  A noisy stabilized Kuramoto-Sivashinsky equation is analyzed by stochastic
decomposition. For values of control parameter for which periodic stationary
patterns exist, the dynamics can be decomposed into diffusive and transverse
parts which act on a stochastic potential. The relative positions of stationary
states in the stochastic global potential landscape can be obtained from the
topology spanned by the low-lying eigenmodes which inter-connect them.
Numerical simulations confirm the predicted landscape. The transverse component
also predicts a universal class of vortex like circulations around fixed
points. These drive nonlinear drifting and limit cycle motion of the underlying
periodic structure in certain regions of parameter space. Our findings might be
relevant in studies of other nonlinear systems such as deep learning neural
networks.
