Characterization of the spatial complex behavior and transition to chaos in flow systems

We introduce a “spatial” Lyapunov exponent to characterize the complex behavior of non-chaotic but convectively unstable flow sytems. This complexity is of spatial type and is due to sensitivity to the boundary conditions. We show that there exists a relation between the spatial-complexity index we define and the comoving Lyapunov exponents. In such systems the transition to chaos, i.e., the occurrence of a positive Lyapunov exponent, can manifest itself in two different ways. In the first case (from neither chaotic nor spatially complex behavior to chaos) one observes the typical scenario; i.e., as the system size grows up the spectrum of the Lyapunov exponents gives rise to a density. In the second case (when the chaos develops from a convectively unstable situation) one observes only a finite number of positive Lyapunov exponents.