#### Milestone Year

### 2013

#### Symbolic dynamics: making sense out of chaos

A chaotic dynamical system is a system whose state evolves in time
according to a deterministic rule: A system at state x will evolve after
one time unit into the state f(x), where f is some specified function.
After two time units, the system will arrive to state f(f(x)), after three
time units to f(f(f(x))) and so on. For "chaotic" systems, these
calculations cannot be done reliably, even with the help of strong
computers, because of the phenomenon of sensitivity to initial conditions:
The slightest errors in the initial state x, or round-off errors in
subsequent steps, can lead to large errors in the prediction of future
states very fast.

"Symbolic dynamics" is a technique for simplifying the law of motion by
means of a suitable change of coordinates in such a way that it becomes
much easier to apply and to study. The technique was applied successfully
in special cases in the sixties. In 2013 WIS scientists showed that it can
be applied to general "chaotic" differentiable dynamical systems
of dimension two.