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