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 Legend: Green =stable manifold Red = unstable manifold

Matlab experimental results

Controlling chaos when the equations determining the system are not known Let Z 1, Z 2,…,Z n 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.