Spatially extended systems yield complex pattems arising from the coupled dynamics of its different regions. In this paper we introduce a matrix computational operator (the so-called RSV operator), .T A, for the characterization of asymmeuic amplitude fragmentation in extended systems. For a given matrix of ,amplitudes this operation results in an asymmetric-triangulation field composed by L points and I straigth lines. The parameter (I -L)/L is a new quantitative measure of the local complexity defined in terms of the asymmetry in the gradient field of the amphtudes. This asymmetric fragmentation parameter is a measure of the degree of structural complexity and characterizes the localized regions of a spatially extended system and symmetry breaking along the evolution of the system. For the case of a random field, in the real domain, which has total asymmetry, this asymmetric fragmentation parameter is expected to have the highest value and this is used to normalize the values for the other cases. Here, we present a detailed description of the operator ~.,4 and some of the fundamental conjectures that arises from its application in spatio-temporal asymmetric patterns.
We also compare the performance of correlation length, entropies and RSV operators applied mainly in non-equilibrium plasma extended systems [1]. The complex regimes we study are stochasticity, symmetry breaking, chaoticity and localized turbulence. The main result is the high performance of the complex entropy and RSV operator to quantify non-finear amplitude fragmentation and localized turbulence in spatio-temporal dynamics.