1. Athens nov 20121 Benjamin Busam, Julius Huijts, Edoardo Martino ATHENS - Nov 2012 Control of Chaos - Stabilising chaotic behaviour

2. Athens nov 20122 Chaos in a nutshell Small change in initial condition  Huge difference in results Deterministic systems, impossible to predict See: [CT]

3. Athens nov 20123 Control of Chaos Stabilisation –Suppression –Synchronisation See: [Fe], [BG]

4. Athens nov 20124 Control of Chaos Stabilisation –Suppression –Synchronisation See: [Fe], [SA] ?

5. Athens nov 20125 Controlling Methods 1.Pyragas Method Delayed Feedback Control 2.OGY-Method Short explanation See: [AF]

6. Athens nov 20126 Pyragas See: [Fe], [SA] Desired Orbit

7. Athens nov 20127 Pyragas See: [Fe], [SA] System X(t) Y(t) u u(t)=G[Y 0 -Y(t)]

8. Athens nov 20128 Pyragas See: [Fe], [SA] System X(t) Y(t) u u(t)=G[Y(t-T)-Y(t)]

9. Athens nov 20129 Pyragas See: [Fe], [SA] System X(t) Y(t) u u(t)=G[Y(t-T)-Y(t)] Only need to know T

10. Athens nov 201210 Controlling Methods 1.Pyragas Method Delayed Feedback Control 2.OGY-Method Short explanation See: [AF]

11. Athens nov 201211 OGY method Objective Reach equilibrium by small perturbation. Why it will work Large number of low-period orbits Ergodicity : trajectory visits neighborhood. Chaotical system is sensible to small perturbation

12. Athens nov 201212 OGY method Steps Determinate the low period orbit embedded in the chaotic set. Determinate the stable orbit or point embedded in the attractor. Apply small perturbation to stabilize the system.

13. Athens nov 201213 OGY method System: x(t +1) = f (x(t),u(t)) When u(t)= u` (constant) x(t) passes by x` infinite times. Equilibrium point x` in the attractor. Problem: Find a control law u(t)=h(x(t)) that stabilizes the system. x: analyzed parameter u: tunable parameter

14. Athens nov 201214 OGY method 1.Restriction on u: Small perturbation [u-δ;u+δ] δ«|u| 2.Approximation of x(t +1) = f (x(t),u(t)): Linear approximation: dx(t +1) = Adx(t) + bdu(t) Where A=∂f/ ∂x| x`,u` b=∂f/ ∂u| x`,u` Control law: du(t) = kdx(t) → dx(t +1) = (A+bk)dx(t) k depends on the physics of the system

15. Athens nov 201215 OGY method OGY control law: u(t)=h(x(t))= u’ If |x(t) – x’|>ε u’ + k(x(t)-x’) If |x(t) – x’|≤ ε { Far from the stable point (curve)Near the stable point (curve) See: [1], [2]

16. Athens nov 201216 OGY method ‹t›: transient timeγ>0 γ: depends on dimension Probability curve moves to neighbors: →‹t›=1/P(ε)≈ε -1 ≈δ -1 ‹t›≈δ -γ How long will it take? See: [BG]

17. Athens nov 201217 Duffing Oscillator See: [We], [YT] driving force damping restoring force

18. Athens nov 201218 Duffing Oscillator See: [We], [Ka] driving force damping restoring force Poincaré section of the duffing oscillator

19. Athens nov 201219 D.O. - Phase Portrait See: [SA]

20. Athens nov 201220 D.O. - Control control term

21. Athens nov 201221 D.O. - Control control term

22. Athens nov 201222 D.O. - Noise See: [SA] noise

23. Athens nov 201223 D.O. - Noise See: [SA] noise

24. Athens nov 201224 D.O. - Noise See: [SA] noise

25. Athens nov 201225 Control of laser chaos See: [HH]

26. Athens nov 201226 Control of laser chaos See: [HH]

27. Athens nov 201227 Control of laser chaos See: [HH]

28. Athens nov 201228 Control of laser chaos See: [HH]

29. Athens nov 201229 Conclusion 1.Pyragas Method 2.OGY-Method 3.Applications

30. Athens nov 201230 Any questions?

31. Athens nov 201231 Practical Chaos control Situation: Toroidal cell in vertical position full of liquid Lower half in heater Two thermometer at 3 and 9 o’clock Chaos in the fluid: Situation Chaos: ΔT changes chaotically →Fluid dynamics equation Convective flux See: [BG]

32. Athens nov 201232 Practical Chaos control Control by feedback: Controlling the ΔT (decreasing oscillation amplitude) by applying perturbation to heater proportional to ΔT. Chaos in the fluid: Control See: [BG]

33. Athens nov 201233 From chaos to order Chaotical systems can become non chaotical: Fireflies http://www.youtube.com/watch?gl=IT&hl=it&v=sROKYelaWbo Rules: Fireflies have their own clock Try to synchronize with ones next to it Result: Up to the parameter synchronization is possible See: [YT2]

34. Athens nov 201234 Bibliography [AF]B.R. Andrievskii, A.L. Fradkov, Control of Chaos: Methods and Applications, I. Methods, Automation and Remote Control, Vol. 64, No.5, 2003, pp. 673-713. [BG]S. Boccaletti, C. Grebogi, Y.-C. Lai, H. Mancini, D. Maza, The control of chaos: theory and applications Physics Report 329 2000, pp. 103-197. [CM]Fireflies, INFN http://oldweb.ct.infn.it/~cactus/laboratorio/Fireflies.html, 2012-11-22. [CT]Chaos theory and global warming: can climate be predicted? http://www.skepticalscience.com/print.php?r=134, 2012-11-22. [Fe]R. Femat, G. Solis-Perales, Robust Synchronization of Chaotic Systems via Feedback, LNCIS, Springer 2008, pp.1-3. [HH]H. Haken, light, volume 2, laser light dynamics North-Holland 1985, chapter 8. [Ka]T. Kanamaru (2008), Duffing oscillator, Scholarpedia, 3(3):6327 http://www.scholarpedia.org/article/Duffing_oscillator, 2012-11-22. [Py]K. Pyragas, Continuous control of chaos by self-controlling feedback, Physics LettersA 170, North-Holland 1992, pp. 421-428. [SA]H. Salarieh, A. Alasty, Control of stochastic chaos using sliding mode method, Journal of Computational and Applied Mathematics, Vol. 225, Elsevier 2009, pp. 135-145.

35. Athens nov 201235 Bibliography [We]E.W. Weisstein, Duffing Differential Equation, MathWorld – A Wolfram Web Resource, http://mathworld.wolfram.com/DuffingDifferentialEquation.html, 2012-11-22. [YT2] Youtube, fireflies sync http://www.youtube.com/watch?gl=IT&hl=it&v=sROKYelaWbo 2012-11-22. [1]People waiting at bus stop http://worldteamjourney.files.wordpress.com/2012/06/people_waiting_at_bus_stop_42-16795068.jpg 2012-11-22. [2]Autostop http://www.digi.to.it/public/autostop%281%29.jpg 2012-11-22.