Application: Targeting & control

d=0 d>2d=1 d≥2 d>2

Challenging!

No so easy!

References

Hand book of Chaos ControlSchoell and Schuster (Wiley-VCH, Berlin, 2007)

Possible motions

Stochastic

Nonlinear Partial Differential Equation: Solitons

Fixed Point

Chaos Control

Fixed point

Periodic

Chaotic

?

?

?

Heart Activity: Periodic

Chaos to Periodic: Heart Attack

Christini D J et al. PNAS 98, 5827(2001)

Chaos to Fixed Point solution: Laser

Chaos Control

Difficulty due to Nonlinearity

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)

Chaos to Periodic motion (OGY-method)

Ott, Grebogi and Yorke, Phys. Rev. Lett. 64, 1196 (1990)

Stabilizing UPOs !!

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

Chaos to Periodic motion (Pyragas-method)

K. Pyragas, Phys. Lett. A 172, 421(1992)

Chaos to Periodic motion (Pyragas-method)

Chaos to Periodic motion (Pyragas-method)

Chaos to Fixed Point solution

K. Bar-Eli, Physica D 14, 242 (1985)

Interaction

X= (X)

Y= (Y)

What will be effect of interaction ??

.

.

X= (X)+FX(, X, Y)Y= (Y)+GY(/, Y, X)

..

Interactions

Instantaneous Delayed Instantaneous Delayed

F [, X11, X2] F [, X11, Y2]

F [, X11(t), X2(t)]F [, X1(t-), X2(t)] F [, X1(t-), Y2(t)]F [, X11(t), Y2(t)]

b

a

dtXtXF )](),(,[

Oscillation Death

Instantaneous Delayed Instantaneous Delayed

F [, X11, X2] F [, X11, Y2]

F [, X11(t), X2(t)] F [, X1(t-), X2(t)]F [, X1(t-), Y2(t)]F [, X11(t), Y2(t)]

b

a

dtXtXF )](),(,[

Nonidentical

Identical/Nonidentical

Systems

X= (X)

Y= (Y).

?

Fixed PointPeriodicQuasiperiodicChaotic

Generalized synch.Stochastic ResonanceStabilizationStrange nonchaotic……

SynchronizationRiddling,Phase-flipAnomalous

Individual InteractingForced

Amplitude Death……

Analysis of coupled systems

EffectInteraction

-- Instantaneous-- Delayed-- Integral-- Conjugate-- …….-- Linear -- Nonlinear-- …..-- Diffusive-- One way-- ……

-- Synchronization-- Hysteresis-- …..-- Riddling-- Hopf -- Intermittency-- …..-- Phase-flip-- Anomalous-- Amplitude Death-- ……

Effect of interaction: Amplitude Death(No Oscillation)

Oscillators derive each other to fixed point and stop their oscillation

Experimental verification

Reddy, et al., PRL, 85, 3381(2000)

Experiment: Coupled lasers

LD1

LD2

PD1

PD2

DC bias 1V1, OSC

V2, OSC

L1

L2

A2

A1

-

-

DC bias 2

Attn1

Attn2

LD1

LD2

PD1

PD2

DC bias 1V1, OSC

V2, OSC

L1

L2

A2

A1

-

-

DC bias 2

Attn1

Attn2

M.-Y. Kim, Ph.D. Thesis, UMD,USAR. Roy, (2006);

Amplitude Death:- possible FPs

F

Coupled chaotic oscillators

)(1 Xf),,()()(

2111 xxFXfdt

tdx

)()(

21 Xfdt

tdy

)()(

31 Xfdt

tdz

),,()()(

2112 xxFXfdt

tdx

)()(

22 Xfdt

tdy

)()(

32 Xfdt

tdz

O1

O2

)(1 Xf

X*=(x1*,x2*,y1*,y2*,z1*,z2*)Constants

Strategy for selecting F(X)

)(1 Xf),,()()(

2111 xxFXfdt

tdx

)()(

21 Xfdt

tdy

)()(

31 Xfdt

tdz

Design : F(, x1, x2)= (x1-) exp[g(X)]

Not good: (1) F(, x1, x2)= (x1-) (x2-) (2) - F(, x1*, x2*)

Strategy for selecting X*

)(1 Xf)exp()()()(

2111 xxXfdt

tdx

)()(

21 Xfdt

tdy

)()(

31 Xfdt

tdz

For desired x1* =: find y1*() and z1*() from uncoupled systems

a

xyayx *1

*1*1*1 0

cx

bzcxzb

*1

*1*2*2 0)(

Examples

Parameter space

-- unbounded-- Periodic-- Fixed point

N - oscillators

Chaos to Chaos

Adaptive methods

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

Chaos to Chaos : Adaptive methods

),,( tXFX

)( * PP

P=desired measure/value