Our goal is to study statistics of interface propagation in spatially extended systems. We are interested in the particular situation when a chaotic or noisy state invades and unstable, ``supercooled'' or quenched state.
We will investigate both those examples from the point of view of front invasion into unstable state. Synchroniced states in chaotic systems are unstable, due to positive Lyapunov exponents. One can then predict a linear spatial spreading speed of small, spatially localized disturbances. In simple, non-chaotic systems, there is a dichotomy of linear, pulled front invasion and nonlinear, pushed invasion; see review article by van Saarloos. It is known that relaxation to pushed fronts is fast, exponential in time, whereas relaxation to pulled front is slow, with algebraic rates.
We are interested in characterizing such dichotomies in chaotic systems. We will therefore study numerically statistics of the front propagation in examples with pushed and pulled fronts. In the pushed case, we intend to derive a simple model inspired by models for the effect of noise on pulse propagation; see for instance reference below.