Critical phenomenon, crisis and transition to spatiotemporal chaos

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Critical phenomenon, crisis and transition to
spatiotemporal chaos

• Kaifen HE
• Inst. Low Ener. Nucl. Phys.
• Beijing Normal Univ.,
• Beijing China

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What is STC?

• TC state two orbits starting from adjacent
  points separate exponentially, Luaponov exponent.
• Does STC refer to any chaotic state in
  space-time-dependent system?
• In numerical simulations it has been found that a
  wave state can be temporally chaotic but
  spatially coherent.
• With STC we refer to those states where the
  spatial coherence is destroyed.
• To solve mechanism for the onset to STC, a key
  problem is how the spatial coherence is
  destroyed.
• How to judge an STC?

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What causes the loss of spatial coherence
?

• Landaus picture (
  ) and Ruelle-Takens picture ( )
  of turbulence. Landaus picture is not supported
  by experimental results, and Ruelle-Takens route
  can only explain chaotic motion in systems with
  few free dimensions.
• Consequently, there must be another process which
  can destroy the spatial coherence----crisis?
• There is evidence to show that there can be
  different routes to STC.

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Model equation

• We use the driven/damped nonlinear drift-wave
  equation as the model
• The unperturbed equation is non-integrable, with
  the driving and damping, it shows very rich
  phenomena (with pseudospectral method).
• The unperturbed equation has solitary wave-like
  solution, a harmonic wave driving.

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fully-developed turbulence?
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Relation to nonlinear dynamics in time-dependent
systems

• Nonlinear dynamics of ST-system is closely
  related to that of the T-system.
• A steady wave (SW) is a fixed point in the
  Fourier space in the moving frame following it.
• One can make stability analysis for the fixed
  point under perturbation as we usually do in
  T-systems.
• If studying response of an SW to a perturbation
  wave (PW) in the moving frame, the results agree
  qualitatively with that by solving the pde.

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The equation in moving frame
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Steady wave

• The equation has SW solutions
• With expansion
• mode amplitudes and phases of an SW, can
  be solved in the reference frame following the SW
  from the SW equation

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Mode equations of steady wave
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Hystereses of steady wave energy

• Wave energy of SW forms groups of hystereses.
• An SW state can be stable or unstable.
• Numerical simulation shows that all the
  complicated behaviours of the system seem in
  connection with the groups of hystereses.
• Does this fact suggests that the states at the
  negative tangency branch plays an important role
  in the complexity?

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Bistability of wave energy
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Hystereses of wave energy
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Another two groups of hystereses
in other ranges of
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State diagram in parameter space,
clearly They are associated with groups of
hystereses respectively,and winding number
bifurctions
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An example of multi-hystereses
here F is in the form of coherent structure of
solitary wave
Its SW shows multi-hystereses, which is
associated with winding number bifurcation
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4th
A driven/damped KdV equation shows multi-hysteres
es
3rd
2nd
1st
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1st
winding number bifurcation along multi-hystereses
2nd
3rd
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Perturbation wave

• If an SW is perturbed by a PW, in the moving
  frame the PW motion is governed by

(Linear dispersion)
(Nonlinear dispersion)
(Self-nonlinearity)
s
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Linear response of PW to SW
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Fourier space of SSW and PW

• The SW and PW can be expanded respectively in
  Fourier space
• In linear approximation of PW we assume

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Eigen equation of PW

• If the self-nonlinearity in the PW equation is
  neglected, one gets an eigen equation of PW
• From which the eigen motion of the PW modes can
  be solved.
• In general PW eigenvalues are complex conjugated
  (the motion is allowed in both directions
  relative to SSW).

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Schematic plot for the SSW and eigen motion of
PW
Due to reflection at the SW, the dispersion of PW
is altered
?
?
?
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Hopf instability
Resonance of two internal modes and Hopf
instability
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Resonance of one internal mode with applied
frequency (Doppler shift) and Saddle instability
In the middle branch of hysteresis the
eigenfrequency zero----no characteristic frequenc
y is allowed to pass through the system
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variation of eigenfrequency along a hysteresis
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Vanishing of mode eigenfrequency and saddle
instability
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Variation of eigenvalues along a hysteresis of
the KdV model, one can also see vanishing of
k2 mode eigen Frequency.
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The physics of saddle instability

• In the moving frame, saddle instability occurs
  when a mode eigen frequency is 0.
• That is, in the lab frame the mode eigen
  frequency is .
• It indicates that the mode eigenfrequency is
  resonant with the applied frequency .
• However, due to scattering at the SSW, the PW
  mode eigenfrequency has been changed nonlinearly.

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Saddle steady wave

• When an SW is unstable due to saddle instability,
  we call it as saddle steady wave (SSW).
• SSW locate at the negative tangency branch of the
  hystereses.
• The internal modes one-by-one become resonant
  with the applied frequency, which causes groups
  of hystereses and superstructures in the state
  diagram.

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Reconnection of eigenvalues and Hopf
bifurcation ------real parts
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Reconnection of eigenvalues in Hopf
birfurcation ------- imaginary parts
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The new frequency of the Hopf bifurcation appeari
ng in the energy and wave patterns
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Nonlinear N-mode and P-mode

• Linear N-mode and P-mode in a stream of velocity
  v, due to Doppler shift,
  
• N-mode
• There is evidence to show that
• Saddle-bifurcation corresponds to transition of
  nonlinear N-mode to P-mode
• Hopf-bifurcation corresponds to exchange of
  energy types of a pair of N-mode and P-mode

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Linear dispersion in the moving frame
set
We have
When varying from 0-1, we come across the
critical point at which the eigenfrequency of
mode k changes sign for our parameter, the
positions for k1,2,3,4 are 0.78,0.465,0.279,0.17
9 corresponding to the superstructures
What happens when nonlinear dispersion is
included?
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Nonlinear dispersion

• In linear case ( ), mode eigenfrequency
  is only allowed to transport in one direction.
• When nonlinear dispersion is considered, the
  eigenfrequency is allowed to transport in both
  directions relative to the SW.
• One may use eigenvector to define a N-mode and
  P-mode

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If define mode eigenvector as
Then N-mode and P-mode can be defined according to
or
According to this definition, the middle branch
just corresponds to the situation when the two
eigenvectors of the resonance mode k2 are equal.
In the lower branch N, upper branch P
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Eigenvectors of k2 mode along the 1st
hysteresis for the KdV model
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Stable and unstable orbits of SSW

• Set
• by neglecting the self-nonlinear term, one can
  find two stable and two unstable orbits of an SSW
  from the PW equation.
• An important feature of the SO/UO orbits are
  they correspond to constant mode phases,
  respectively, and the difference between two
  stable/unstable orbits is .

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Unstable orbit
Stable orbit
Saddle point
SO and UO of a saddle point
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Stable orbits and unstable orbits of SSW
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Effect of relative phase difference between PW
and SSW

• This result indicates that there are two (group
  of ) relative phase differences between the PW
  modes and SSW, at which the PW mode amplitudes
  are excited by the SSW, there are another two
  groups
• This knowledge is very important for us to
  understand the turbulent behaviors.

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Self-nonlinearity of PW is included
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Perturbation wave including self-nonlinear term
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Mode equations of PW
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Gap soliton in periodic potential

• In a system with periodic potential, forbidden
  gap appears in the linear spectrum. Linear wave
  with the frequency in the gap is not allowed to
  pass through the system.
• However, if the dielectric constant is assumed to
  depend on the local field intensity (nonlinear),
  the radiation can be transmitted through the
  system with an envelop of a soliton shape.

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Forbidden gap in the linear spectrum in a system
with periodic potential.
When nonlinearity is considered, the
radiation is allowed to pass through, it forms a
gap soliton.
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Gap solitary wave and coexisting steady waves

• In our case we have a periodic potential
  in the system, there is a forbidden gap in the
  linear spectrum (for saddle instability, no any
  characteristic frequency is allowed to transmit
  into the system), what happens when
  self-nonlinearity of PW is included?
• Similar to gap soliton, we find that may
  build up a coherent structure on periodic
  potential of SSW
• The structure added on the SSW to
  form a new SW coexisting with the old SSW
• The new SW is a gap solitary wave in the sense
  that its energy locates about in the gap of the
  hysteresis.

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The PW wave energy tends to a nonzero fixed
point, the PW modes construct a gap solitary
wave trapped in the SSW.
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Hysteresis and gap states Eg locates at the
forbidden gap of the linear wave Depending
on initial conditions the system may settle to
different gap states
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Gap solitary wave building up on the states at
middle and upper branches
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Physics of gap soliton-like structure

• The structure can be well fitted into
• ordinary soliton is a result of balance between
  dispersion and nonlinearity
• Likewise, our solitary-like structure is also a
  result of balance between dispersion and
  nonlinearity, the difference here is our
  dispersion is nonlinear, it includes the
  contribution from the interaction between the PW
  and SW.

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Two fixed points in the system

• For given parameters, now we have two fixed
  points (two SW) one is unstable (SSW), the
  other one can be stable (the gap solitary wave).
• The gap solitary wave can also become unstable.
• What happens then?

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The new gap SW may also lose its
stability Leading to very complicated wave
patterns
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Crisis due to heteroclinic tangency

• Transition to the STC is due to a crisis.
• Since the system has a saddle point (SSW), it has
  SO and UO.
• The new gap SW forms a bifurcation center.
• when PW attractor expands, it collides to the
  unstable orbit the old attractor disappears and
  a new attractor is created.
• So it is a crisis due to heteroclinic tangency.

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Crisis due to collision of PW attractor to UO of
SSW
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Spatially regular and STC wave patterns

• Before the crisis the wave is temporally chaotic,
  spatially it is regular (SR).
• After the crisis the wave patterns becomes
  irregular both in time and in space(STC).
• There is a sharp transition from the SR to STC in
  the time evolution when the parameters are fixed.
• It is a critical phenomenon in parameter space.
  With fixed we find a critical at which
  the wave pattern transits from the SR to STC.

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Onset of STC
In time evolution the transition to the STC
occurs suddenly.
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Spatially regular and temporally chaotic wave
pattern
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Spatiotemporally chaotic wave pattern
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How to distinguish SR and STC

• Spatial spectrum can be a good choice.
• In our case the spatial spectrum of SR displays
  exponential law, that of STC displays a power
  law.
• spatial correlation ?

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Spectra of SR and STC
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Disappeare of the old attractor and creation of
a new attractor. It is a typical phenomenon of
crisis
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Sudden change from the SR to STC in the time
evolution
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Why there is a sudden transition in the time
evolution?

• What does it mean by collision in physics?
• Why the transition occurs suddenly in the time
  evolution?
• Why the critical transition time depends so
  sensitively on, e.g. initial conditions?
• What triggers the transition?
• Does the virtual saddle SW play a role in the
  transition?

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pattern resonance and onset of transition to STC

• For fixed an onset of transition from
  SR to STC is observed.
• An unstable SSW can not be realized, however, in
  certain condition if the realized waveform
  evolves to nearly the same shape of the SSW, the
  onset of transition to the STC occurs.
• We call this phenomenon as pattern resonance.

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Pattern resonance and onset of crisis
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Pattern resonance and onset of crisis
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Difference between realized pattern with
unrealized SSW virtual pattern
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Distance between realized wave and SSW
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Virtual pattern of SSW and its role in the
dynamics

• Its significant that an SSW can trigger a
  dynamic transition when it is occasionally
  realized, although in general it is a virtual
  wave pattern, invisible.
• With pattern resonance we can explain why the
  critical transition time is so sensitive to the
  initial conditions.
• In experiments we should also concern those
  states which normally can not be observed, but
  can realize occasionally.

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Pattern resonance and nonlinear frequency
resonance

• At the pattern resonance, PW amplitude is very
  small, in this case linear approximation is
  valid.
• Therefore the saddle instability occurs, PW
  amplitude increases exponentially, leading to
  transition.
• In saddle instability a mode has zero
  eigenfrequency in moving frame, indicating in lab
  frame the eigenfrequency is .
• So it is actually a nonlinear frequency
  resonance.
• Nonlinear---dispersion is altered due to SSW.

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Critical phenomenon of the transition in
parameter space

• The transition to the STC displays as a critical
  phenomenon in parameter space
• For example, for the critical
  transition point is
• What causes this critical behavior?

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Energy representation

• Self-energy of SW mode
• Interaction energy between SW and PW modes
• Self-energy of PW mode

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Stable orbit as a bounder of the PW orbit
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Stable orbit as a bounder of PW orbit
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Stable orbit---bounder of PW

• The UO of SSW causes crisis, what is the effect
  of SO?
• Stable orbit serves as a bounder of the motion of
  PW mode, respectively.

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The transition as a critical phenomenon in
parameter space

• The transition is a critical phenomenon in
  parameter space, transition parameter,
  .
• k1 mode is a key mode in the transition as a
  result of mode-mode couplings---slaving?
• Can we define an order parameter to describe the
  critical phenomenon?
• What is the law when approaching to the critical
  point?

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Symmetry change at the transition point

• Before transition of k1 mode is
  stronger on the l.h.s.
• Approaching to the critical point, the orbit
  becomes more and more symmetric.
• Beyond the critical point, on
  the r.h.s. suddenly becomes much more stronger
  than on the l.h.s.

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Variation of the symmetry of k1 PW orbit when
approaching to the critical point
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Sudden change of symmetry after the crisis
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Slaving effect

• K1 mode slaves the other modes, in transition to
  STC, only k1 mode is found to have changed the
  symmetry behavior.
• The other modes do not show qualitative different
  behavior before and after the transition.

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Trapped state of k2 PW mode before and after
crisis
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Time-averaged behaviours

• Define a quantity as the sum of the time-averages
  for the N and P interaction mode energies

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Critical behavior of a time-averaged quantity in
parameter space
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Behaviors of two different time averages when
approaching to the critical point
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Power law of the order parameter

• When approaching to critical point , S?0.
• simple time-average does not show such
  behavior.
• S displays a power law when approaching to .
• It transits discontinuously to a finite positive
  value when across .
• The transition is an first order transition.

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Power law of when
approaching
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Distrubution of k-1-4 PW modes in an SR state
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Distribution of k1-4 PW modes in an STC Before
the crisis
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Distribution of k1-4 PW modes in the STC after
the crisis
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Trapped to free of k1 mode

• Before transition k1 PW partial wave is trapped
  by that of SSW, like a potential well.
• After transition it can be free from the
  trapping.
• Transitions in both the mode amplitude and phase
  (1) the amplitude surpasses that of the SSW, (2)
  the phase transits from vibrating to a
  combination of vibrating and whirlings

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Trapped to free of k1 PW partial wave
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Evolution of amplitude and phase of k1 mode
Vibrating Whirling vibrating
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K2 PW partial wave before and after crisis
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Effect of UO in turbulence

• Unstable saddle orbits play an important role
  also in the turbulent motion after the
  transition.
• UO correspond constant mode phases.
• Before transition k1 mode phase moves in between
  of two singular phases.
• After transition it can cross the two singular
  phases frequently, where averagely positive
  growth rate of the mode amplitude is caused.

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Variation of growth rate of k1 mode amplitude
with phase in SR state
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Variation of growth rate of k1 mode amplitude
with phase in STC state
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Conclusion

• The nonlinear wave dynamics can be well explained
  on the basis of PW interacting with SW, the
  dispersion of PW is altered due to existence of
  SW.
• Saddle steady wave plays a critical role in the
  transition to turbulence.
• The symmetry change of PW causes the critical
  phenomenon in parameter space.
• The pattern resonance with the virtual pattern of
  SSW triggers the crisis.
• In SR all the PW modes are trapped by that of
  SSW.
• In STC k1 mode becomes free of the trapping, it
  crosses the UO frequently and strongly excited at
  the UO phases.

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Thank you!