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

gvidal@alumni.unav.es