1. Synchronization State Between Pre-turbulent Hyperchaotic Attractors G. Vidal H. Mancini Universidad de Navarra

2. Introduction Main Objective:  Study the synchronization states in hyperchaotic attractors. Motivation:  Gap in the literature.  Numerical and theoretical studies for preparing an experiment.

3. 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).

4. Experimental System B-M convective time dependent pattern. PitchforkHopf Heteroclinic Connection [3] T.Ondarçuhu et al., “Dynamical patterns in Bénard-Marangoni convection in a square container”, Phys. Rev. Lett. 70, 3892 (1993).

5. The Model How can we model this pattern? d x = d·cosα α

6. The Model Symmetries in D 4 system mρmρ mρ2mρ2 mρ3mρ3 ρ m [1] R. Hoyle, Pattern Formation, Cambridge Univ. Press (2006).

7. 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).

8. 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).

9. Synchronization 2 identical systems are coupled with different initial conditions. System 1: Projection (x,y)System 1: Projection (z,w)

10. Synchronized System Coupled Oscillators Simplified Model

11. Synchronized System Coupled Oscillators Simplified Model

12. 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 ε x Lyapunov Exponents

13. Synchronized System What happens outside the window? Coupling Strength ε x Lyapunov Exponents [2] J. Bragard, G. Vidal, C. Mendoza, H. Mancini, S. Boccaletti, Chaos suppression through asymmetric coupling, Chaos, 17, 043107 (2007).

14. Synchronized System Phase Planes ε x = 5.0 Plane x 2 vs y 2 Plane x 1 vs x 2 Plane w 2 vs z 2 Plane z 1 vs z 2

15. Synchronized System What happens inside the window? Coupling Strength ε x Lyapunov Exponents [2] J. Bragard, G. Vidal, C. Mendoza, H. Mancini, S. Boccaletti, Chaos suppression through asymmetric coupling, Chaos, 17, 043107 (2007).

16. Synchronized System Phase Planes ε x = 0.5 Plane x 2 vs y 2 Plane x 1 vs x 2 Plane w 2 vs z 2 Plane z 1 vs z 2

17. Synchronized System Sum of Positive LE Coupling Strength ε x Sum of Positive LE (mean values)

18. Synchronized System Synchronization Window Sum of Positive LE Coupling Strength ε x

19. Is this a general behaviour? Generalized Synchronization arises from Complexity Reduction?  We can compare this result with other systems, such as, Chen or Lü.

20. Hyperchaotic Chen Based on Chen System Sum of Positive LE Coupling Strength ε x

21. Hyperchaotic Lü Based on Lü System Sum of Positive LE Coupling Strength ε x

22. 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.

23. 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.

24. Publicity… Networks 08  Complex Systems, Spatio-Temporal Patterns, Networks…  http://fisica.unav.es/networks2008/default.html http://fisica.unav.es/networks2008/default.html  gvidal@alumni.unav.es