In this paper we present a framework in which one can rigorously study the effect of spatio-temporal noise on traveling waves, stationary patterns, and oscillations that are invariant under the action of a finite-dimensional set of continuous isometries (such as translation or rotation). This formalism can accommodate patterns, waves, and oscillations in reaction-diffusion systems and neural field equations. To do this, we define the phase by precisely projecting the infinite-dimensional system onto the manifold of isometries. We outline a precise stochastic differential equation for the phase. The phase is then used to show that the probability of the system leaving the attracting basin of the manifold after an exponentially long period of time (in ε-2, the magnitude of the noise) is exponentially unlikely.
All Science Journal Classification (ASJC) codes
- Applied Mathematics
- pattern formation
- phase reduction
- traveling waves