Controlling Chaos
Journal presentation by Vaibhav
Madhok
Outlook

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Background of chaos
OGY Method to Control chaos
Experiment to Control chaos
Modification made by DN
Chaos Control by self-controlling
feedback
Conclusion
References
Chaos

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
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Defining chaos,
Essential features of trajectories,
Phase space picture,
Geometry, Strange attractors,
Topology, Fractals.
Chaos
Chaos is aperiodic long-term
behavior in a deterministic system
that exhibits sensitive dependence
on initial conditions.
 Obeys deterministic laws of evolution
 Long-term aperiodic behavior
 Sensitive dependence or the “butterfly
effect”
The Butterfly effect
Exponential divergence of nearby trajectories
t
 ~ 0 e
t
The plot
There are ‘n’ Liapunov exponents for
an n dimensional system.
Example

A game of pinballs
2
1
3
Analytical model
Differential equations
Necessary condition for chaos is
nonlinearity.
Not a sufficient condition.
Example: Damped and driven pendulum:


x  y,

y  cy  sin x  F cos z,
z  t
c  0.05,
  0.7,
The phase space approach

x  y,

y  cy  sin x  F cos z,
z  t
c  0.05,
  0.7,
The system is
periodic for F
values:0.4,0.5, 0.9
but chaotic for
0.6,0.7 and 1.0
Geometry of the attractor

Chaotic trajectories typically are
confined to a bounded region, yet they
separate from their neighbors
exponentially fast.
What is happening?
Geometry of the attractor

Process of stretching and folding….
The pastry map.
The cross section is a
cantor set!
Geometry of the attractor

1)
2)
3)
An attractor is a closed set such that
Any trajectory that starts on A,
remains on A.
A attracts an open set of initial
conditions
A is minimal: No proper subset of A
satisfies A and B.
Few attractors
Lorenz Attractor
dx/dt = sigma (y-x)
dy/dt = rho x - y - xz
dz/dt = xy - beta z
Rossler Attractor
dx/dt = - (y+z)
dy/dt = x + a y
dz/dt = b + z(x-c)
Geometry of the attractor
Geometry of the attractor is typically
associated with fractals…
1) Fractals have fine structure at arbitrary
scales.
2) Dimension can be a non integer.

Koch Curve
The cantor set
Branches
Divergence of nearby trajectories on the Lorenz attractor.
Small set of
initial conditions
Evolution and
diverging of
trajectories.
Final state shows sensitive
dependence on initial
Essence of OGY Method



Butterfly effect
revisited…….
Balancing a stick on
your Palm
In principle, small
perturbations can be
given to any system
wide parameter of
the system.
Poincare sections



Poincare surface of the
section technique
Fixed point
Periodic orbit in the map
as repeated iteration
Poincare section
Theoretical analysis of the
OGY method
Z  f (Z , p ),
Z  f (Z , p ),
Z  Z  A(Z  Z )  B( p  p ),
( p  p )   K (Z  Z )
 Z  ( A  BK ) Z .
n 1
n
*
*
n 1
*
n
0
n
*
n
T
n
n
0
*
T
n 1
n
0
Analysis

g  Z
/
*
p

1
p Z (p )
*
Z
n 1
Assume the fixed point initially
to be the origin.

pg
n
 [ u eu
f
u
  s es
f ][ Z
s
n

p g]
n
Equations for control
The condition to be on
the stable manifold
next iteration:
f .Z
u
p   (
n
u
0
n 1
1

1
)
(Z n .
u
(1)
f
u
) / g.
The parameter p has a
threshhold, above which we
cannot work.
f
u
(2)
Time to achieve control
The control is activated if Zn falls in a narrow strip:
Z
u
n
 Z*
Where ,
Z
Z
u
n
s
n

f .Z
f .Z

u
n
,
s
n
,
Hence ,
Z

*
p
(1   u ) g .
1
*
f
Time to achieve control:
  
p
*

u
Numerical example
Considering a Henon map:
xn1  A  xn  B y ,
2
n
y
n 1

x,
n
B  0.3,
AA
0
p
p is the control
parameter.
Verification of the power law :
Two important issues

Delayed coordinate embedding

How to find fixed points?

Application on a magnetoelastic ribbon
Delay coordinate embedding

x   ( y  x),

y  rx y  xz,

z   bz  xy .
 10,r  28,b 8 / 3
Time series of measuring
variable x(t) with t=0.05
time steps.
Reconstruction of the Lorenz
attractor

v
i
 ( xi , xi l ,..... xi  ( m 1)l ).
Here: l=3 and m=3.
How to obtain fixed points?
Observe 2 successive
points close to each
other in the time series.
Z
n 1
ZF  M(
 X n2



n 1
 X n 1 
 X n 1

Z n1  X 
n 

 X F

Z F X 
 F
Z
Z Z
n
F
),
Experiment…controlling the
magneto-elastic ribbon
Young’s Modulus of the
ribbon is changed by
applying H(t).
Finding fixed points and
linearization
1) Linearize the time series data to
find M, hence the eigenvalues and
eigenvectors

n 1
   M (    ),
F
n
F


  X n2
n 1
 X n 1 


 n1   X n1
 Xn 


 F   X F
 X F

2) Change Hdc and find the precise location
of fixed point and find vector g :

g  Z
*
/
p

1
p Z (p )
*
Experiment
Time series for Xn for
Hdc=0.112Oe,
Hac=2.050 Oe, and
f=.85Hz
Switching between different
periods
Modification made by DN
The Basic idea is that if the parameter p is changing,
The Poincare map depends on both Pn and Pn-1 result.

i 1
 P(
 ,P ,P)
i 1
i
  n1  M  n  w
p
n
i
 v
p
n 1
Optimal value of Pn is given by demanding:
f

u
.
p
i

i 1

0
u
f u .u
f
u
.

i

f u .v
.
f u .u
p
i 1
,
Modification made by DN
Further modification gives:
  n  2  M   n  w
2
f
u
.

i2
and

p
i 1
 0.
0
p
n 1
 (v  Mw)
p
n
 Mv
p
n 1
,
Modification of the equations
The Duffing attractor
The Duffing equation:

x1  x ,
2

x2
 dx2  x1  x1  f cos  x3 ,
3

x3  1
Attractor and cross section
for d=0.2, f=36,w=0.661
Application to the Duffing
oscillator
Delayed self controlling
feedback
1) Small time continuous
perturbation
2) External force control

y  P( y, x)  F (t ),

x  Q ( x, y )
F (t )  K [ y (t )  y (t )]  KD(t )
i
External Force control
a) Rossler system
dx/dt = - (y+z)
dy/dt = x + 0.2 y+F(t)
dz/dt = 0.2 + z(x-5.7)
k=0.4
b) Lorenz system

x   ( y  x),

y  rx y  xz,

z   bz  xy .
 10,r  28,b 8 / 3
Delayed feedback control
The control equation
F (t )  K[ y(t  )  y(t )] KD(t )
Delayed feedback control
a) Rossler system
dx/dt = - (y+z)
dy/dt = x + 0.2 y+F(t)
dz/dt = 0.2 + z(x-5.7)
K=0.2
t=17.5
Conclusion
1) Chaos can be controlled using small perturbations to
the system.
2) System dynamics need not be known.
3) Different periods can be stabilized in the same
system in the same parameter range.
4) Noise in the system can again throw it out of control
5) Control can be obtained in the form of feedback.
References
References


Nonlinear Dynamics and Chaos, Steven
H. Strogatz
Classical Dynamics of particles and
systems:Thornton, Marion