controlling chaos! dylan thomas and alex yang. why control chaos? one may want a system to be used...
TRANSCRIPT
Controlling Chaos!
Dylan Thomas and Alex Yang
Why control chaos?
One may want a system to be used for different purposes at different times
Chaos offers flexibility (ability to switch between behaviors as circumstances change)
Small changes produce large effects
How is it done?
Chaotic systems can be controlled by using the underlying non-linear deterministic structure.
Exploit extreme sensitivity to initial conditions
Use small, appropriately timed changes to bring the system onto the stable manifold of an unstable orbit
Famous examples
Chaotic ribbon
Lorentz equations
ISEE-3/ICE and the n body problem
Two methods Ott, Grebogi, Yorke: modify parameters of the system to move the
stable manifold to the current system state
Garfinkel et. al. (Proportional perturbation feedback): force the system onto the stable manifold by a small perturbation
The logistic map
The Hénon map
Variation of a parameter in the Hénon map
-0.95 -0.9 -0.85 -0.8 -0.75 -0.7 -0.65 -0.6 -0.55 -0.5 -0.45-0.95
-0.9
-0.85
-0.8
-0.75
-0.7
-0.65
-0.6
-0.55
-0.5
-0.45
a=0a=0.01
a=0.02a=0.03
a=0.04a=0.05
a=0.06a=0.07
a=0.08a=0.09
a=0.1a=0.11a=0.12
a=0.13a=0.14a=0.15a=0.16a=0.17a=0.18a=0.19a=0.2
Legend:Green =stable manifoldRed = unstable manifold
Matlab experimental results
0 200 400 600 800 1000 1200-1.4
-1.38
-1.36
-1.34
-1.32
-1.3
-1.28
-1.26
0 200 400 600 800 1000 1200-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Controlling chaos when the equations determining the system are not known
Let Z1, Z2,…,Zn be a trajectory, or a series of piercing of a Poincare surface-of-section
If two successive Zs are close, then there will be a period one orbit Z* nearby
Find other such close successive pairs of points, which will exist because orbits on a strange attractor are ergodic.
Perform a regression to estimate A, an approximation of the Jacobian matrix, and C, a constant vector.
For period 2 points, proceed the same way, for pairs (Zn, Zn+2)
Altering the dynamics of arrythmia
Cardiac tissue
Neurons
Schiff et al. removed and sectioned the hippocampus of rats (where sensory inputs and distributed to the forebrain) and perfused it with artificial cerebrospinal fluid.