Document 6523209

DYCOEC Days, Besançon, November 8- 10, 2010
Luis Aguirre
How and why t he analysis of a dynamics
can depend on t he choice of t he observable
Christophe Letellier
Space reconst ruct ion wit h derivat ive coordinat es
Equivalence ?
Original space
Reconstructed space
Measurement function
z
x
f x
x
m
R
s
hx
s
R
Z
y
x
X
Projection h: Rm
Differential embedding
X1 s
X2
s
Xm
s(m
1)
X
F X
X
Rm
Y
R
A model in a canonical form
X1 X 2
X2
X3
Xm
F X 1 , X 2 ,..., X m
Takens t heorem
Physicists do not take care (enough) this assumption
If h is generic then
defines a diffeomorphism between
t he original phase space and t he reconstructed phase
space, provided that m is greater than or equal to 2d+1.
Floris Takens
A safe reconstruction is thus always assumed to be obtained
This implicitely means «does not depend on the chosen observable »
In practice, nobody takes care about the variable
Nearly impossible t o get a global model from variable z
of the Rössler system
Estimating correlation dimension is not always easy
We invoked the observability concept from the control theory
Condit ions f or a f ull observabilit y
Original phase space
Reconstructed space
Change of variables
x
z
y z
y x ay
z b z x c
x
y
X
Y
y
y x ay
Z
y
ax
a2 1 y z
y
X
Y
Y
Z
Z
F X ,Y , Z
X
Diffeormophic equivalence if Det J
Det J
y
0
1
Det 0
0
a a2 1
0
1
1
Y
0
a
That is the case for the Rössler system when observed through variable y(t)
The system is fully observable
Observabilit y mat rix
Original phase space
- f(x) : Rm Rm is the dynamical system
x f x
st h x
where
- h(x) : Rm R is the measurement function
Between the original and the reconstructed spaces, change of variables
X
L0f h x
st
1
f
Y
L h x
st
Z
L2f h x
st
where L jf h x
L jf 1 h x
x
f x
are Lie derivatives
Observability matrix = Jacobian matrix of the change of variables
L0f h x
x
Os x
m 1
f
L
J
h x
x
System f(x) is said observable if all
init ial condit ions xi and xj are
distinguishable
with
r espect
to
measured variable s(t), that is,
h(xi) h(xj) iff xi xj
Observabilit y coef f icient s
For a nonlinear system, observability depends on
- the location in phase space
- the chosen variable
Quantification using observability coefficients computed along a trajectory
s
x
1
T
T
t 0
T
O
s Os , x t
min
max
OsT Os , x t
where max[OsTOs, x(t)] designates the
eigenvalue max of matrix OsTOs estimated
at point x(t)
end
ep
d
!
y
e
a
l
b
m
Observability
coefficients
a
ysis en var i
l
a
n
s
A
Non observable h0e chos(x)
1 Observable
t
n
o
x=0.88
y=1.0
z=0.44
The best
The poorest !
Est imat ing correlat ion dimension
Log I
ved
o
r
p
im
n
le!
o
b
i
t
a
i
a
r
Est im nge of va
ha
c
a
by
Slope
Slope
Measurements:
I nt ensit y I
Case of a CO2 laser with modulated losses
Observabilit y f or a CO2 laser wit h modulat ed losses
A two- level laser model forced by a sinusoidal signal
where D is the inversion population
the cavity damping rate
the pump parameter
the modulation amplitude, and the modulation frequency
the population inversion relaxation rate
It becomes a 4D autonomous model
with observability coefficients
I3
0.31 and
D3
0.29
Observabilit y t rough a change of variable
Applying the change of variable I
with
x3
log I = x
0.79
observability improved from 0.31 to 0.79
Quality of t he correlation dimension est imat ion depends on
the choice of the observable
Synchronizat ion & observabilit y
Complete synchronization is possible using
variable y of the Rössler system
variable x of the Lorenz system
es
c
i
o
l ch
a
c
i
w
ir
Emp know ho
=
Used variable y of the Rössler system for phase synchronization
Our assumption Synchronizability depends (at least partly) on
the observability of t he or iginal dynamics
through the coupling variable
Synchronizat ion bet ween non- ident ical syst ems
Coupling two non-identical Rössler systems
where
1,2
s
used for detuning the two systems:
coupling term
=
2-
1=0.04
= 0 for a coupling variable
j (i=j) = 0 otherwise
i
Phase coherence of the Rössler system
Phase coherent (a=0.42)
Phase non- coherent (a=0.556)
Complet e synchronizat ion: variable y
Phase coherent
Phase non- coherent
complete synchronization is obtained at lower cost for
COHERENT attractors than for NON- COHERENT attractors
Complet e synchronizat ion: variable x
coherent
non- coherent
only obtained in the neighborhood of a homoclinic situation
= the neighborhood of the inner fixed point is visited
Complet e synchronizat ion: variable x
Neighborhood of t he inner fixed point = long phases during which the
dynamics remains observable
interrupted by bursts during which the dynamics is non observable
Enough to obtain
complete synchronization
a=0.455
Complet e synchronizat ion: variable z
Complete synchronization never obtained
Hyperchaotic behavior
Complet e synchronizat ion: variable z
Phase coherent
Phase non- coherent
measures the distance between the attractor and t he boundary
of the attraction basin
Phase synchronizat ion
easy to check for coherent attractors
But har d in non coherent cases due t o t he difficulty to define
accurately a phase
Attractor organized around two
foci, each one defining a
phase
proposed to use the curvature, implying that both foci are taken
into account although a single revolution is described!
Phase synchronizat ion
The non-coherent Rössler attractor is bounded by a genus- one torus
the phase should be defined
according to this torus
2 k
a cr oss-section can serve
as a reference
where ( Tk)=2 k (rad), that is, 2
around the inner fixed point
when
=
2-
1
per revolution
0, then « cross- section synchronized »
Phase synchronizat ion
Phase coherent (a=0.398)
Phase non- coherent (a=0.556)
Variable x
Variable x
The best
Variable y
Variable y
Variable z
The poorest!
Coupling strength
Coupling strength
in agreement with the observability coefficients
good, then it depends on the dynamics
When observability is
poor, then synchronization is never obtained
Conclusion
The measurement function determines the observability
and so the quality of the reconstructed phase portrait
Many techniques depend on observabilty
Global modelling
Synchronization
Estimating correlation dimensions
Estimating Shannon entropy (from continuous time series)
An open question
Is it possible t o have a t echnique not depending on t he
observability coefficients, that is, on t he measurement
function ?
Reference
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