Application: Targeting & control
Motion
Stationary/
Fixed point
d=0
Periodic
Quasiperiodic
d=1
d≥2
Strange
Nonchaotic
No so easy!
Challenging!
d>2
Strange
Chaotic
d>2
References
Hand book of Chaos Control
Schoell and Schuster (Wiley-VCH, Berlin, 2007)
Possible motions
Stochastic
Fixed Point
Nonlinear Partial Differential Equation: Solitons
Chaos Control
?
Periodic
?
?
Fixed point
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=f (X)
.
Y=y (Y)
What
will
be
effect
of
interaction
??
.
.X=f (X)+FX(e,/ X, Y)
Y=y (Y)+GY(e , Y, X)
Interactions
Interaction
Similar
integral
Conjugate
b
F [e, X1, X2]
F [e, X1, Y2]
 F [ e , X ( t ), X ( t  t )] d t
a
Instantaneous
F [e, X1(t), X2(t)]
Delayed
Instantaneous
F [e, X1(t-t), X2(t)]
F [e, X1(t), Y2(t)]
Delayed
F [e, X1(t-t), Y2(t)]
Oscillation Death
Interaction
Similar
integral
Conjugate
b
F [e, X1, X2]
F [e, X1, Y2]
 F [ e , X ( t ), X ( t  t )] d t
a
Instantaneous
F [e, X1(t), X2(t)]
Delayed
Instantaneous
F [e, X1(t-t), X2(t)]
F [e, X1(t), Y2(t)]
Nonidentical
Identical/Nonidentical
Delayed
F [e, X1(t-t), Y2(t)]
Systems
Individual
Forced
?
Interacting
X=f (X)
.
Y=y (Y)
Fixed Point
Periodic
Quasiperiodic
Chaotic
Generalized synch.
Stochastic Resonance
Stabilization
Strange nonchaotic
…
…
Synchronization
Riddling,
Phase-flip
Anomalous
Amplitude Death
…
…
Analysis of coupled systems
Interaction
-- Instantaneous
-- Delayed
-- Integral
-- Conjugate
-- …….
-- Linear
-- Nonlinear
-- …..
-- Diffusive
-- One way
-- ……
Effect
-- 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
DC bias 1
-
-
LD1
LD2
DC bias 2
L1
L2
V1, OSC
PD1
A1
PD2
A2
V2, OSC
Attn1
Attn2
M.-Y. Kim, Ph.D. Thesis, UMD,USA
R. Roy, (2006);
Amplitude Death:- possible FPs
F
Coupled chaotic oscillators
O1
dx 1 ( t )
dt
dy 1 ( t )
 f 1 ( X )  F ( e , x1 , x 2 )
 f2 ( X )
dt
dz 1 ( t )
dt
dx 2 ( t )
O2
dt
dy 2 ( t )
dt
dz 2 ( t )
dt
 f1 ( X )
 f3 ( X )
 f 1 ( X )  F ( e , x1 , x 2 )
 f1 ( X )
 f2 ( X )
 f3 ( X )
X*=(x1*,x2*,y1*,y2*,z1*,z2*)
Constants
Strategy for selecting F(X)
dx 1 ( t )
dt
dy 1 ( t )
dt
 f 1 ( X )  F ( e , x1 , x 2 )
 f2 ( X )
dz 1 ( t )
dt
 f1 ( X )
 f3 ( X )
Design :
F(e, x1, x2)= e(x1-a) exp[g(X)]
Not good:
(1) F(e, x1, x2)= e(x1-a) (x2-b)
(2) - F(e, x1*, x2*)
Strategy for selecting X*
dx 1 ( t )
dt
dy 1 ( t )
dt
dz 1 ( t )
dt
 f 1 ( X )  e ( x1  a ) exp( b  x 2 )  f 1 ( X  )
 f2 ( X )
 f3 ( X )
For desired x1* =a:
find y1*(a) and z1*(a) from uncoupled systems
x1*  ay 1*  0  y1*  
x1*
b  z 2 * ( x 2 *  c )  0  z 1* 
a
b
x1*  c
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

X  F ( X ,  ,t)

  e (P  P)
*
P=desired measure/value