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.