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Application: Targeting & control

Application: Targeting & control. d =0. d =1. d≥2. d>2. d>2. No so easy!. Challenging!. References. Hand book of Chaos Control Schoell and Schuster (Wiley-VCH, Berlin, 2007). Possible motions. Stochastic. Fixed Point. Nonlinear Partial Differential Equation: Solitons.

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Application: Targeting & control

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

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

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

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

  11. Chaos to Periodic motion (OGY-method) • 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

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

  13. Chaos to Periodic motion (Pyragas-method)

  14. Chaos to Periodic motion (Pyragas-method)

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

  16. Interaction . X=f (X) . Y=y (Y) What will be effect of interaction ?? . . X=f (X)+FX(e, X, Y) Y=y (Y)+GY(e/, Y, X)

  17. Interactions F [e, X1, X2] F [e, X1, Y2] F [e, X1(t-t), X2(t)] F [e, X1(t), Y2(t)] F [e, X1(t-t), Y2(t)] F [e, X1(t), X2(t)]

  18. Oscillation Death F [e, X1, X2] F [e, X1, Y2] F [e, X1(t-t), Y2(t)] F [e, X1(t), X2(t)] F [e, X1(t-t), X2(t)] F [e, X1(t), Y2(t)] Nonidentical Identical/Nonidentical

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

  20. 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 -- ……

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

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

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

  24. Amplitude Death:- possible FPs F

  25. Coupled chaotic oscillators O1 O2 X*=(x1*,x2*,y1*,y2*,z1*,z2*) Constants

  26. Strategy for selecting F(X) Design : F(e, x1, x2)= e(x1-a) exp[g(X)] Not good: (1)F(e, x1, x2)= e(x1-a) (x2-b) (2) - F(e, x1*, x2*)

  27. Strategy for selecting X* For desired x1* =a: find y1*(a) and z1*(a) from uncoupled systems

  28. Examples

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

  30. N - oscillators

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

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

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