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Synchronization State Between Pre-turbulent Hyperchaotic Attractors
G. Vidal
H. Mancini
Universidad de Navarra
Introduction
Main Objective: Study the synchronization states in hyperchaotic
attractors. Motivation:
Gap in the literature. Numerical and theoretical studies for preparing an
experiment.
Introduction
Object of study: Bénard-Marangoni convection pattern. Codimension-2 Takens-Bogdanov Bifurcation
under square symmetry [1]. Method:
Lyapunov Exponents (LE) to detect synchronization states [2].
Phase Planes (PP) to characterize the synchronization state.
[1] R. Hoyle, Pattern Formation, Cambridge Univ. Press (2006).[2] J. Bragard et al., Chaos suppression through asymmetric coupling, Chaos, 17, 043107 (2007).
Experimental System B-M convective time dependent pattern.
Pitchfork HopfHeteroclinic Connection
[3] T.Ondarçuhu et al., “Dynamical patterns in Bénard-Marangoni convection in a square container”, Phys. Rev. Lett. 70, 3892 (1993).
The Model
How can we model this pattern?d
x = d·cosα
α
The Model
Symmetries in D4 systemmρ mρ2
mρ3
ρ
m
[1] R. Hoyle, Pattern Formation, Cambridge Univ. Press (2006).
Equation System
Mathematical Model 4 variables 9 parameters
[4] D. Armbruster, “Codimension-2 bifurcation in binary convection with square symmetry” pp 385-398, in Non-linear Evolution of Spatio-Temporal Structures in Dissipative Continuous Systems F.Busse and L. Kramer, Ed. Plenum Press, New York (1990).
Bifurcation System The equation system shows different dynamics
according to the parameter values.
a
μ
G.B. Midlin et al. “Comparison of Data from Bénard-Marangoni Convection in a Square Container with a Model Based on Symmetry Arguments”, IJBC, 4 (5) 1121 (1994).
Synchronization
2 identical systems are coupled with different initial conditions.
System 1: Projection (x,y) System 1: Projection (z,w)
Synchronized System Coupled Oscillators
Simplified Model
Synchronized System Coupled Oscillators
Simplified Model
Synchronized System
Using LE for detecting complete synchronization window.
[2] J. Bragard, G. Vidal, C. Mendoza, H. Mancini, S. Boccaletti Chaos suppression through asymmetric coupling, Chaos, 17, 043107 (2007).
Coupling Strength εx
Lyap
unov
Exp
onen
ts
Synchronized System
What happens outside the window?
Coupling Strength εx
Lyap
unov
Exp
onen
ts
[2] J. Bragard, G. Vidal, C. Mendoza, H. Mancini, S. Boccaletti, Chaos suppression through asymmetric coupling, Chaos, 17, 043107 (2007).
Synchronized System Phase Planes εx = 5.0
Plane x2 vs y2
Plane x1 vs x2
Plane w2 vs z2
Plane z1 vs z2
Synchronized System
What happens inside the window?
Coupling Strength εx
Lyap
unov
Exp
onen
ts
[2] J. Bragard, G. Vidal, C. Mendoza, H. Mancini, S. Boccaletti, Chaos suppression through asymmetric coupling, Chaos, 17, 043107 (2007).
Synchronized System Phase Planes εx = 0.5
Plane x2 vs y2
Plane x1 vs x2
Plane w2 vs z2
Plane z1 vs z2
Synchronized System
Sum
of
Pos
itive
LE
Coupling Strength εx
Sum of Positive LE (mean values)
Synchronized System
Synchronization WindowS
um o
f P
ositi
ve L
E
Coupling Strength εx
Is this a general behaviour?
Generalized Synchronization arises from Complexity Reduction? We can compare this result with other systems,
such as, Chen or Lü.
Hyperchaotic Chen
Based on Chen SystemS
um o
f P
ositi
ve L
E
Coupling Strength εx
Hyperchaotic Lü
Based on Lü SystemS
um o
f P
ositi
ve L
E
Coupling Strength εx
Conclusions
The system’s complexity is reduced when the coupling strength is adjusted into a Lyapunov Exponents window.
In TB system complete synchronization without
chaos suppression exists for values of coupling parameter inside the window.
The window in the LE also appears in the other systems studied together with a complexity reduction.
Future Works
We are exploring if generalized synchronization in the LE window is a universal behavior.
We are still looking for a minimum number of space-time sample points in order to synchronize two experiments.
We are exploring to use an entrainment (or synchronization) test to validate the matching between the model and the experiment.
Publicity…
Networks 08
Complex Systems, Spatio-Temporal Patterns, Networks…
http://fisica.unav.es/networks2008/default.html
