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

Abstract

We introduce a ``spatial'' Lyapunov exponent to characterize the complex behavior of non chaotic but convectively unstable flow systems. 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 these systems the transition to chaos, i.e. the appearing of a positive Lyapunov exponent, can take place in two different ways. In the first one (from neither chaotic nor spatially complex behavior to chaos) one has the typical scenario; that is, as the system size grows up the spectrum of the Lyapunov exponents gives rise to a density. In the second one (when the chaos develops from a convectively unstable situation) one observes only a finite number of positive Lyapunov exponents.