1 / 32

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.

Download Presentation

Application: Targeting & control

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related