1. Application: Targeting & control d=0d>2d=1d≥2d>2 Challenging! No so easy!

2. References Hand book of Chaos Control Schoell and Schuster (Wiley-VCH, Berlin, 2007)

3. Possible motions Stochastic Nonlinear Partial Differential Equation: Solitons Fixed Point

4. Chaos Control Fixed point Periodic Chaotic ? ? ?

5. Heart Activity: Periodic

6. Chaos to Periodic: Heart Attack Christini D J et al. PNAS 98, 5827(2001)

7. Chaos to Fixed Point solution: Laser

8. Chaos Control Difficulty due to Nonlinearity

9. Chaos ? Sensitive to initial conditions? UPOs: Unstable Periodic Orbits : Skeleton of Chaotic motion How to find UPOs: Lathrop and Kostelich Phys. Rev. A, 40, 4028 (1998)

10. Chaos ? Sensitive to initial conditions? UPOs: Unstable Periodic Orbits : Skeleton of Chaotic motion  exp  h  Number of UPOs of period T : h=Topological Entropy

11. Chaos to Periodic motion (OGY-method) Ott, Grebogi and Yorke, Phys. Rev. Lett. 64, 1196 (1990) Stabilizing UPOs !!

12. Chaos to Periodic motion (OGY-method)

13. Find the accessible parameter Represent system by Map Find the periodic orbit/point Find the maximum range of parameter which is acceptable to vary Fixed point should vary with change of parameter

14. Chaos to Periodic motion (Pyragas-method) K. Pyragas, Phys. Lett. A 172, 421(1992)

15. Chaos to Periodic motion (Pyragas-method)

17. Chaos to Fixed Point solution K. Bar-Eli, Physica D 14, 242 (1985)

18. Interaction X=  (X) Y=  (Y) What will be effect of interaction ??.. X=  (X)+F X ( , X, Y) Y=  (Y)+G Y (  /, Y, X)..

19. Interactions 1 F [ , X 1, X 2 ] 1 F [ , X 1, Y 2 ] 1 F [ , X 1 (t), X 2 (t)] F [ , X 1 (t-  ), X 2 (t)] F [ , X 1 (t-  ), Y 2 (t)] 1 F [ , X 1 (t), Y 2 (t)]

20. Oscillation Death 1 F [ , X 1, X 2 ] 1 F [ , X 1, Y 2 ] 1 F [ , X 1 (t), X 2 (t)]F [ , X 1 (t-  ), X 2 (t)] F [ , X 1 (t-  ), Y 2 (t)] 1 F [ , X 1 (t), Y 2 (t)] Nonidentical Identical/Nonidentical

21. Systems X=  (X) Y=  (Y). ? Fixed Point Periodic Quasiperiodic Chaotic Generalized synch. Stochastic Resonance Stabilization Strange nonchaotic … Synchronization Riddling, Phase-flip Anomalous Individual Interacting Forced Amplitude Death …

22. Analysis of coupled systems Effect Interaction -- Instantaneous -- Delayed -- Integral -- Conjugate -- ……. -- Linear -- Nonlinear -- ….. -- Diffusive -- One way -- …… -- Synchronization -- Hysteresis -- ….. -- Riddling -- Hopf -- Intermittency -- ….. -- Phase-flip -- Anomalous -- Amplitude Death -- ……

23. Effect of interaction: Amplitude Death (No Oscillation) Oscillators derive each other to fixed point and stop their oscillation

24. Experimental verification Reddy, et al., PRL, 85, 3381(2000)

25. Experiment: Coupled lasers M.-Y. Kim, Ph.D. Thesis, UMD,USA R. Roy, (2006);

26. Amplitude Death:- possible FPs F

27. Coupled chaotic oscillators O1O1 O2O2 X*=(x 1*,x 2*,y 1*,y 2*,z 1*,z 2* ) Constants

28. Strategy for selecting F(X) Design : F( , x 1, x 2 )=  (x 1 -  ) exp[g(X)] Not good: (1) F( , x 1, x 2 )=  (x 1 -  ) (x 2 -  ) (2) - F( , x 1*, x 2* )

29. Strategy for selecting X * For desired x 1* =  : find y 1* (  ) and z 1* (  ) from uncoupled systems

30. Examples

31. Parameter space -- unbounded -- Periodic -- Fixed point

32. N - oscillators

33. Chaos to Chaos Adaptive methods Yang, Ding,Mandel, Ott, Phys. Rev. E,51,102(1995) Ramaswamy, Sinha, Gupte, Phys. Rev. E, 57, R2503 (1998)

34. Chaos to Chaos : Adaptive methods P=desired measure/value