A new methodology to characterize nonlinear systems is described. It is based on the measurement over the time series of two quantities: the "Dynamical order" and the "Self-correlation". The averaged "Scalar" and "Perpendicular" products are introduced to measure these quantities. While this approach can be applied to general nonlinear systems, the aim of this work is to focus on the characterization and modeling of chaotic systems. In order to illustrate the method, applications to a two-dimensional chaotic system and the modeling of real telephony traffic series are presented. Three important aspects are discussed: the use of the averaged "Scalar" product as supplement of the "Lyapunov exponent", the use of the averaged "Perpendicular" product as a refinement of the "Mutual information" and the reduction of m-dimensional systems to the study of only one dimension. This new conceptual framework introduces a perspective to characterize real and theoretical processes with a unifying method, irrespective of the system classification.
Probabilistic characterization of nonlinear systems under α-stable white noise via complex fractional moments