m$.}
\end{center}
\end{figure*}
Let us now consider the {\it chain 1} with one defect. When the first site ($3'$- end)
is a defect pair, the melting temperature is about 433 K. The melting temperature
further reduces to 432 K if the $2^{nd}$ pair is a defect pair. However there is
something interesting to note after this. When the location of defect is $4^{th}$
pair onward (towards $5'$- end) the melting temperature reduces to 430 K and
remains constant till we reach $13^{th}$ site. As we reach on the $5'$ end,
the $T_m$ again increases. This complete cycle displays a necklace kind of
plot as shown in Fig. \ref{fig04}.
The variation in the melting temperature when the defect is anywhere
between 4 to 13 is negligible. When the number of defects
in the chain is increased from 1 to 4 the width or plateau (where there
is no change in the melting temperature of the molecule) decreases ($\sim$8-12).
In order to explore more about the nature of denaturation, we investigate
other chains that have different distribution of base pairs.
Consider \textit{chain 2} with single defect. This chain is having
alternate $AT/GC$ pairs. As shown in Fig. \ref{fig04}, the symmetry about
the middle is lost. For this chain, the location of defect site, whether
it is $AT$ or $GC$ pair, is important. The energy landscape of this chain is not smooth
over the complete length because of the difference in the dissociation energies
of $AT$ and $GC$ pairs. For single defect that move from position 1 to 16, $T_m$ shows
a zig-zag pattern and $T_m$ varies between a range of 496 K to 487 K.
When we consider two consecutive defects in the chain this pattern is lost
because of the loss in the sequence heterogeneity. This can be thought of as
reviving the homogeneous structure of the DNA molecule with the dual pair
having an average of $AT$ and $GC$ pair's dissociation energies.
However, the necklace pattern obtained for this case is not as symmetric
as observed for the {\it chain 1} since the end pairs are not same.
Remember at $3'$- end there is an $AT$ pair while on $5'$- end,
there is a $GC$ pair. For this chain, $T_m$ varies between
485 K to 476 K. The zig-zag pattern is retained when three consecutive defects
are introduced. However, the $T_m$ is lower as compared to the chain
with one defect. Again with four consecutive defects an asymmetric necklace
is observed with short plateau.
Let us consider \textit{chain 3},
that is having $50AT+50GC$ pairs in the sequence, with one defect.
In this case, we obtain a hook kind of structure in the plot.
A sudden drop in the $T_m$ is observed in the middle of the chain
(on $8\;\&\; 9$ pair) at the interface of \textit{GC} \& \textit{AT} pair.
The smoothness at the interface increases with increase in the number of defects
in the chain. As the number of defect increases, on the interface the effect of
presence of $AT$ and $GC$ pairs diminishes. Next is, \textit{chain 4} that is having a
random distribution of \textit{AT/GC} pairs. Due to random distribution of $AT$ and
$GC$ pairs, the energy landscape is also random. Hence the fluctuation in the
values of $T_m$ should also be random. This is observed in the figure.
The random pattern of the plot varies with number of defects in the molecule.
Let us observe the single defect on 4,5,\& 6 sites. While $T_m$ is 488 K for $4^{th}$
site, it is 495 K for $5^{th}$ \& $6^{th}$ sites. For two consecutive
defects in the molecule, this is averaged to 478 K, \textit{i.e.}. This is
because of the indistinguishability of \textit{AT} and \textit{GC} pair.
Similarly, for three (consecutive) defects in the molecule, the high barrier
on \textit{11} \& {\it 12} sites, is lost. The pattern observed for this
chain is closer to the real sequences.
\section{Force Induced transitions}
\label{force}
The replication process is initiated by the force exerted
by DNA polymerase on a segment of DNA chain (Owcarzy fragment). The replication
starts at some site which is called replication origin \cite{branzei,kaufmann}
and the replication fork propagates bidirectionally. The defect or mismatch
pair(s) may slow or stall the replication process. In the case if the
mismatch repair system does not work properly cell may die \cite{paivi}.
Mathematically one can model the replication as the force applied on
an end of the DNA chain \cite{somen}. Physics of opening of
chain due to thermal fluctuation and mechanical forces is completely different
\cite{hatch,huguet,ritort}. Thus, the study on the mismatch in the sequence
and its role in the replication process is an interesting way to
look into the physics of a complex mechanism. In this section, we
discuss the force induced unzipping in DNA molecules in presence of defect(s).
The modified Hamiltonian for the DNA that is pulled mechanically from an
end is,
\begin{equation}
\label{eq8}
H = \sum_{i=1}^N\left[\frac{p_i^2}{2m}+ V_M(y_i) \right] +
\sum_{i=1}^{N-1}\left[W_S(y_i,y_{i+1})\right] -F\cdot y_1
\end{equation}
where the force $F$ is applied on the $1^{st}$ pair \cite{ns03}.
Whereas in thermal denaturation, the opening is due to increase in the
entropy of the system, for mechanically stretched DNA chain the opening is
enthalpic. The thermodynamic quantities, of interest, from the modified
Hamiltonian can be calculated using Eq. \ref{eqn5} \& \ref{eqn6}.
Here we consider the same four chains that we considered for thermal
denaturation studies. All the base pairs of dsDNA that is kept in
a thermal bath share equal amount of energy. In the case when the chain
is pulled from an end, the amount of force decreases from the pulling point
to the other end of the chain. Thus the location of defect(s) should have
different impact on the opening of the chain that is subjected to a
mechanical pull from an end.
\begin{figure*}[t]
\begin{center}
\includegraphics[height=5in, width=6.5in]{fig05.eps}
\caption{\label{fig05} (Color online)
The value of critical force, $F_c$, for different chains having
different number of defects as well as their location. We show both
the cases when force is applied on $3'$-end (solid lines with circle)
and on the $5'$-end (dashed lines with square). Only for the case when
the defect site is in the middle of the chain is there no change in
$F_c$ for these two case. The critical forces for pure
\textit{ chains 1, 2, 3 \& 4} is 4.54, 6.96, 6.98, and 6.97 pN
respectively.}
\end{center}
\end{figure*}
Let us consider {\it chain 1} with one defect.
As shown in the Fig. \ref{fig05} when the force is applied on the
$3'$ -end and the defect pair is $1^{st}$ pair ($3'$ -end), the
critical force reduces to 4.04 pN from 4.54 pN. This value further
decreases to 3.99 pN when the defect pair is $2^{nd}$ pair.
When the defect is located between 3-13 pairs, there is no change in
the value of critical force, it is $\sim$3.97 pN. This means that
the base pair (defected) in this section of the chain have similar
response to the applied force, irrespective of their location.
The defect pair means a loop in the chain which will increase
the entropy of the chain. From the results, this is clear that
the loop contributes to the opening of the chain in addition
to the applied force and end entropy. However,
as the defect location is somewhere between 14 to 16,
contribution of bubble in the entropy of the chain is negligible.
Hence the critical force increases. In this case too, we observe a
necklace pattern. The pattern obtained here is not as symmetric as
observed for thermal melting of the same chain with single defect.
The reason for this difference lies in the nature of the chain opening in
these two cases. We consider now the opening of the chain in another
condition. The force is applied on $3'$ -end and defect pair is the
$5'$ -end. In this case, the critical force is 4.02 pN which is less
than for the previous case (where the force is on $3'$ -end and defect
pair is also $3'$ -end) where $F_c$ is 4.04 pN. The difference is
about 0.02 pN. The reason for this reduction is the difference
in the end entropies for these two cases. In case when the defect
end is $5'$- end, the entropy of this open end contributes to the opening.
While for the first case, when the defect end is the $3'$ -end
(the force is also on this end), the contribution from the $5'$ -end
will be less as it is an intact pair. Hence we need slightly higher
force to open the chain for the first case.
We obtained similar results for this chain with more defects ($m = 2,3,4$). For
all the investigations whenever $m>1$, all the defects are
consecutive defects. As the number of defects increases in the chain,
the difference in the $F_c$ for two different cases is greater.
In order to verify our arguments, we calculate the probabilities of
opening of the base pairs for these two cases.
The probability of opening of the $i^{th}$ pair, in a sequence is
defined as \cite{ns2011}:
\begin{equation}
P_i = \frac{1}{Z_c}\int_{y_0}^{\infty} dy_i \exp\left[-\beta H(y_i, y_{i+1})\right] Z_j
\end{equation}
where
\begin{equation}
Z_j = \int_{-\infty}^{\infty} \prod_{j=1, j\neq i}^N dy_j
\exp\left[-\beta H(y_j, y_{j+1})\right]
\end{equation}
while $ Z_c $ is the configurational part of the partition function
defined as in eq.\ref{eqn5}. For $y_0$, we have taken a value of
2 ${\rm \AA}$. To avoid the overflow of figures, we choose to display
the surface plot for {\it chain 1} with 4 defects, see Fig. \ref{fig06}.
We observe that the difference in the critical force for these two
cases is $\sim 0.04$ pN.
\begin{figure*}[t]
\begin{center}
\includegraphics[height=3.0in,width=3.21in]{fig06a.eps}
\includegraphics[height=3.0in,width=3.21in]{fig06b.eps}
\caption{\label{fig06} (Color online)
The density plots to show the difference in the opening of homogeneous
DNA molecule ({\it chain 1 with four defects}) when force is applied on
$3'$- end. (Left) When defect pairs are 1-4 ($3'$ end).
(Right) When the four defect pairs are on the $5'$ end (13-16).
The difference in the critical force $F_c$ for the two cases is
observed here ({\it more clearly in the zoomed version}).
In order to open 50\% of the base pairs, the $F_c$ for first case (left)
is 2.96 pN while for the second case (right) it is 2.92 pN. }
\end{center}
\end{figure*}
Let us consider, the {\it chain 2} (alternating $AT$ and $GC$ pairs).
In this case, we have four possible combinations of force and
defect locations. First, when force is applied on $3'$ -end
and the defect is also at the $3'$ -end. Second one is
when force is applied on $3'$ -end and defect is at $5'$ -end. The other two cases
are the alternate combinations of these two. If we fix the location
of applied force at $3'$ -end and change the defect locations,
we find that for single defect the difference in $F_c$ is $\sim 0.3$ pN.
This is because of the difference in the entropy contribution from
the two ends. In one case the end is $AT$ while in another case it is $GC$.
If we fix the defect location and change the applied force locations
from $3'$ -end to $5'$ -end, we find that the difference in
$F_c$ is $\sim 0.02$ pN. The same argument which we gave for {\it chain 1}
with single defect is valid here too. Now consider this
chain with two defects. When the force is applied on $3'$ -end and the
defect locations are $3'$ -end and $5'$ -end, the difference in $F_c$
is $\sim 0.04$ pN. This chain with two defects can be thought of as a
homogeneous chain (of $AT + GC$ block) with single defect. However,
the ends in this chain can be either $AT$ or $GC$ and hence we get a
different pattern at the ends as compare to {\it chain 1}.
Similar kind of feature is observed for the same chain with four defects.
For the {\it chain 3},
the difference in the energy of $AT$ and $GC$ pair is clearly visible.
In this case, this is important on which end the force is applied.
When the force is applied on $3'$ -end and the defect locations are
$3'$ -end and $5'$ -end the difference in the $F_c$ is $\sim 0.32$ pN.
The \textit{chain 4} is a chain with random distribution of $AT$ and $GC$
pairs. Since the distribution is random the energy landscape of $AT$ and
$GC$ pairs will play an important role in the opening of chain with different
locations of defect. The unzipping behavior of this chain displays some of
the features of all the three chains that we considered above.
In case when the defect(s) are in the middle of the chain,
the change in the value of critical force is negligible, {\it i.e.}, it
does not matter from which end the chain is pulled.
\section{Conclusions}
\label{result}
In the present work, we have studied the role of defect(s) on the thermal
as well as mechanical denaturation of DNA molecule. It is known that
the defects delay the replication process which may further cause
the cell death and hence may lead to initiation of cancer.
Motivated by the experimental studies, we considered four
different kind of DNA molecules. These molecules have different
numbers of \textit{AT} and \textit{GC} pairs and the distribution of
these pairs along the chain is also different. We have considered all the chains with
$m$ number of defects, where $m$ varies from 1-4. Here we assumed that for
$m>1$ all the defects are in a block. For the equilibrium calculations,
we used PBD model and found the denaturation point in thermal as well
as in constant force ensembles. For the homogeneous chain, we found that
there is a segment (4-12) of the chain where $T_m$ is unaffected by the
location of the defect in the chain. In case of heterogeneous
chain, there is no plateau but it matters on a location whether there is
an \textit{AT} pair or a \textit{GC} pair. When we compared the opening in two
ensembles for homogeneous chain we observed that there is a striking difference.
While for the homogeneous chain we obtained a symmetric necklace kind of
plot in thermal ensemble, this was missing in force ensemble. This validates
the role of finite end entropy of the homogeneous chain in the denaturation
of the DNA molecule. For the thermal melting the ends have less
impact on the opening because of the fact that each base pair shares the
same amount of thermal energy. There only the sequence of $AT/GC$ pairs
matters.
For the chain that is pulled from an end by some force, it is important
for all kinds of chains (with defect) whether the force is applied on
$3'$- end or $5'$- end. For unzipping in constant force ensemble we
considered four possible cases. First two are when force is applied
on $3'$- end and the defect locations are either on $3'$- or $5'$- ends.
Similarly other two combinations are when the force is applied on $5'$- end
and defect locations are either $3'$- or $5'$- ends. In all these
cases, the nature of end pair is important. For the chain with
alternate $AT$ and $GC$ sequence we observed that in addition to the
ends the interface of defect and intact pair affect the opening of the
chain. The interfaces for this chain with two defects are either of
$AGA$ or $GAG$ kind. Hence there is a difference in the critical force
for the four cases. To show the importance of ends in the
opening we calculated the probabilities of opening for the
homogeneous chain with four defects. Here we considered two cases;
one when the force is applied on $3'$- end and defects are either at
$3'$- end or at $5'$- end. When the defect location is $3'$- end,
the end entropy is suppressed and hence we obtained a slightly higher critical
force for this case. The studies on \textit{chain 4}
are closer to the real chain as it has random sequence of
\textit{AT} and \textit{GC} pairs. The force profile (Fig. \ref{fig05})
shows the weak and strong sections of the chain. As a future of this work,
one can study the opening of DNA molecule in both the ensemble as a
function of time and the exact delay in the opening can be predicted.
This is an attempt to understand the defect and their effect on the
replication process. However, the real picture would be clearer
from non-equilibrium studies.
How the cell decides which segment of DNA with a mismatch in the sequence can be
repaired or which would be destroyed will be an interesting area of
future studies. Is there any role of free energies of the sequence?
The time evolution of this kind of molecule may provide some
useful information.
\section*{Acknowledgement}
We are thankful to Y. Singh and S. Kumar, Department of Physics, Banaras Hindu
University, India, for useful discussions. We acknowledge the financial support
provided by Department of Science and Technology, New Delhi [SB/S2/CMP-064/2013]
and University Grant Commission, New Delhi, India for BSR fellowships to AS.
|
{
"timestamp": "2015-09-29T02:16:22",
"yymm": "1509",
"arxiv_id": "1509.08195",
"language": "en",
"url": "https://arxiv.org/abs/1509.08195"
}
|
\section{Introduction}
\begin{quotation}
``\ldots of course, the motion of the system tends to
move away from repellers. Nonetheless a repeller might be
important because, for example, it might describe a separatrix
that serves to divide two different attractors or two different
types of motion.'' \emph{Kadanoff and Tang\rf{KT84}}.
\end{quotation}
The 1984 Kadanoff and Tang investigation of strange repellers was prescient
in two ways. First, at the time it was not obvious why anyone would care
about ``repellers,'' as their dynamics would be transient. Today, much of the
research in turbulence focuses on repellers. In particular, significant
effort is invested in understanding the state space regions of shear-driven
fluid flows that separate laminar and turbulent
regimes\rf{TI03,SYE05,SchEckYor07,duguet07,AvMeRoHo13},
and these ``separatrices'' indeed often
appear to be strange repellers.
Kadanoff and Tang's study was quantitative, and modest by today's standards:
they computed escape rates for a family of 3-dimensional mappings in terms of
their unstable periodic orbit s (`repulsive cycles'), while today corresponding
computations are carried out for very high-dimensional ($\sim$100,000
computational degrees of freedom), numerically accurate discretizetions of
Navier-Stokes flows\rf{GHCW07,channelflow,openpipeflow}. In light of the
heuristic nature of their investigation, their second insight was remarkable:
they were the first to posit the {\em exact} weight for the contribution of an
unstable periodic orbit $p$ to an average computed over a strange
repeller (or attractor):
\[
{1}
/
{\left|\det \left( {\bf 1}-\jMps_p(x) \right)\right| }
\]
(here $\jMps_p(x)$ is the Jacobian matrix of linearized flow, computed along
the orbit of a periodic point $x$). While, at the time, they were aware only
of Bowen's (1975) work\rf{bowen}, today this formula is a cornestone of the
modern periodic orbit theory of chaos in deterministic flows\rf{DasBuch},
based on zeta functions of Smale (1967)\rf{smale}, Gutzwiller
(1969)\rf{gutzwiller71}, Ruelle (1976)\rf{Ruelle76a,Ruelle76} and their
cycle expansions (1987)\rf{inv,AACI,AACII,CBook:appendHist}. Much has
happened since -- in particular, the formulas of periodic orbit theory for
3-dimensional dynamics that they had formulated in 1983 are today at the core
of the challenge very dear to Kadanoff, a dynamical theory of
turbulence\rf{Christiansen97,GHCW07}. For that to work, many extra moving
parts come into play. We have learned that the convergence of cycle
expansions relies heavily on the flow topology and the associated symbolic
dynamics, and that understanding the geometry of flows in the state space is
the first step towards extending periodic orbit theory to systems of high or
infinite dimensions, such as fluid flows. It turns out that taking care of
the symmetries of a nonlinear fluid flow is also a difficult problem. While
one can visualize dynamics in 2 or 3 dimensions, the state space\ of these flows
is high-dimensional, and symmetries -both continuous and discrete- complicate
the flow geometry as each solution comes along with all of its symmetry
copies. In this contribution to Leo Kadanoff memorial volume, we develop new
tools for investigating geometries of flows with symmetries, and illustrate
their utility by applying them to a spatiotemporally chaotic Ku\-ra\-mo\-to-Siva\-shin\-sky\ system.
Originally derived as a simplification of the complex Ginzburg-Landau
equation\rf{KurTsu76} and in study of flame fronts\rf{siv}, the Ku\-ra\-mo\-to-Siva\-shin\-sky\
is perhaps the simplest spatially extended dynamical system that exhibits
spatiotemporal chaos. Similar in form to the Navier-Stokes equations, but much easier
computationally, the Ku\-ra\-mo\-to-Siva\-shin\-sky\ partial differential equation (PDE) is a
convenient sandbox for developing intuition about
turbulence\rf{Holmes96}. As for the Navier-Stokes, a state of the Ku\-ra\-mo\-to-Siva\-shin\-sky\ system is
usually visualized by its shape over configuration space
(such as states shown in \reffig{f-ksconf}). However, the function space
of allowable PDE fields is an infinite-dimensional state space, with the
instantaneous state of the field a point in this space. In spite of the
state space\ being of high (and even infinite) dimension, evolution of the
flow can be visualized, as generic trajectories are 1-dimensional
curves, and numerically exact solutions such as equilibria and periodic
orbits are points or closed loops, in any state space\ projection.
There are many choices of a ``state space.'' Usually one starts out with the
most immediate one: computational elements used in a
finite-dimensional discretization of the PDE studied. As the Ku\-ra\-mo\-to-Siva\-shin\-sky\ system
in one space dimension, with periodic boundary condition, is equivariant
under continuous translations and a reflection, for the case at hand the
natural choice is a Fourier basis, truncated to a desired numerical
accuracy. This is still a high-dimensional space: in numerical work
performed here, $30$-dimensional. For effective visualizations, one
thus needs to carefully pick dynamically intrinsic coordinate frames, and
projections on them\rf{SCD07,GHCW07}.
Such dynamical systems visualisations of turbulent flows, complementary
to the traditional spatio-temporal visualizations, offer invaluable
insights into the totality of possible motions of a turbulent fluid.
However, symmetries, and especially continuous symmetries, such as
equvariance of the defining equations under spatial translations, tend to
obscure the state space\ geometry of the system by their preference for
higher-dimensional invariant $N$-tori solutions, such as equivariant equilibria\ and equi\-vari\-ant periodic orbit s.
In order to avoid dealing with such effects of continuous symmetry, a
number of papers%
\rf{Christiansen97,RCMR04,ReCi05,lanCvit07,ReChMi07} study the Ku\-ra\-mo\-to-Siva\-shin\-sky equation\
within the flow-invariant subspace of solutions symmetric under
reflection. However, such
restrictions to flow-invariant subspaces miss the physics of the problem:
any symmetry invariant subspace is of zero measure in the full state space,
so a generic turbulent trajectory explores the state space\ \emph{outside} of
it. Lacking continuous-symmetry reduction schemes, earlier papers
on the geometry of the Ku\-ra\-mo\-to-Siva\-shin\-sky\ flow in the full state space%
\rf{ksgreene88,AGHks89,KNSks90,SCD07} were restricted to the study of
the smallest invariant structures: equi\-lib\-ria, their stable/unstable
manifolds, their heteroclinic connections, and their bifurcations under
variations of the domain size.
Stationary solutions are important for understanding the state space\
geometry of a chaotic attractor, as their stable manifolds typically
set the boundaries of the strange set. The Lorenz attractor is the best
known example\rf{Williams79} and Gibson \etal\rf{GHCW07} visualizations
for the plane Couette flow are so far the highest-dimensional setting,
where this claim appears to hold. In this paper we turn our attention to
(relative) periodic orbit s, which --unlike unstable equi\-lib\-ria-- are embedded within the
strange set, and are expected to capture physical properties of an
ergodic flow.
\refRefs{Christiansen97,lanCvit07}, restricted to the
reflection-invariant subspace of the Ku\-ra\-mo\-to-Siva\-shin\-sky\ flow, have succeeded in
constructing symbolic dynamics for several system sizes. In these
examples, short periodic orbit s have real Floquet multipliers, with very thin
unstable manifolds, around which the longer periodic orbit s are organized by means
of nearly 1\dmn\ Poincar\'e \ return maps.
In this paper we study the unstable manifolds of equi\-vari\-ant periodic orbit s of Ku\-ra\-mo\-to-Siva\-shin\-sky\ system in
full state space, with no symmetry restrictions. In contrast to the
flow-invariant subspace considered in \refrefs{Christiansen97,lanCvit07},
the shortest equi\-vari\-ant periodic orbit\ of the full system that is stable for small system
sizes ($L < 21.22$) has a complex leading Floquet multiplier. This
renders the associated unstable manifold 2-dimensional. Elimination of
the marginal directions, the space and time translation symmetries,
by a `slice' and a {Poincar\'e section} conditions, together with a novel
reduction of the spatial reflection symmetry, enables us to study here
this 2-dimensional unstable manifold. We compute and visualize the
unstable manifold of the shortest periodic orbit\ as we increase the system size
towards the system's transition to chaos.
Summary of our findings is
as follows:
At the system size $L \approx 21.22$, the shortest periodic orbit\ undergoes a torus
bifurcation\rf{hopf42} (also sometimes referred to as
the Neimark-Sacker bifurcation\rf{Neimark59,Sacker65},
if the flow is studied in a {Poincar\'e section}), which gives birth to a
stable 2-torus. As the system size is increased, this torus first
goes unstable, and is eventually destroyed by
the bifurcation into stable and unstable pair of period-$3$ orbits, to which the
unstable manifold of the parent orbit is heteroclinically connected.
As the system size is increased further, the stable period-$3$ orbit goes
unstable, then disappears, and the dynamics becomes chaotic.
Upon a further increase of the system size, the unstable period $3$ orbit
undergoes a symmetry-breaking bifurcation, which introduces richer
dynamics as the associated unstable manifold has connections to both
drifting (relative) and non-drifting periodic orbit s.
We begin by a short review of the Ku\-ra\-mo\-to-Siva\-shin\-sky\ system in the next section, and
review continuous symmetry reduction by {first Fourier mode slice} method in
\refsect{s-SymmRedCont}. The main innovation introduced in this paper is the
invariant polynomial discrete symmetry reduction method described in
\refsect{s-SymmRedDiscr}. The new symmetry reduction method is applied to and
tested on the Ku\-ra\-mo\-to-Siva\-shin\-sky\ system in \refsect{s-UnstMan}, where the method
makes it possible to track the
evolution of the periodic orbit s unstable manifolds through the system's
transition to chaos. We discuss the implications of our results and
possible future directions in \refsect{s-Discuss}.
\section{Ku\-ra\-mo\-to-Siva\-shin\-sky\ system and its symmetries}
\label{s-kse}
We study the Ku\-ra\-mo\-to-Siva\-shin\-sky equation\ in one space dimension
\begin{equation}
u_\zeit = -u\,u_\conf
-u_{\conf \conf}-u_{\conf \conf \conf \conf} \,,
\label{e-ks}
\end{equation}
with periodic boundary condition $u(\conf,\zeit)=u(\conf+L, \zeit)$. The
real field $u(\conf, \zeit)$ is the ``flame front'' velocity\rf{siv}. The
domain size $L$ is the bifurcation parameter for the system, which
exhibits spatiotemporal chaos for sufficiently large $L$: see
\reffig{f-ksconf}\,(e) for a typical spatiotemporally chaotic trajectory
of the system at $L=22$.
\begin{figure}[h]
\centering
\begin{overpic}[height=0.30\textwidth]{ksRecycled-ConfEq2}
\put (-0.75,-1) {(a)}
\end{overpic} \quad
\begin{overpic}[height=0.30\textwidth]{ksRecycled-ConfTW1}
\put (-0.75,-1) {(b)}
\end{overpic} \quad
\begin{overpic}[height=0.30\textwidth]{ksRecycled-ConfPpo}
\put (-0.75,-1) {(c)}
\end{overpic} \quad
\begin{overpic}[height=0.30\textwidth]{ksRecycled-ConfRpo}
\put (-0.75,-1) {(d)}
\end{overpic} \quad
\begin{overpic}[height=0.30\textwidth]{ksRecycled-ConfErgodic}
\put (-0.75,-1) {(e)}
\end{overpic}
\caption{
Examples of invariant solutions of the Ku\-ra\-mo\-to-Siva\-shin\-sky\ system and the
chaotic
flow visualized as the color coded amplitude of the scalar field
$u(\conf, \zeit)$:
(a) Equi\-lib\-rium\ $E_1$,
(b) travelling wave\ $TW_1$,
(c) Pre-periodic orbit\ with period $T=32.4$,
(d) Equi\-vari\-ant periodic orbit\ with period $T=33.5$ .
(e) Chaotic flow.
Horizontal and vertical axes correspond to space
and time respectively.
System size $L=22$. The invariant solutions and their labels are
taken from \refref{SCD07}.
}
\label{f-ksconf}
\end{figure}
We discretize the Ku\-ra\-mo\-to-Siva\-shin\-sky\ system by Fourier expanding the field
\(
u(\conf, \zeit) = \sum_k \tilde{u}_k (\zeit) e^{i q_k \conf}
\,,
\)
and expressing \refeq{e-ks} in terms of Fourier modes as an infinite set
of ordinary differential equations (ODEs)
\begin{equation}
\dot{\tilde{u}}_k = ( q_k^2 - q_k^4 )\, \tilde{u}_k
- i \frac{q_k}{2} \!\sum_{m=-\infty}^{+\infty} \!\!\tilde{u}_m \tilde{u}_{k-m}
\,,\quad
q_k = \frac{2 \pi k}{L}
\,.
\label{e-Fks}
\end{equation}
Ku\-ra\-mo\-to-Siva\-shin\-sky equation\ is \emph{Galilean invariant}: if $u(\conf,\zeit)$ is a solution, then
$v+u(\conf-v\zeit,\zeit)$, with $v$ an arbitrary constant velocity, is also a
solution. In the Fourier representation \refeq{e-Fks}, the {Galilean
invariance} implies that the zeroth Fourier mode $\tilde{u}_0$ is decoupled
from the rest and time-invariant. Hence, we exclude $\tilde{u}_0$ from the
state space\ and represent a Ku\-ra\-mo\-to-Siva\-shin\-sky\ state $u=u(\conf,\zeit)$ by the Fourier
series truncated at $k\!=\!N$, \ie, a $2N$\dmn\ real valued state space\
vector
\begin{equation}
\ssp = (b_1, c_1, b_2, c_2, \dots, b_N, c_N)\,,
\label{e-Statesp}
\end{equation}
where $b_k = \Re[\tilde{u}_k]$, $c_k = \Im [\tilde{u}_k]$. One can rewrite
\refeq{e-Fks} in terms of this real valued state space\ vector, and express
the truncated set of equations compactly as
\begin{equation}
\dot{\ssp} = \vel (\ssp )
\,.
\label{e-ODE}
\end{equation}
In our numerical work we use a pseudo-spectral formulation of
\refeq{e-ODE}, described here in \refappe{s-Stability}, and in
detail in the appendix of \refref{SCD07}.
Spatial translations $u(\conf, \zeit) \rightarrow u(\conf + \delta \conf,
\zeit)$ on a periodic domain correspond to $\SOn{2}$ rotations
\(
\ssp \to \matrixRep(\LieEl (\theta))\,\ssp
\)
in the Ku\-ra\-mo\-to-Siva\-shin\-sky\ state space, with the
matrix representation
\begin{equation}
\matrixRep(\LieEl(\theta)) = \mathrm{diag}\left[\,{}
R(\theta),\, R(2 \theta),\, \ldots,\,
R (N \theta)\,\right]\,,
\label{e-DSO2}
\end{equation}
where
$\theta = 2 \pi \delta \conf / L$
and
\[
R(k \theta) = \begin{pmatrix}
\cos k\theta & -\sin k\theta \\
\sin k\theta & ~\cos k\theta
\end{pmatrix}
\]
are $[2\!\times\!2]$ rotation matrices. Ku\-ra\-mo\-to-Siva\-shin\-sky\ dynamics commutes with
the action of \refeq{e-DSO2}, as can be verified by checking that
\refeq{e-ODE} satisfies the equivariance relation
\begin{equation}
\vel (\ssp) = \matrixRep^{-1} (\LieEl(\theta))
\vel(\matrixRep(\LieEl(\theta)) \ssp)
\,.
\ee{eqvRelat}
By the translation symmetry of the Ku\-ra\-mo\-to-Siva\-shin\-sky\ system, each solution of
PDE \refeq{e-ks}
has infinitely many dynamically equivalent copies that can be obtained by
translations \refeq{e-DSO2}. Systems with continuous symmetries thus tend
to have higher-dimensional invariant solutions: equivariant equilibria\ (traveling waves)
and equi\-vari\-ant periodic orbit s. A \emph{travelling wave} evolves only along the continuous symmetry
direction
\[
\ensuremath{\ssp_{\!_{TW}}} (\zeit) = \matrixRep(\LieEl( \zeit\,\dot{\theta}_{tw}) ) \,
\ensuremath{\ssp_{\!_{TW}}} (0) \,,
\]
where $\dot{\theta}_{tw}$ is a constant \phaseVel, and the suffix
$\scriptsize{tw}$ indicates that the solution is a ``traveling wave.''
A \emph{equi\-vari\-ant periodic orbit} recurs exactly at a symmetry-shifted location after
one period
\begin{equation}
\ensuremath{\ssp_{\!_{RPO}}} (0) = \matrixRep(\LieEl(- \theta_{rp} ) )
\, \ensuremath{\ssp_{\!_{RPO}}} (\period{rp}) \,.
\label{e-RPO}
\end{equation}
\refFig{f-ksconf}\,(b) and (d) show space-time visualizations
of a Ku\-ra\-mo\-to-Siva\-shin\-sky\ travelling wave\ and a equi\-vari\-ant periodic orbit. The sole
dynamics of a travelling wave\ is a constant drift along the continuous symmetry
direction, while a equi\-vari\-ant periodic orbit\ shifts by amount $\theta_{rp}$ for each repeat of
its period, and traces out a torus in the full state space.
The Ku\-ra\-mo\-to-Siva\-shin\-sky equation\ \refeq{e-ks} has no preferred direction, and is thus also
equivariant under the \emph{reflection} symmetry $u(\conf, \zeit)
\rightarrow - u (- \conf, \zeit)$: for each solution drifting left, there
is a reflection-equivalent solution which drifts right. In terms of
Fourier components, the reflection $\sigma$ acts as complex conjugation
followed by a negation,
whose action on vectors in state space\ \refeq{e-Statesp} is represented by
the diagonal matrix
\begin{equation}
\matrixRep(\sigma) = \mathrm{diag}\left[\,-1,\, 1,\,-1,\, 1,\,
\ldots, \,-1,\, 1 \right]
\,,
\label{e-DR}
\end{equation}
which flips signs of the real components $b_i$.
Due to this reflection symmetry, the Ku\-ra\-mo\-to-Siva\-shin\-sky\ system can also have
strictly non-drifting equi\-lib\-ria\ and (pre-)\-periodic orbit s. An \emph{equi\-lib\-rium} is a
stationary solution
\(
\ensuremath{\ssp_{\!_{EQ}}} (\zeit) = \ensuremath{\ssp_{\!_{EQ}}} (0)
\,.
\)
A \emph{periodic orbit} $p$ is periodic with period \period{p},
\(
\ssp_p(0) = \ssp (\period{p})
\,,
\)
and a \emph{pre-periodic orbit} is a equi\-vari\-ant periodic orbit\
\begin{equation}
\ensuremath{\ssp_{\!_{PPO}}} (0) = \matrixRep(\sigma)\,\ensuremath{\ssp_{\!_{PPO}}} (\period{pp})
\label{e-PPO}
\end{equation}
which closes in the full state space\ after the second repeat,
hence we refer to it here as `pre-periodic'.
In \reffig{f-ksconf}\,(a) we show equi\-lib\-rium\ $E_1$ of Ku\-ra\-mo\-to-Siva\-shin\-sky equation\ (so labelled in
\refref{SCD07}).
If we were to take the mirror image of \reffig{f-ksconf}\,(a) with
respect to $\conf = 0$ line, and then interchange red and blue colors,
we would obtain the same solution; all equi\-lib\-ria\ belong to
the flow-invariant subspace of solutions invariant under the
reflection symmetry of the Ku\-ra\-mo\-to-Siva\-shin\-sky equation.
Similar to equilibria, time-periodic solutions of the Ku\-ra\-mo\-to-Siva\-shin\-sky equation\ that
are not repeats of pre-periodic ones \refeq{e-PPO} also
belong to the reflection-invariant subspace. See
\rf{Christiansen97,RCMR04,ReCi05,lanCvit07,ReChMi07}
for examples of such solutions.
\refFig{f-ksconf}\,(b) shows a pre-periodic solution of the Ku\-ra\-mo\-to-Siva\-shin\-sky\
system: dynamics of the second period can be obtained
from the first one by reflecting it. Both equi\-lib\-ria\ and pre-periodic orbit s have
infinitely many copies that can be obtained by
continuous translations, symmetric across the shifted symmetry line,
$\LieEl(\theta) \sigma \LieEl(-\theta)$.
Note that reflection $\sigma$ and translations
$\LieEl(\theta)$ do not commute:
\(
\sigma\,\LieEl(\theta) = -\LieEl(\theta)\,\sigma
\,,
\)
or, in terms of the generator of translations, the reflection reverses the
direction of the translation,
\(
\sigma\,\Lg = -\Lg\,\sigma
\).
Let $\flow{\zeit}{\ssp}$ denote the finite-time flow induced by
\refeq{e-ODE}, and let $\ensuremath{\ssp_{\!_{PPO}}}$ belong to a pre-periodic orbit\ defined by
\refeq{e-PPO}. Then the shifted point
$\ensuremath{\ssp_{\!_{PPO}}}' = \matrixRep(\LieEl(\theta))\,\ensuremath{\ssp_{\!_{PPO}}}$ satisfies
\[
\flow{\period{p}}{\ensuremath{\ssp_{\!_{PPO}}}'} = \matrixRep(\LieEl(\theta))
\matrixRep(\sigma)
\matrixRep(\LieEl(-\theta))
\,\ensuremath{\ssp_{\!_{PPO}}}'
\,.
\]
In contrast, a equi\-vari\-ant periodic orbit\ \refeq{e-RPO} has a distinct reflected copy
$\ensuremath{\ssp_{\!_{RPO}}}' = \matrixRep(\sigma) \ensuremath{\ssp_{\!_{RPO}}}$ with the reverse phase shift:
\[
\ensuremath{\ssp_{\!_{RPO}}}' (0) = \matrixRep(\LieEl(\theta_p ) )
\, \ensuremath{\ssp_{\!_{RPO}}}' (\period{p})
\,.
\]
In order to carry out our analysis, we must first eliminate all these
degeneracies. This we do by symmetry reduction, which we
describe next.
\section{Symmetry reduction}
\label{s-SymmRed}
A group orbit of state $\ssp$ is the set of all state space\ points
reached by
applying all symmetry actions to $\ssp$.
\emph{Symmetry reduction} is any coordinate transformation that maps
each group orbit to
a unique state space\ point \sspRefRed\ in the symmetry-reduced state space.
For the \On{2}\ symmetry considered here, we achieve this in two steps:
We first reduce continuous translation symmetry of the system by
method of slices, and then reduce the remaining reflection symmetry by
constructing an invariant polynomial basis.
To the best of our knowledge, Cartan\rf{CartanMF} was first to use
method of slices\ in purely differential geometry context and early
appearances of slicing methods in dynamical systems literature
are works of Field\rf{Field80} and Krupa\rf{Krupa90}.
Our implementation of the method of slices\ for \SOn{2}
symmetry reduction follows \refref{BudCvi14}. For a more exhaustive
review of the literature we refer the reader to \refref{DasBuch}.
Invariant polynomial or `integrity' bases\rf{gatermannHab,ChossLaut00}
are a standard tool\rf{Hilbert93,Noether15} for orbit space reduction.
They work very well in low
dimensions\rf{ChossLaut00,GL-Gil07b,SiCvi10,BuBoCvSi14}, but in high
dimensions integrity bases are high-order polynomials of the original
state space\ coordinates, accompanied by large numbers of nonlinear
{syzygies} that confine the
symmetry-reduced dynamics to lower-dimensional manifolds. These make the
geometry of the reduced state space\ complicated and hard to work with for
applications we have in mind here, such as visualizations of unstable
manifolds of invariant solutions. Even with the use of computer
algebra\rf{gatermannHab}, constructing an \On{2}-invariant integrity
basis becomes impractical for systems of dimension higher than $\sim 12$.
In spatio-temporal and fluid
dynamics applications the corresponding $n$ (Fourier series truncation)
is easily of order 10-100. The existing methods for construction of
such integrity bases are neither feasible for higher\dmn\ state space
s\rf{SiminosThesis} (we need to reduce symmetry for $10^5$-$10^6$\dmn\
systems\rf{GHCW07,ACHKW11}), nor helpful for reduced state space\
visualizations ($m$-th Fourier coefficient is usually replaced by a
polynomial of order $m$).
Here we avoid constructing such high-order \On{2} polynomial integrity
bases by a hybrid approach. We reduce the continuous symmetry by the
first Fourier mode slice\ in \refsect{s-SymmRedCont}, and \emph{then} reduce the remaining
reflection symmetry by a transformation to invariant polynomials in
\refsect{s-SymmRedDiscr}. The resulting polynomials are only
second order in the original state space\ coordinates, with no syzygies.
\subsection{{\SOn{2}} symmetry reduction}
\label{s-SymmRedCont}
Following \refref{BudCvi14}, we reduce the \SOn{2} symmetry of the
Ku\-ra\-mo\-to-Siva\-shin\-sky equation\ by implementing the \emph{first Fourier mode slice} method, \ie, by rotating the Fourier
modes as
\begin{equation}
\sspRed(\zeit) = \matrixRep(\LieEl(\phi(\zeit))^{-1})\,\ssp(\zeit)
\,,
\label{e-fFslice}
\end{equation}
where $\phi (\zeit) = \arg (\tilde{u}_1 (\zeit))$
is the phase of the first Fourier mode.
This transformation exists
as long as the first mode in the Fourier expansion \refeq{e-fFslice} does
not vanish, $b_1^2+c_1^2 > 0$,
and its effect is to fix the phase of the first Fourier mode to zero for
all times, as illustrated in \reffig{f-fFslice}. The
\SOn{2}-reduced state space\ is one dimension lower than the full
state space, with coordinates
\begin{equation}
\sspRed = (\hat{b}_1, 0, \hat{b}_2, \hat{c}_2,
\ldots \hat{b}_N, \hat{c}_N)\,.
\label{e-sspRed}
\end{equation}
The dynamics within the first Fourier mode slice\ is given by
\begin{equation}
\dot{\sspRed} = \velRed (\sspRed)
= \vel(\sspRed) - \frac{\dot{c}_1}{\hat{b}_1} \,\Lg \sspRed
\,,
\label{e-velRed}
\end{equation}
where $\Lg$ is the generator of infinitesimal \SOn{2} transformations,
$\matrixRep(\LieEl(\theta)) = \exp \Lg \theta$, and $\dot{c}_1$ is full
state space\ orbit's out-of-slice velocity, the second element of the
velocity field \refeq{e-ODE}.
Symmetry-reduced state space\ velocity \refeq{e-velRed} diverges
when the amplitude $\hat{b}_1$ of the first Fourier mode tends
to $0$. If $\hat{b}_1$ were $0$, then the transformation
\refeq{e-fFslice} would no longer be uniquely defined. However,
our experience had been such that this does not happen for
generic trajectories of a chaotic system; and the singularity
in the vicinity of $\hat{b}_1 = 0$ can be regularized by a
time-rescaling transformation\rf{BudCvi14}.
For further details we refer the reader to
\refrefs{DasBuch,BudCvi14,BuBoCvSi14}.
\begin{figure}[h]
\floatbox[{\capbeside\thisfloatsetup{capbesideposition={left,center},capbesidewidth=0.4\textwidth}}]{figure}[\FBwidth]
{\caption{ A sketch of the full state space\ trajectory $\ssp(\zeit)$ (blue
and red) projected onto the first Fourier mode subspace $(b_1, c_1)$,
with rotation phases $\phi(\zeit_{1})$, $\phi(\zeit_{2})$
at times $\zeit_1$ and $\zeit_2$, see \refeq{e-fFslice}.
In this 2\dmn\ projection we are looking at the symmetry-reduced
state space\ ``from the top''; the symmetry-reduced orbit is confined to
the horizontal half-axis $(\hat{b}_1 > 0,\, \hat{c}_1=0)$ , and the
remaining $2N\!-\!2$ coordinates are all projected onto the origin.
}\label{f-fFslice}}
{
\setlength{\unitlength}{0.5\textwidth}
\begin{picture}(1,0.75)%
\put(0,0){\includegraphics[width=\unitlength]{NBBunrot}}%
\put(0.001,0.5){\color[rgb]{0,0,0}\makebox(0,0)[lb]{\smash{$\ssp(\zeit_2)$}}}%
\put(0.18,0.32){\color[rgb]{0,0,0}\makebox(0,0)[lb]{\smash{$\ssp(\zeit_1)$}}}%
\put(0.37,0.32){\color[rgb]{0,0,0}\makebox(0,0)[lb]{\smash{$\phi (\zeit_1)$}}}%
\put(0.95,0.08){\color[rgb]{0,0,0}\makebox(0,0)[lb]{\smash{$b_1$}}}%
\put(0.42,0.07){\color[rgb]{0,0,0}\makebox(0,0)[lb]{\smash{$\sspRed(\zeit_1)$}}}%
\put(0.77,0.07){\color[rgb]{0,0,0}\makebox(0,0)[lb]{\smash{$\sspRed(\zeit_2)$}}}%
\put(0.28,0.70){\color[rgb]{0,0,0}\makebox(0,0)[lb]{\smash{$c_1$}}}%
\put(0.59264739,0.58){\color[rgb]{0,0,0}\makebox(0,0)[lb]{\smash{$\phi (\zeit_2)$}}}%
\put(0.58,0.07){\color[rgb]{0,0,0}\makebox(0,0)[lb]{\smash{$\ssp(\zeit_0)$}}}%
\end{picture}
}
\end{figure}
\subsection{{\On{2}} symmetry reduction}
\label{s-SymmRedDiscr}
Our next challenge is to devise a transformation from \refeq{e-sspRed} to
discrete-symmetry-reduced coordinates, where the equivariance under
reflection is also reduced.
Consider the action of reflection on the \SOn{2}-reduced state space. In
general, a slice is an arbitrarily oriented hyperplane, and action of the
reflection $\sigma$ can be rather complicated: it maps points within the
{slice hyperplane} into points outside of it, which then have to be rotated
into the slice.
However, the action of $\sigma$ on the first Fourier mode slice\ is particularly simple.
Reflection operation $\matrixRep(\sigma)$ of \refeq{e-DR} flips the sign of the
first \SOn{2}-reduced state space\ coordinate in \refeq{e-sspRed}, \ie, makes
the phase of the first Fourier mode $\pi$. Rotating back into the slice
by \refeq{e-fFslice}, we find that within the first Fourier mode slice, the reflection
acts by alternating the signs of even (real part) and odd (imaginary
part) Fourier modes:
\begin{eqnarray}
\matrixRepRed (\sigma) &=& \matrixRep(\LieEl(- \pi))
\matrixRep(\sigma) \nonumber \\
&=& \mathrm{diag}\left[\,1,\,-1,\,-1,\, 1,
\,1,\,-1,\,-1,\, 1,\, 1,\,
\ldots \right]
\,.
\label{e-DRRed}
\end{eqnarray}
The action on the slice coordinates
(where we for brevity omit all terms whose signs do not
change under reflection) is thus
\begin{eqnarray}
&\matrixRepRed (\sigma)& (\hat{b}_2, \hat{c}_3, \hat{b}_4,
\hat{c}_5, \hat{b}_6, \hat{c}_7, \ldots )
\nonumber \\
&& = (-\hat{b}_2, -\hat{c}_3, -\hat{b}_4,
-\hat{c}_5, -\hat{b}_6, -\hat{c}_7, \ldots )
\,.
\label{e-EvenOdd}
\end{eqnarray}
Our task is now to construct a transformation to a set of
coordinates invariant under (\ref{e-EvenOdd}).
One could declare a half of the symmetry-reduced state space\ to be a
`fundamental domain'\rf{DasBuch}, with segments of orbits that exit it
brought back by reflection, but this makes orbits appear discontinuous
and the dynamics hard to visualize. Instead, here we shall reduce the
reflection symmetry by constructing polynomial invariants of coordinates
(\ref{e-EvenOdd}). Squaring (or taking absolute value of) each
sign-flipping coordinate in (\ref{e-EvenOdd}) is not an option, since
such coordinates would be invariant under every individual sign change of
these coordinates, and that is not a symmetry of the system. We are
allowed to impose \emph{only one} condition to reduce the 2-element group
orbit of the discrete reflection subgroup of \On{2}.
How that can be achieved is suggested by Miranda and
Stone\rf{GL-Mir93,GL-Gil07b} reduction of $\Ztwo$ symmetry
\(
(x, y, z) \rightarrow (-x, -y, z)
\)
of the Lorenz flow. They construct the
symmetry-reduced ``proto-Lorenz system'' by transforming coordinates
to the polynomial basis
\begin{equation}
u = x^2 - y^2 \,,\quad v = 2xy\,, \quad z = z
\,.
\label{e-InvPolLorenz}
\end{equation}
The $x$ coordinate can be recovered from $u$ and $v$ of
\refeq{e-InvPolLorenz} up to a choice of sign, \ie, up to the original
reflection symmetry. We extend this approach in order to achieve a
$2\!-\!\mbox{to}\!-\!1$ symmetry reduction for Ku\-ra\-mo\-to-Siva\-shin\-sky\ system:
we construct the first coordinate from
squares, but then `twine' the successive sign-flipping terms
\(
(\hat{b}_2, \hat{c}_3, \hat{b}_4,
\hat{c}_5, \hat{b}_6, \hat{c}_7, \ldots )
\)
into second-order invariant polynomials basis set
\begin{eqnarray}
&& (p_2, p_3, p_4, p_5, p_6, \ldots) \nonumber\\
&& = (\hat{b}_2^2 - \hat{c}_3^2, \,
\hat{b}_2 \hat{c}_3, \,
\hat{b}_4 \hat{c}_3, \,
\hat{b}_4 \hat{c}_5, \,
\hat{b}_5 \hat{c}_6, \,
\ldots)
\,.
\label{e-RefInvs}
\end{eqnarray}
The original coordinates can be recovered
recursively by the $1\!-\!\mbox{to}\!-\!2$ inverse transformation
\begin{eqnarray}
b_2 &=& \pm \sqrt{\frac{p_2 + \sqrt{p_2^2 + 4 p_3^2}}{2}} \nonumber \\
c_3 &=& p_3 / b_2\,,\quad
b_4 \,=\, p_4 / c_3\,,\quad
c_5 \,=\, p_5 / b_4\,,\quad\cdots
\,.
\nonumber
\end{eqnarray}
To summarize: we first reduce the group orbits generated by the continuous
\SOn{2}\ symmetry subgroup by implementing the {first Fourier mode slice} \refeq{e-fFslice},
and then reduce the group orbits of the discrete 2-element reflection subgroup
by replacing the sign-changing coordinates (\ref{e-EvenOdd}) with the
invariant polynomials (\ref{e-RefInvs}).
The final \On{2} symmetry-reduced coordinates are
\begin{equation}
\sspRefRed =
(\hat{b}_1, 0,
\hat{b}_2^2 - \hat{c}_3^2,
\hat{c}_2,
\hat{b}_3,
\hat{b}_2 \hat{c}_3,
\hat{b}_4 \hat{c}_3,
\hat{c}_4,
\hat{b}_5,
\ldots )\,.
\label{e-sspRefRed}
\end{equation}
Here pairs of orbits related by reflection $\sigma$ are mapped into a
single orbit, and $\hat{c}_1$ is identically
set to $0$ by continuous symmetry reduction, thus the symmetry-reduced
state space\ has one dimension less than the full state space.
The symmetry-reduced state space\ \refeq{e-sspRefRed} retains all
physical information of the Ku\-ra\-mo\-to-Siva\-shin\-sky\ system:
equivariant equilibria\ and equi\-vari\-ant periodic orbit s of the original system
become equi\-lib\-ria\ and periodic orbit s in the symmetry-reduced state space\
\refeq{e-sspRefRed}, and pre-periodic orbit s close after one period.
For this reason, in what follows we shall refer to both
equi\-vari\-ant periodic orbit s and pre-periodic orbit s as `periodic orbit s', unless we comment on their
specific symmetry properties.
\section{Unstable manifolds of periodic orbit s}
\label{s-UnstMan}
In order to demonstrate the utility, and indeed, the necessity of the
\On{2}\ symmetry reduction, we now investigate the transition to chaos in
the neighborhood of a short Ku\-ra\-mo\-to-Siva\-shin\-sky\ {pre-periodic orbit}, focusing on the parameter range
$L \in [21.0, 21.7]$.
Our method yields a symmetry-reduced velocity field
$\velRefRed (\sspRefRed) = \dot{\sspRefRed}$
and a finite-time flow
$\flowRefRed{\zeit}{\sspRefRed(0)} = \sspRefRed(\zeit)$
in the symmetry-reduced state space\ \refeq{e-sspRefRed}.
Although we can obtain $\velRefRed (\sspRefRed)$ by chain rule, we find
its numerical integration unstable, hence in practice we obtain
$\velRefRed (\sspRefRed)$ and $\flowRefRed{\zeit}{\sspRefRed}$ from the
first Fourier mode slice\ by applying the appropriate \jacobianMs, as described in
\refappe{s-Stability}.
\begin{table}
\caption{\label{t-EeigVals}
The two leading non-marginal Floquet multipliers
$\Lambda = \exp(\period{} \mu + i\theta)$ of periodic orbit s \primeOrb{0},
\primeOrb{1}, \primeOrb{2} for system sizes $L$ studied here. Dash --
indicates that the orbit is not found for the corresponding system size.
}
\begin{center}
\begin{tabular}{c || c c | c c | c c}
& \multicolumn{2}{c}{\primeOrb{0}}
& \multicolumn{2}{c}{\primeOrb{1}}
& \multicolumn{2}{c}{\primeOrb{2}} \\
L & $\mu$ &$\theta$
& $\mu$ &$\theta$
& $\mu$ &$\theta$ \\
\hline
21.25 & $6.443 \times 10^{-4}$ & $\pm 2.177$
& \multicolumn{2}{c |}{--}
& \multicolumn{2}{c}{--} \\
21.30 & $1.839 \times 10^{-3}$ & $\pm 2.158$
& \multicolumn{2}{c |}{--}
& \multicolumn{2}{c}{--} \\
21.36 & $1.839 \times 10^{-3}$ & $\pm 2.158$
& $5.854 \times 10^{-3}$ & $0$
& $-1.623 \times 10^{-3}$ & $\pm 0.3098$ \\
& &
& $- 8.357 \times 10^{-3}$ & $0$
& & \\
21.48 & $7.638 \times 10^{-3}$ & $\pm 2.097$
& $1.307 \times 10^{-2}$ & $ 0 $
& \multicolumn{2}{c}{--} \\
& &
& $- 1.234 \times 10^{-2}$ & $0$
& & \\
21.70 & $1.739 \times 10^{-2}$ & $\pm 2.044$
& $2.521 \times 10^{-2}$ & $0$
& \multicolumn{2}{c}{--} \\
& &
& $4.157 \times 10^{-3}$ & $\pi$
& &
\end{tabular}
\end{center}
\end{table}
At $L=21.0$, the Ku\-ra\-mo\-to-Siva\-shin\-sky\ system has a stable periodic orbit\ \primeOrb{0},
which satisfies
\(
\sspRefRed_{\primeOrb{0}} = \flowRefRed{\period{\primeOrb{0}}}{
\sspRefRed_{\primeOrb{0}}}
\)
for any point $\sspRefRed_{\primeOrb{0}}$ on the periodic orbit\ $\primeOrb{0}$.
Linear stability of a periodic orbit\ is described by the Floquet
multipliers
$\Lambda_i$ and Floquet vectors $\VRefRed_i$,1
which are the eigenvalues and eigenvectors of the \jacobianM\
$\jMpsRefRed_{p}$
of the finite-time flow
$\flowRefRed{\period{p}}{\sspRefRed_{p}}$
\[
\jMpsRefRed_{p} \VRefRed_i = \Lambda_i \VRefRed_i \,.
\]
Each periodic orbit\ has at least one marginal Floquet multiplier $\Lambda_v = 1$,
corresponding to the velocity field direction. When $L < 21.22$,
all other Floquet multipliers of \primeOrb{0}\ have absolute values less
than $1$. At $L \approx 21.22$, leading
complex pair of Floquet multipliers
$\Lambda_{1, 2}$ crosses the unit circle, and the corresponding
eigenplane
spanned by the real and imaginary parts of $\VRefRed_1$ develops
`spiral out' dynamics that connects to a $2$-torus.
\begin{figure}[h]
\centering
\begin{overpic}[height=0.45\textwidth]{PPO1Poincare}
\put (-0.75,-1) {(a)}
\end{overpic} \quad
\begin{overpic}[height=0.45\textwidth]{PsectL21p25}
\put (-0.75,-1) {(b)}
\end{overpic} \quad
\caption{\label{f-Psect}
(a) Pre-periodic orbit\ \primeOrb{0}\ (red), its velocity field
$\velRefRed(\sspRefRed_{\primeOrb{0}})$ at the starting point
(green), orthogonal vectors that span the eigenplane
corresponding to the leading Floquet vectors (blue) and
the {Poincar\'e section} hyperplane (gray, transparent).
(b) Spiral-out dynamics of a single trajectory
in the {Poincar\'e section} projected onto $(e_1, e_2)$ plane,
system size $L=21.25$.
}
\end{figure}
In order to study dynamics within the neighborhood of $\primeOrb{0}$, we
define a {Poincar\'e section} as the hyperplane of points
$\sspRefRed_{\PoincS}$
in an open
neighborhood of $\sspRefRed_{\primeOrb{0}}$, orthogonal to the tangent
$\velRefRed (\sspRefRed_{\primeOrb{0}})$ of the orbit at the {Poincar\'e section} point,
\begin{equation}
(\sspRefRed_{\PoincS} - \sspRefRed_{\primeOrb{0}})
\cdot \velRefRed (\sspRefRed_{\primeOrb{0}}) = 0
\quad \mbox{and} \quad
|| \sspRefRed_{\PoincS} - \sspRefRed_{\primeOrb{0}} || < \alpha
\,,
\label{e-Psect}
\end{equation}
where $||.||$ denotes the Euclidean (or $L2$) norm, and the threshold
$\alpha$ is empirically set to $\alpha = 0.9$ throughout.
The locality condition in \refeq{e-Psect} is a computationally
convenient way
to avoid Poincar\'e section border\rf{atlas12,DasBuch}, defined
as the set of points $\sspRefRed_{\PoincS}^*$ that satisfy the hyperplane
condition
$(\sspRefRed_{\PoincS}^* - \sspRefRed_{\primeOrb{0}})
\cdot \velRefRed (\sspRefRed_{\primeOrb{0}}) = 0$
, but their orbits do not intersect this hyperplane
transversally, \ie\
$\velRefRed (\sspRefRed_{\PoincS}^*)
\cdot \velRefRed (\sspRefRed_{\primeOrb{0}}) = 0$.
From here on, we study the discrete time dynamics induced by the
flow on the Poincar\'e section \refeq{e-Psect}, as visualized
in \reffig{f-Psect}\,(a).
In \reffig{f-Psect} and the rest of the state space\ projections of this
paper, projection bases are constructed as follows:
Real and imaginary parts of the Floquet vector $\VRefRed_{1}$ define
an ellipse
$\Re [\VRefRed_{1}] \cos \phi + \Im [\VRefRed_{1}] \sin \phi $
in the neighborhood of $\sspRefRed_{\primeOrb{0}}$, and we pick as the first two
projection-subspace spanning vectors the principal axes of this ellipse.
As the third vector we take the velocity field
$\velRefRed (\sspRefRed_{\primeOrb{0}})$, and the projection bases
$(e_1, e_2, e_3)$ are found by orthonormalization of these vectors
via the Gram-Schmidt procedure. All state space\ projections are centered
on $\sspRefRed_{\primeOrb{0}}$, \ie, $\sspRefRed_{\primeOrb{0}}$ is
the origin of all {Poincar\'e section} projections.
As an example, we follow a single trajectory starting from
$\sspRefRed_{\primeOrb{0}} + 10^{-1} \Re[\VRefRed_1] $ as it connects to the
2-torus surrounding the periodic orbit\ in \reffig{f-Psect}\,(b).
For \reffig{f-Psect}\,(b) and
all figures to follow, the two leading non-marginal Floquet multipliers
of $\primeOrb{0}$, $\primeOrb{1}$ and $\primeOrb{2}$ are listed in
\reftab{t-EeigVals}.
For system size $L = 21.25$ the complex unstable
Floquet multiplier pair is nearly marginal, $|\Lambda_{1,2}|= 1.00636$,
hence the spiral-out is very slow.
\begin{figure}[h]
\centering
\includegraphics[width=0.9\textwidth]{UnstMan21p30png}
\caption{\label{f-UnstMan21p31a}
Unstable manifold (gray) of $\primeOrb{0}$ on the {Poincar\'e section}
\refeq{e-Psect} and an individual trajectory (red)
within, system size $L=21.30$.
}
\end{figure}
Assume that $\delta \sspRefRed(0)$ is a small perturbation to
$\sspRefRed_{\primeOrb{0}}$ that lies in the plane spanned by
$(\Re[\VRefRed_1], \, \Im[\VRefRed_1])$.
Then there exists a coefficient vector $c = (c_1, c_2)^T$, with
which we can express
$\delta \sspRefRed (0)$ in this plane as
\begin{equation}
\delta \sspRefRed (0) = W c \,,
\label{e-Perturb}
\end{equation}
where $W = [\Re[\VRefRed_1], \, \Im[\VRefRed_1]]$
has real and imaginary
parts of the Floquet vector $\VRefRed_1$ on its columns.
Without a loss of generality, we can rewrite $c$ as
$ c = \delta r R( \theta) c^{(1)} $, where
$ c^{(1)} = (1, 0)^T$ and
$R(\theta)$ is a $[2\!\times\!2]$ rotation matrix.
Thus \refeq{e-Perturb} can be expressed as
$\delta \sspRefRed (0) = \delta r W R(\theta) c^{(1)}$.
In the linear approximation, discrete time dynamics
$\delta \sspRefRed (n \period{\primeOrb{0}})$ is given by
\begin{equation}
\delta \sspRefRed (n \period{\primeOrb{0}}) = |\Lambda_1|^n \delta r W
R(\theta - n \arg \Lambda_1)
c^{(1)} \,,
\label{e-DiscreteTime}
\end{equation}
which can then be projected onto the {Poincar\'e section}
\refeq{e-Psect} by acting from the left with the projection operator
\begin{equation}
\mathbb{P} (\sspRefRed_{\PoincS}) =
\matId - \frac{\velRefRed(\sspRefRed_{\PoincS}) \otimes
\velRefRed(\sspRefRed_{\primeOrb{0}})}{
\inprod{\velRefRed(\sspRefRed_{\PoincS})}{
\velRefRed(\sspRefRed_{\primeOrb{0}})}
} \,, \label{e-ProjPsect}
\end{equation}
computed at $\sspRefRed_{\PoincS} = \sspRefRed_{\primeOrb{0}}$.
In \refeq{e-ProjPsect},
$\otimes$ denotes the outer product.
Defining
$\delta \sspRefRed_{\PoincS} \equiv
\mathbb{P} (\sspRefRed_{\PoincS}) \delta \sspRefRed$
for a small perturbation
$\delta \sspRefRed$ to the point $\sspRefRed_{\PoincS}$
on the Poincar\'e
section, discrete
time dynamics of $\delta \sspRefRed_{\PoincS}$ in
the {Poincar\'e section} is given by
\begin{equation}
\delta \sspRefRed_{\PoincS} [n] = |\Lambda_1|^n \delta r W_{\PoincS}
R(\theta - n \arg \Lambda_1)
c^{(1)} \,,
\label{e-DiscreteTimePsect}
\end{equation}
where
$W_{\PoincS} = [\Re[\VRefRed_{1, \PoincS}], \, \Im[\VRefRed_{1, \PoincS}]]
= \mathbb{P}(\sspRefRed_{\primeOrb{0}}) W$,
and $n$ is the discrete time variable counting returns to the
{Poincar\'e section}. In the {Poincar\'e section}, the solutions
\refeq{e-DiscreteTimePsect} define ellipses which expand and rotate
respectively by factors of $|\Lambda_1|$ and $\arg \Lambda_1$ at each
return. In order to resolve the unstable manifold, we start trajectories
on an elliptic band parameterized by $(\delta,\phi)$, such that the
starting point in the band comes to the end of it on the first return,
hence totality of these points cover the unstable manifold in the linear
approximation. Such set of perturbations are given by
\begin{equation}
\delta \sspRefRed_{\PoincS} (\delta, \phi) =
\epsilon |\Lambda_1|^\delta W_{\PoincS} R(\phi) c^{(1)}
\,, \quad \mbox{where } \delta \in [0, 1) \,, \,
\phi \in [0, 2 \pi )
\,,
\label{e-InitUnstMan}
\end{equation}
and $\epsilon$ is a small number. We set $\epsilon = 10^{-3} $ and
discretize \refeq{e-InitUnstMan}
by taking $12$ equidistant points in $[0, 1)$ for $\delta$ and
$36$ equidistant points in $[0, 2 \pi)$ for $\phi$ and integrate
each
$\sspRefRed_{\primeOrb{0}} + \delta \sspRefRed_{\PoincS} (\delta, \phi)$
forward in time. \refFig{f-UnstMan21p31a} shows the unstable
manifold of $\primeOrb{0}$ resolved by this procedure at system size
$L=21.30$, for which the torus surrounding $\primeOrb{0}$
appears to be unstable
as the points approaching to it first slow down and then
leave the neighborhood in transverse direction.
In order to illustrate this better, we marked
an individual trajectory in \reffig{f-UnstMan21p31a} color red.
In \reffig{f-UnstMan21p31b} we show initial points that go
into the calculation, and their first three returns in order
to illustrate the principle of the method.
\begin{figure}[h]
\floatbox[{\capbeside\thisfloatsetup{capbesideposition={left,center},capbesidewidth=0.3\textwidth}}]{figure}[\FBwidth]
{\caption{Initial points (black) on the {Poincar\'e section} for
unstable manifold computation and their first (red),
second (green), and third (blue) returns. Inset: zoomed out
view of the initial points and their first three returns.}\label{f-UnstMan21p31b}}
{
\begin{overpic}[width=0.6\textwidth]{UnstMan21p31ret3Zoompng}
\put (16, 12){\includegraphics[height=0.28\textwidth]{UnstMan21p31ret3png}}
\end{overpic}
}
\end{figure}
As we continue increasing the system size, we find that
at $L \approx 21.36$, trace of the invariant torus disappears
and two new periodic orbit s $\primeOrb{1}$ and $\primeOrb{2}$ emerge in the
neighborhood of $\primeOrb{0}$. Both of these orbits appear as period~3
periodic orbit s in the Poincar\'e map. While
$\primeOrb{1}$ is unstable
(found by a Newton search),
$\primeOrb{2}$ is initially
stable with a finite basin of
attraction. The unstable manifold of $\primeOrb{0}$ connects
heteroclinically to the stable manifolds of $\primeOrb{1}$ and
$\primeOrb{2}$.
As we show in
\reffig{f-Connectionsa}, resolving the unstable manifold of
$\primeOrb{0}$ enables us to locate these heteroclinic connections
between periodic orbit s. Note that 1-dimensional
stable manifold of $\primeOrb{1}$ separates the unstable manifold of
$\primeOrb{0}$ in two pieces. Green and blue orbits in
\reffig{f-Connectionsa} appear to be at two sides of this
invariant boundary: while one of them converges to $\primeOrb{2}$,
the other leaves the neighborhood to explore other parts of the
state space\ that are not captured by the Poincar\'e section,
following the unstable manifold of $\primeOrb{1}$.
\begin{figure}[h]
\centering
\includegraphics[width=0.9\textwidth]{UnstMan21p36png}
\caption{\label{f-Connectionsa}
Unstable manifold (gray) of $\primeOrb{0}$ on the {Poincar\'e section}
\refeq{e-Psect} at $L=21.36$. Colored dots
correspond to different individual trajectories within
the unstable manifold, with qualitatively different
properties. Diamond shaped markers correspond to the
period-3 orbits $\primeOrb{1}$ (magenta) and $\primeOrb{2}$ (cyan).
}
\end{figure}
As the system size is increased, $\primeOrb{2}$ becomes
unstable at $L \approx 21.38$. At $L \approx 21.477$ the two complex
unstable Floquet multipliers collide on the real axis and at
$L \approx 21.479$ one of them crosses the unit circle.
After this bifurcation, we were no longer able to continue this
orbit. At $L = 21.48$, the spreading of the $\primeOrb{0}$'s
unstable manifold becomes more dramatic, and its boundary is set
by the 1-dimensional unstable manifold of $\primeOrb{1}$, as shown in
\reffig{f-Connectionsb}. We compute the unstable manifold of
$\primeOrb{1}$ similarly to \refeq{e-InitUnstMan}, by integrating
\begin{equation}
\sspRefRed_{\PoincS} (\delta) = \sspRefRed_{\primeOrb{1}, \PoincS} \pm
\epsilon \Lambda_1^\delta \VRefRed_{1, \PoincS}
\,, \quad \mbox{where } \delta \in [0, 1) \,.
\label{e-InitUnstMan1D}
\end{equation}
$\Lambda_1$ and $\VRefRed_{1}$ in \refeq{e-InitUnstMan1D} are the
unstable Floquet multiplier and the corresponding Floquet vector
of $\sspRefRed_{\primeOrb{1}}$, and the initial conditions
\refeq{e-InitUnstMan1D} cover the unstable manifold of
$\sspRefRed_{\primeOrb{1}}$ in the linear approximation.
\begin{figure}[h]
\centering
\includegraphics[width=0.9\textwidth]{UnstManPPO4L21p48png}
\caption{\label{f-Connectionsb}
Unstable manifold of $\primeOrb{0}$ (gray) and two orbits
(black and green) within at $L=21.48$. Red points lie on
the 1-dimensional unstable manifold of $\primeOrb{1}$
(magenta).
}
\end{figure}
A negative real Floquet multiplier of \primeOrb{1} crosses the unit
circle at $L \approx 21.6$ leading to ``drifting'' dynamics
in the associated unstable direction.
Such ``symmetry-breaking'' bifurcations of equi\-vari\-ant periodic orbit s with
$\Ztwo$ symmetry are ubiquitous in many physical settings:
Earlier examples are studies of reduced-order models of
convection\rf{KnoWei81}, forced pendulum\rf{DBHL82}, and
Duffing oscillator\rf{NoFre82}, which reported that symmetry
breaking bifurcations precede period doubling route to chaos.
A key observation was made by Swift and Wiesenfeld\rf{SwiWie84},
who showed in the context of periodically driven damped pendulum
that Poincar\'e map associated with the symmetric system is the
second iterate of another ``reduced'' Poincar\'e map, which
identifies symmetry-equivalent points. They then argue that
$\Ztwo$-symmetric periodic orbits generically
do not undergo period doubling bifurcations when a single
parameter of the system varied. More recent
works\rf{MarLopBla04,BlMaLo05} adapt \refref{SwiWie84}'s
reduced Poincar\'e map to fluid systems in order to study their bifurcations
in the presence of symmetries. For a review of the
symmetry-breaking bifurcations in fluid dynamics, see \refref{CraKno91}.
As in the previous cases, in order
to investigate the dynamics of the system at this stage, we
compute and visualize the unstable manifold of \primeOrb{1}.
Similarly to
\refeq{e-InitUnstMan} and \refeq{e-InitUnstMan1D}, the 2-dimensional
unstable manifold of \primeOrb{1}
is approximately covered by initial conditions
\begin{equation}
\sspRefRed_{\PoincS} (\delta, \phi) =
\sspRefRed_{\primeOrb{1}, \PoincS} +
\epsilon \left[
|\Lambda_1|^\delta \cos \phi \, \VRefRed_{1, \PoincS}
+ |\Lambda_2|^\delta \sin \phi \, \VRefRed_{2, \PoincS}
\right]
\label{e-InitUnstMan2DReal}
\end{equation}
where $\delta \in [0, 1) \, , \; \phi \in [0, 2 \pi)$.
At system size $L=21.7$, we set $\epsilon = 10^{-3}$ and
discretize \refeq{e-InitUnstMan2DReal} by choosing
$10$ and $36$ equally spaced values for $\delta$ and $\phi$,
respectively. First $38$ returns of orbits generated
according to \refeq{e-InitUnstMan2DReal} are shown in
\reffig{f-ConnectionsL21p7a} as red points along with
the unstable manifold of \primeOrb{0} (gray). Note that,
unlike \reffig{f-Connectionsb},
in \reffig{f-ConnectionsL21p7a} there is no clear
separation on the unstable manifold of \primeOrb{0}.
This is because the
connection of \primeOrb{0}'s unstable manifold to
\primeOrb{1} is no longer captured by the {Poincar\'e section}
\refeq{e-Psect} after the unstable manifold of \primeOrb{1}
becomes 2-dimensional.
Yet, unstable manifold of \primeOrb{1}
still shapes that of \primeOrb{0}.
\begin{figure}[h]
\centering
\includegraphics[width=0.9\textwidth]{UnstManPPO4L21p7png}
\caption{\label{f-ConnectionsL21p7a}
Unstable manifolds of $\primeOrb{0}$ (gray) and
$\primeOrb{1}$ (red) on the Poincar\'e
section \refeq{e-Psect} at $L=21.7$.
Magenta, cyan, and yellow diamond markers respectively indicate the
{Poincar\'e section} points of $\primeOrb{1}$, $\primeOrb{3}$, and
$\primeOrb{4}$.
Green and black dots correspond to two individual orbits
started on the linear approximation to the unstable manifold
of $\primeOrb{1}$, which visit neighborhoods of
$\primeOrb{3}$ and $\primeOrb{4}$ respectively.
}
\end{figure}
Since the leading Floquet exponent $\mu_1$ of
\primeOrb{1} is approximately an order of magnitude larger than $\mu_2$
(see \reftab{t-EeigVals}), unstable manifold of \primeOrb{1}
appears as if it is 1-dimensional in \reffig{f-ConnectionsL21p7a}.
However, it is absolutely crucial to study this manifold in $2$
dimensions as different initial conditions in this 2-dimensional
manifold connect to the regions of state space\ with qualitatively
different dynamics. In order to illustrate this point, we have
marked two individual trajectories on the unstable manifold of
\primeOrb{1} with black and green in \reffig{f-ConnectionsL21p7a}.
After observing that these orbits have nearly recurrent dynamics,
we ran Newton searches in their vicinity and found two new
periodic orbits $\primeOrb{3}$ and $\primeOrb{4}$,
marked respectively with cyan and yellow diamonds
on \reffig{f-ConnectionsL21p7a}. In the full state space\
$\primeOrb{3}$ is a pre-periodic orbit\
\refeq{e-PPO}, whereas $\primeOrb{4}$ is a equi\-vari\-ant periodic orbit\ \refeq{e-RPO}
with a non-zero drift. We show a time segment of the
orbit marked green on \reffig{f-ConnectionsL21p7a}
without symmetry reduction, as
color-coded amplitude of the scalar field $u(\conf, \zeit)$ in
\reffig{f-ConnectionsL21p7b}\,(a).
For comparison we also show two repeats of
\primeOrb{1} (bottom) and \primeOrb{4} (top) in
\reffig{f-ConnectionsL21p7b}\,(b). \refFig{f-ConnectionsL21p7b}
suggests that this orbit leaves the neighborhood of
\primeOrb{1} following a heteroclinic connection to \primeOrb{4}.
\begin{figure}[h]
\floatbox[{\capbeside\thisfloatsetup{capbesideposition={left,center},capbesidewidth=0.4\textwidth}}]{figure}[\FBwidth]
{\caption{(a) Space-time visualization of a segment of the orbit marked
green on \reffig{f-ConnectionsL21p7a} as it leaves the neighborhood of $\primeOrb{1}$
and enters the neighborhood of $\primeOrb{4}$.
(b) Space-time visualizations of $\primeOrb{1}$ (bottom) and
$\primeOrb{4}$ (top).}\label{f-ConnectionsL21p7b}}
{
\begin{overpic}[height=0.45\textwidth]{KSHetCon21p7conf}
\put (0.5,-1) {(a)}
\end{overpic} \quad
\begin{overpic}[height=0.45\textwidth]{KSHetCon21p7POsconf}
\put (0.5,-1) {(b)}
\end{overpic}
}
\end{figure}
In \reffig{f-ConnectionsL21p7a}, some of the red points appear on
the unstable manifold of \primeOrb{0}. These points corresponds to
trajectories that leave the unstable manifold of \primeOrb{1}, come back
after exploring other parts of the state space\ and follow unstable manifold
of \primeOrb{0}. We could have excluded these points by showing shorter
trajectories for higher values of $\delta$ in \refeq{e-InitUnstMan2DReal}
in \reffig{f-ConnectionsL21p7a}, however we chose not to do so
in order to stress that visualizations of unstable manifolds of periodic orbit s
are not restricted to the dynamics within a small neighborhood of a periodic orbit,
but in fact they illuminate the geometry of the flow in a finite part of
the strange attractor.
An interesting feature of the bifurcation scenario studied here is the
apparent destabilization of the invariant torus before its breakdown.
Note that in \reffig{f-UnstMan21p31a} the trajectories within
the unstable manifold of \primeOrb{0} diverge in normal direction from the
region that was inhabited by a stable 2-torus for lower values of $L$. This
suggest that the invariant torus has become normally
hyperbolic\rf{Feniche71}.
This torus could be computed by the method of \refref{LCC06}, but our
goal here is more modest, what we have computed already amply
demonstrates the utility of our \On{2} symmetry reduction.
Note also that the
stable period-3 orbit $\primeOrb{2}$ in \reffig{f-Connectionsa} has a finite basin
of attraction, and the trajectories which do not fall into it leave its
neighborhood. In typical scenarios involving generation of stable
- unstable pairs of periodic orbits within an invariant torus (see {e.g.}\
\refref{arnold82}), the torus becomes a heteroclinic connection between the
periodic orbit pair. Here the birth of the period-3 orbits appears to
destroy the torus.
\section{Summary and future directions}
\label{s-Discuss}
The two main results presented here are:
1) a new method for reducing the \On{2}-symmetry of PDEs,
and
2) a symmetry-reduced state space\ Poincar\'e \ section visualization of
1- and 2-dimensional unstable manifolds of Ku\-ra\-mo\-to-Siva\-shin\-sky\ periodic orbit s.
Our method for the computation of unstable manifolds is
general and can find applications in many other ODE and PDE settings.
The main idea here is a generalization of Gibson \etal\rf{GHCW07} method
for visualizations of the unstable manifolds of equi\-lib\-ria, originally applied
to plane Couette flow, a setting much more complex then the current
paper. All our computations are carried
out for the full Ku\-ra\-mo\-to-Siva\-shin\-sky equation\ \refeq{e-ks}, in 30 dimensions, and it is
remarkable how much information is captured by the 2- and 3-dimensional
projections of the \On{2} symmetry-reduced {Poincar\'e section s} - none of
that structure is visible in the full state space.
The Ku\-ra\-mo\-to-Siva\-shin\-sky\ \On{2} symmetry reduction method described here might require
modifications when applied to other problems. For example,
for PDEs of space dimensions larger then one,
there can be more freedom in choosing the phase fixing
condition \refeq{e-fFslice}.
This indeed is the case for shear flows with both
homogeneous (streamwise and spanwise translation invariant) and inhomogeneous
(wall-normal) directions. When adapting the first Fourier mode slice\ method
to such problems, one should experiment with the dependence
of the phase fixing condition on the inhomogeneous coordinate
such that the slice fixing phase is uniquely defined for
state space\ regions of interest; see chapter 3 of
\refref{BudanurThesis} for details.
\refRef{WiShCv15} makes this choice for pipe
flow by taking a `typical state' in the turbulent flow, setting all
streamwise Fourier modes other than the first one to zero, and using
this state as a ``slice template''.
Another point to be taken into
consideration for canonical shear flows is that their symmetry
group is $\SOn{2} \times \On{2}$. So far, continuous
symmetry reduction in pipe flows\rf{ACHKW11,WiShCv15}
were confined to settings, where an imposed symmetry in conjugacy
class of spanwise reflection disallows spanwise rotations.
When no such restriction is present, one needs two
conditions for fixing both streamwise and spanwise translations.
These conditions must be chosen such that the order at which
continuous symmetries are reduced does not matter. For
direct products of commuting $\SOn{2}$ symmetries, this is
a straightforward task and outlined in section 3 of
\refref{BudanurThesis}. An application of these ideas to the
pipe flow is going to appear in a future publication\rf{BudHof17}.
Furthermore, while invariant polynomials similar to (\ref{e-RefInvs}) can
be constructed for any problem with a reflection symmetry, an intermediate
step is necessary if the action of reflection $\sigma$ symmetry is not
the sign flip of a subset of coordinates. In that case, one should first
decompose the state space\ into symmetric and antisymmetric subspaces by
computing
$\ssp_S = (1/2) [\ssp + \matrixRep (\sigma) \ssp]$ and
$\ssp_A = (1/2) [\ssp - \matrixRep (\sigma) \ssp]$, respectively,
and construct invariants analogous to (\ref{e-RefInvs}) for elements of
$\ssp_A$ that are not strictly zero.
Generalizations of this approach to richer discrete symmetries, such
as dihedral groups, remains an open problem, with potential
application to systems such as the Kolmogorov
flow\rf{PlaSirFit91,Faraz15}.
Bifurcation scenarios similar to the one studied here are
ubiquitous in high-dimensional systems. For example, Avila
\etal\rf{AvMeRoHo13} study of transition to turbulence in pipe flow, and
Zammert and Eckhardt's study of the plane Poiseuille flow\rf{ZamEck15}
both report torus bifurcations of equi\-vari\-ant periodic orbit s along transitions to chaos. We
believe that the methods presented in this paper can lead to a
deeper understanding of these scenarios.
While unstable manifold visualizations of periodic orbit s in the
symmetry-reduced state space\ illustrates bifurcations of these orbits,
our motivation for
investigating such objects is not a study of bifurcations, but ultimately
a partition of the turbulent flow's
state space\ into qualitatively different regions, and
construction of the corresponding symbolic dynamics.
\refFig{f-ConnectionsL21p7a} and \reffig{f-ConnectionsL21p7b} demonstrate
our progress in this direction:
we are able to identify symmetry breaking heteroclinic connections from
non-drifting solutions to the drifting ones.
Such observations would have been very hard to make without reducing
symmetries of the system, since each equi\-vari\-ant periodic orbit\ has a reflection copy,
corresponding to a solution drifting in the other direction; and each
such solution has infinitely many copies obtained by translations.
\begin{acknowledgements}
We are grateful to
Xiong Ding,
Evangelos Siminos,
Simon Berman,
and
Mohammad Farazmand
for many fruitful discussions.
\end{acknowledgements}
|
{
"timestamp": "2016-11-09T02:06:32",
"yymm": "1509",
"arxiv_id": "1509.08133",
"language": "en",
"url": "https://arxiv.org/abs/1509.08133"
}
|
"\\section*{Acknowledgements}\nWe thank Jurgen Schukraft for providing valuable feedback on an early(...TRUNCATED)
| {"timestamp":"2016-03-10T02:13:18","yymm":"1509","arxiv_id":"1509.07939","language":"en","url":"http(...TRUNCATED)
|
"\\section{Introduction}\n\nOver the past few decades, the hidden simplicity of $\\mathcal{N} = 4$ s(...TRUNCATED)
| {"timestamp":"2015-09-29T02:14:02","yymm":"1509","arxiv_id":"1509.08127","language":"en","url":"http(...TRUNCATED)
|
"\\section{Conclusion} \\label{sec:conclusion}\nThis paper contributed to the problem of active simu(...TRUNCATED)
| {"timestamp":"2016-08-30T02:06:04","yymm":"1509","arxiv_id":"1509.08155","language":"en","url":"http(...TRUNCATED)
|
"\\section{Introduction}\n\n \n\nBy their very definition, local martingales are\n``almost'' marting(...TRUNCATED)
| {"timestamp":"2017-03-03T02:04:32","yymm":"1509","arxiv_id":"1509.08280","language":"en","url":"http(...TRUNCATED)
|
"\\section{Introduction}\n\nIn this work we investigate the reliability of the Kirkwood approximatio(...TRUNCATED)
| {"timestamp":"2015-09-29T02:04:27","yymm":"1509","arxiv_id":"1509.07924","language":"en","url":"http(...TRUNCATED)
|
"\\chapter*{Abstract}\n\nAbstract to go here.\n\n\n\\chapter*{Acknowledgements} \n\nThis thesis sim(...TRUNCATED)
| {"timestamp":"2015-09-29T02:17:21","yymm":"1509","arxiv_id":"1509.08223","language":"en","url":"http(...TRUNCATED)
|
"\\section*{Introduction}\n\n\n\\noindent Quasidiagonality was first introduced by Halmos for sets o(...TRUNCATED)
| {"timestamp":"2016-08-05T02:00:54","yymm":"1509","arxiv_id":"1509.08318","language":"en","url":"http(...TRUNCATED)
|
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