Identifying the dynamics of complex spatio-temporal systems by …mocenni/automatica2011_chiara.pdf · Identifying the dynamics of complex spatio-temporal systems by spatial recurrence

Identifying the dynamics of complexspatio-temporal systems by spatial

recurrence properties

Chiara Mocenni

Department of Information EngineeringCentre for the Study of Complex Systems

University of Siena

[email protected]

in collaboration work with A. Facchini and A.Vicino

Chiara Mocenni Automatica.it, September 7-9, 2011, Pisa

Outline of the talk

Complex spatio-temporal dynamical systems;State space reconstruction from time series andspatio-temporal time series;Recurrence plots: definition and measures;DET − ENT diagram for the classification of complex2D spatio-temporal systems;Structural changes in time and space dynamics;Application to the Complex Ginzburg-LandauEquation;Application to the Schnackenberg reaction-diffusionsystem;Conclusions and future research.


Spatio-temporal complex systems

Spatially extended systems may exhibit irregularbehavior both in space and time leading tospontaneous emergence of spatial patterns: Turingstructures, traveling and spiral waves, turbulence.Reaction-diffusion equations have been used fordescribing the main physical mechanisms leading tosuch phenomena.One main and still investigated problem is dealingwith a partially unknown system of which only theobservations of some of its spatial variables areavailable.


State-space reconstruction

Reconstructing the state space of a dynamicalsystem consists with identifying its dynamics using aset of measurements.The starting point of the embedding theorem1 for timeseries is that in nonlinear systems every observedvariable include, in an unknown way, the informationof all the others.The concept of recurrence is strictly related to that ofdynamical systems, as originally stated by Poincaré.

1Takens F., “Detecting strange attractors in turbulence”, LectureNotes in Math. Springer New York (1981).


The case of time series

Given a time series [s1, . . . , sN ], where si = s(i∆t)and ∆t is the sampling time, the system dynamicscan be reconstructed using the theorem of Takensand Mañe.The reconstructed trajectory X is expressed as amatrix in which each row is a phase space vector

xi = [si , si+τ , . . . , si+(DE−1)τ ],

i = 1, . . . ,N − (DE − 1)τ , where DE is the embeddingdimension.


1D Recurrence Plot

The Recurrence Plot (RP), proposed for the first timeby Eckmann et al. (1987), is a visual tool able toidentify temporal recurrences in multidimensionalphase spaces.In the RP, any recurrence of state i with state j ispictured on a boolean matrix expressed by:

RDEi,j = Θ(�− ||xi − xj ||) , (1)

where xi,j ∈ RDE are embedded vectors, i , j ∈ N, Θ(·)is the Heaviside step function and � is an arbitrarythreshold.


Examples of Recurrence Plot

0 200 400 600 800 10000

100

200

300

400

500

600

700

800

900

1000

time (in samples)

tim

e (

in s

am

ple

s)

2000 2100 2200 2300 2400 2500 2600 2700 2800 2900 30002000

2100

2200

2300

2400

2500

2600

2700

2800

2900

3000

time (in samples)

tim

e (

in s

am

ple

s)

Figure: Recurrence Plots of periodic, random and chaoticsignals.


Spatio-temporal time series

Analogously to time series, it can be assumed thatthe evolution of a certain region of a spatiallydistributed complex system depends in some way byall the other regions.The problem of understanding the dynamics ofspatio-temporal dynamical system may beinvestigated by identifying the spatial staterecurrences in the spatial domain.


Spatial Recurrence Plots

Given a d dimensional cartesian system, then-dimensional RP2, 3 is

R~ı,~ = Θ(�− ||~x~ı − ~x~||)

where~ı = i1, i2, . . . , id is the d-dimensional coordinatevector and ~x~ı is the associated phase-space vector.

The Line of Identity is given by R~ı,~ = 1, ∀~ı = ~, and isrepresented by an hypersurface.

2N. Marwan, J. Kurths and P. Saparin, “Generalised recurrenceplots analysis for spatial data”, Phys. Lett. A, 360, pp. 545-551 (2007)

3D. B. Vasconcelos, S. R. Lopes, R. L. Viana and J. Kurths,”Spatial recurrence plots“, Physical Review E, 73, pp. 1-10 (2006)


Application to 2D systems

The discretized solutions of a 2D spatio-temporal systemat fixed time can be represented by two-dimensionalcartesian objects (images) composed of scalar values,therefore the GRP

Ri1,i2,j1,j2 = Θ(�− ||xi1,i2 − xj1,j2||)

defines a four dimensional RP containing atwo-dimensional LOI plane, where xi1,i2 identifies a pixel ofthe image.

Two states are recurrent if the associated pixels xi1,i2 andxj1,j2 are within the threshold �.

The line structures become 2-dimensional.Chiara Mocenni Automatica.it, September 7-9, 2011, Pisa

Recurrence Rate (RR)

Analogously to the one dimensional case, we define theGeneralized Recurrence Quantification Analysis (GRQA)measures based on the histogram P(l) of the line lengths:

RR =1

N4

N∑i1,i2,j1,j2

Ri1,i2,j1,j2 =1

N4

N∑l=1

lP(l).

RR is the fraction of recurrent points with respect to thetotal number of possible recurrences. It is a densitymeasure of the RP.


Determinism (DET )

DET =

∑Nl=lmin

lP(l)∑Nl=1 lP(l)

,

where lmin is the minimum length considered for thediagonal structures.DET is the fraction of recurrent points forming diagonalstructures with respect to all the recurrences4.

4In the 1D framework, a line of length l indicates that, for l timesteps, the trajectory in the phase space has visited the same region atdifferent times.


Entropy (ENT )

ENT = −N∑

l=lmin

p(l) log p(l), p(l) =P(l)∑N

l=lminP(l)

.

ENT is a measure of the distribution of the diagonal linesin the GRP.It refers to the Shannon entropy with respect to theprobability to find a diagonal line of exactly length l5.

5For periodic signal or uncorrelated noise the value is small(∼ 0.2− 0.8), while for chaotic systems, e.g. Lorenz, ENT ∼ 3− 4.


The spatial recurrence properties

We have proposed to use DET and ENT for theanalysis of spatially distributed dynamical systems bylooking at the spatial recurrence properties of thesystem, and, in particular, by seing the availablesnapshots as solutions of an unknown 2D system ata fixed regime time.The idea is that some signatures of the system maybe identified by evaluating the spatial properties ofthe solution at a fixed time.


Examples: fractals and chemotaxis


Examples: Turing structures in the BZreaction


Examples: chlorophyll distribution in oceans


Examples: periodic patterns

,


Histograms of the line lengths distribution

4 5 6 7 8 9 10 11 12 130

5

10

15

20

25

Line length

log(N

l)(a)

Uniform Noise

Linear fit

0 20 40 60 800

5

10

15

20

25

Line length

log(N

l)

(b)

Turing Patterns

(a) White noise: the line lengths are exponentiallydistributed and the maximum length is short;(b) Turing patterns: In the beginning an exponentialdistribution is found, while in the remaining part thehistogram is more complex.


ENT and DET indicators for the classificationof complex images

DET is a measure of the global appearance of the image:values of determinism larger than 60-70% indicate thatthe image has strong recurrent components;

ENT accounts for the local organization: periodicdistribution of the diagonals shows low ENT values, sincethe distribution is trivial; a random distribution of thediagonal structures produces a low entropy value;

We introduced the the DET − ENT diagram tocharacterize the images according to their recurrenceproperties.


The DET − ENT diagram

0 10 20 30 40 50 60 70 800.5

1

1.5

2

2.5

3

DET

EN

T

(a)

0 10 20 30 40 50 60 70 800.5

1

1.5

2

2.5

3

DET

EN

T

(b)

Turing

Fractals (small)

Fractals (big)

Chlorophyll

Diffusion waves

Random

Periodic

Dict. Discoideum

A

F

E

D

C

B

A

F

E

D

B

C


Detecting changes in the dynamics

Is it now possible to analyze and detect structuralchanges in the spatio-temporal dynamics of a partiallyunknown system using a limited number of information ontemporal evolution of its spatial variable?


Complex Patterns in spatial systems


The Complex Ginzburg-Landau Equation(CGLE)

We use GRQA and the DET − ENT diagram forinvestigating the dynamics of the ComplexGinzburg-Landau Equation.The Complex Ginzburg-Landau Equation displays arich spectrum of dynamical behaviors describing alarge variety of physical systems, such as nonlinearwaves, second order phase transitions,superconductivity, superfluidity, Bose-Einsteincondensation and liquid crystals.It is a prototypical example of pattern formation (seeprevious slide) and presents bifurcations.


The CGLE equation

The CGLE reads:

∂tA = A+(1+ıa)∇2A−(b−ı)|A|2A A(x , y) ∈ C, a,b ∈ R(2)

The first term of the rhs is related to the linear instabilitymechanism leading to oscillation. The second termaccounts for diffusion and dispersion, while the cubic terminsures, for b > 0, the saturation of the linear instabilityand is involved in the renormalization of the oscillationfrequency.In two dimensions, the solutions of the CGLE are familiesof plane waves. Their behavior in the parameter space(a,b) is very complex and still under investigation.


The bifurcation curve in parameter space

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

b

a

Stable spirals

Unstable spirals

0 0.2 0.4 0.6 0.8 1 1.2 1.4−1.5

−1

−0.5

0

0.5

Stable spirals

Transi

tion zon

e

Behavior in the parameter space of the real part of A.Chiara Mocenni Automatica.it, September 7-9, 2011, Pisa

The DET − ENT diagram (1/2)

0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

b

a

(a)

0 10 20 30 40 501.5

2

2.5

3

3.5

4

4.5

DET

EN

T

(b)

S2

S1

Distribution of 80 points in plane (b,a) (a); Clustering (b).Chiara Mocenni Automatica.it, September 7-9, 2011, Pisa

The DET − ENT diagram (2/2)

The three regions are clearly identifiable in theDET − ENT diagram, where the clusters of stableand unstable spirals are clearly separated by anintermediate region corresponding to the transitionzone above the curve S1.The curve S1 itself corresponds to the curve S2 in theDET − ENT diagram and the cluster of the transitionzone lays on this curve.


New experiments and method refinement

The CGLE was integrated in a square domain ofL = 512 points with periodic boundary conditions.A portion of the phase plane ranging froma = [−1.5,1.5] and b = [−1.5,1.5] is considered.Starting from random initial conditions, the wholetrajectory of the system is initially analyzed for eachvalue of a and b.


Temporal evolution of DET and ENT

0 100 200 1000 2000 3000 40000

5

10

15

20

25

30

Iterations

D

0 100 200 1000 2000 3000 40000

0.5

1

1.5

2

2.5

3

3.5

Iterations

E

α=−1, β=−1

α=−1, β=0.1

α=−1, β=1

α=−1, β=−1

α=−1, β=0.1

α=−1, β=1


A sensitivity function

K (b) =

[(∆ENT

∆b

)2+

(∆DET

∆b

)2]1/2.


Clustering

5 10 15 20 25 30 35 401.5

2

2.5

3

3.5

4

D

E

A

B

G


The DET − ENT diagram


movie1.movMedia File (video/quicktime)

Bifurcations detection (1/3)

In the DET − ENT diagram the zones A and B areseparated by a transition zone.The lines bounding the regions A correspond, withvery good agreement, to the line S1: the boundary ofthe convective instability of the spiral waves, alsoknown as the Eckhaus limit.The transition region G separating clusters B and A isfound to separate the regions A and B in the (a,b)plane.



−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

b

a

A

A

B

Line S1, as in [16]G

G

GA

G A



5 10 15 20 25 30 35 401.5

2

2.5

3

3.5

4

D

E

A

B

G

−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

b

a

A

A

B

Line S1, as in [16]G

G

GA

G A

A cluster jump in the DET − ENT diagram corresponds tocrossing a bifurcation line in the parameter space.


The Schnakenberg system

Describes a simple chemical reaction showing limit cyclebehavior and Turing instabilities. The equations reads:

∂tu = γ(k1 − u + u2v) +∇2u,∂tv = γ(k2 − u2v) + d∇2v ,

u(x , y , t), v(x , y , t) ∈ R;x , y are the spatial variables;γ is proportional to the spatial domain size;k1 and k2 depend on the reaction rates;d is the ratio of the diffusions of the two reactants.The critical diffusion coefficient dc depends on k1 and k2.


Detecting Turing bifurcations in theSchnakenberg system (1/2)

9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 150

50

100

d

D

(a)

9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 151

2

3

4

5

d

E

(b)

D*

E*

dc

dc

The critical value dc ∼ 10 is well identified by looking atthe abrupt change of the indicators. For both Determinism(a) and Entropy (b), the saturation values D∗, E∗ arereached after a transient.


Detecting Turing bifurcations in theSchnakenberg system (2/2)

0.1 0.2 0.3 0.4 0.5 0.620

25

30

35

40

45

50

55

60

k1

D*(

k1)

D*(k1)

quadratic fit

k =0.15

k =0.30

k =0.501

1

1

Plot of the saturation values D∗(k1) for k1 ∈ [0.1,0.6].Chiara Mocenni Automatica.it, September 7-9, 2011, Pisa

Conclusions

We proposed the DET − ENT diagram for theanalysis of complex patterns;The method identifies the essential characteristics,including structural changes, of a complexspatio-temporal dynamical system by analyzinginstantaneous spatial measurements at steady state;The application of the GRQA to the solutions of theCGLE and to the Schnackenberg system led to theidentification of bifurcation lines in the parameterspace.


Future Research

Solving inverse problems for the reconstruction ofocean plankton dynamics and turbulent patterns fromremote sensing images.Identification of the dynamics in the field of systemsbiology, such as spatial modeling of tumor growth andcell diseases, brain cancer, where the spatial dataare provided by biopsy and Functional MagneticResonance imaging (FMRi) techniques.


For Further Reading

C. Mocenni, A. Facchini,A. Vicino, “Identifying thedynamics of complex spatio-temporal systems byspatial recurrence properties” Proc. Nat. Academy ofSciences, 107, 8097-8102, 2010.A. Facchini, F. Rossi, and C. Mocenni, “Spatialrecurrence strategies reveal different routes to Turingpattern formation in chemical systems”, Phys. Lett.A, 373:4266-4272, 2009.A. Facchini, C. Mocenni and A. Vicino, “GeneralizedRecurrence Plots for the analysis of images fromspatially distributed systems”, Physica D, vol. 238,pp. 162-169, 2008.


Main Talk

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Identifying the dynamics of complex spatio-temporal systems by …mocenni/automatica2011_chiara.pdf · Identifying the dynamics of complex spatio-temporal systems by spatial recurrence