Nonnormality and the localized control of extended systems

The idea of controlling the dynamics of spatially extended systems using a small number of localized perturbations is very appealing - such a setup is easy to implement in practice. However, when the distance between controllers generating the perturbations becomes large, control fails due to increasing sensitivity of the system to noise and nonlinearities. We show that this failure is due to the fact that the evolution operator for the controlled system becomes increasingly nonnormal as the distance between controllers grows. This nonnormality is the result of control and can arise even for systems whose evolution operator is normal in the absence of control.Comment: 4 pages, 4 figure

Solvable model for spatiotemporal chaos

We show that the dynamical behavior of a coupled map lattice where the individual maps are Bernoulli shift maps can be solved analytically for integer couplings. We calculate the invariant density of the system and show that it displays a nontrivial spatial behavior. We also introduce and calculate a generalized spatiotemporal correlation function