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

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

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

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

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

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

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

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

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

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

Athens nov 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 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 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

Athens nov 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 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]

Athens nov 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]

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

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

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

Athens nov D.O. - Control control term

Athens nov D.O. - Control control term

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

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

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

Athens nov Control of laser chaos See: [HH]

Athens nov Control of laser chaos See: [HH]

Athens nov Control of laser chaos See: [HH]

Athens nov Control of laser chaos See: [HH]

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

Athens nov Any questions?

Athens nov 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]

Athens nov 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]

Athens nov From chaos to order Chaotical systems can become non chaotical: Fireflies Rules: Fireflies have their own clock Try to synchronize with ones next to it Result: Up to the parameter synchronization is possible See: [YT2]

Athens nov 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 [BG]S. Boccaletti, C. Grebogi, Y.-C. Lai, H. Mancini, D. Maza, The control of chaos: theory and applications Physics Report , pp [CM]Fireflies, INFN [CT]Chaos theory and global warming: can climate be predicted? [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): [Py]K. Pyragas, Continuous control of chaos by self-controlling feedback, Physics LettersA 170, North-Holland 1992, pp [SA]H. Salarieh, A. Alasty, Control of stochastic chaos using sliding mode method, Journal of Computational and Applied Mathematics, Vol. 225, Elsevier 2009, pp

Athens nov Bibliography [We]E.W. Weisstein, Duffing Differential Equation, MathWorld – A Wolfram Web Resource, [YT2] Youtube, fireflies sync [1]People waiting at bus stop [2]Autostop