Temporal chaos versus spatial mixing in reaction-advection-diffusion systems

We develop a theory describing the transition to a spatially homogeneous regime in a mixing flow with a chaotic in time reaction. The transverse Lyapunov exponent governing the stability of the homogeneous state can be represented as a combination of Lyapunov exponents for spatial mixing and temporal chaos. This representation, being exact for time-independent flows and equal P\'eclet numbers of different components, is demonstrated to work accurately for time-dependent flows and different P\'eclet numbers.