Control of Chaos Stabilising chaotic behaviour Benjamin Busam

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

Chaos in a nutshell Small change in initial condition Huge difference in results Deterministic systems, impossible to predict See: [CT] Athens nov 2012 2

Control of Chaos • Stabilisation – Suppression – Synchronisation See: [Fe], [BG] Athens nov 2012 3

Control of Chaos • Stabilisation – Suppression – Synchronisation ? See: [Fe], [SA] Athens nov 2012 4

Controlling Methods 1. Pyragas Method Delayed Feedback Control 2. OGY-Method Short explanation See: [AF] Athens nov 2012 5

Pyragas Desired Orbit See: [Fe], [SA] Athens nov 2012 6

Pyragas u(t)=G[Y 0 -Y(t)] u System X(t) See: [Fe], [SA] Athens nov 2012 Y(t) 7

Pyragas u(t)=G[Y(t-T)-Y(t)] u System X(t) See: [Fe], [SA] Athens nov 2012 Y(t) 8

Pyragas u(t)=G[Y(t-T)-Y(t)] u X(t) System Y(t) Only need to know T See: [Fe], [SA] Athens nov 2012 9

Controlling Methods 1. Pyragas Method Delayed Feedback Control 2. OGY-Method Short explanation See: [AF] Athens nov 2012 10

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 Athens nov 2012 11

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. Athens nov 2012 12

OGY method Equilibrium point x` in the attractor. System: x(t +1) = f (x(t), u(t)) x: analyzed parameter u: tunable parameter When u(t)= u` (constant) x(t) passes by x` infinite times. Problem: Find a control law u(t)=h(x(t)) that stabilizes the system. Athens nov 2012 13

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 Athens nov 2012 14

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) See: [1], [2] • Near the stable point (curve) Athens nov 2012 15

OGY method How long will it take? ‹t›≈δ-γ ‹t›: transient time γ>0 γ: depends on dimension →‹t›=1/P(ε)≈ε-1≈δ-1 Probability curve moves to neighbors: See: [BG] Athens nov 2012 16

Duffing Oscillator damping restoring force See: [We], [YT] Athens nov 2012 driving force 17

Duffing Oscillator damping restoring force driving force Poincaré section of the duffing oscillator See: [We], [Ka] Athens nov 2012 18

D. O. - Phase Portrait See: [SA] Athens nov 2012 19

D. O. - Control control term Athens nov 2012 20

D. O. - Control control term Athens nov 2012 21

D. O. - Noise noise See: [SA] Athens nov 2012 22

D. O. - Noise noise See: [SA] Athens nov 2012 23

D. O. - Noise noise See: [SA] Athens nov 2012 24

Control of laser chaos See: [HH] Athens nov 2012 25

Control of laser chaos See: [HH] Athens nov 2012 26

Control of laser chaos See: [HH] Athens nov 2012 27

Control of laser chaos See: [HH] Athens nov 2012 28

Conclusion 1. Pyragas Method 2. OGY-Method 3. Applications Athens nov 2012 29

Any questions? Athens nov 2012 30

Practical Chaos control Chaos in the fluid: Situation: • Toroidal cell in vertical position full of liquid • Lower half in heater • Two thermometer at 3 and 9 o’clock Chaos: ΔT changes chaotically →Fluid dynamics equation Convective flux See: [BG] Athens nov 2012 31

Practical Chaos control Chaos in the fluid: Control by feedback: Controlling the ΔT (decreasing oscillation amplitude) by applying perturbation to heater proportional to ΔT. See: [BG] Athens nov 2012 32

From chaos to order Chaotical systems can become non chaotical: Fireflies Rules: Result: • Fireflies have their own clock Up to the parameter synchronization is possible • Try to synchronize with ones next to it http: //www. youtube. com/watch? gl=IT&hl=it&v=s. ROKYela. Wbo See: [YT 2] Athens nov 2012 33

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 Letters. A 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. Athens nov 2012 34

Bibliography [We] E. W. Weisstein, Duffing Differential Equation, Math. World – A Wolfram Web Resource, http: //mathworld. wolfram. com/Duffing. Differential. Equation. html, 2012 -11 -22. [YT 2] Youtube, fireflies sync http: //www. youtube. com/watch? gl=IT&hl=it&v=s. ROKYela. Wbo 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. Athens nov 2012 35