Análise da autossemelhança de atratores por espectro de dimensão de correlação : releitura de Grassberger-Procaccia

Abstract

The 1980s were crucial for significant advances in the field of dynamical systems, with research focused on the dimensionality of strange attractors (sensitive to initial conditions) and the reconstruction of attractors by time-lagged spaces. In this sense, Takens (1981) and Grassberger-Procaccia (GP) (1983) stand out, guiding a significant part of the scientific papers for the time being. Studying the dimensionality of the phase space is important to determine the range of potential behavior dynamics and to understand the system evolution, since they are related. Similar in nature, the dimensionality of an attractor gives us an estimate of the number of active degrees of freedom in the system. Despite the exceptional results found in the literature, the GP algorithm presents noise, making the graphical analysis an arduous task. Thus, the present work proposes an alternative for analysis called Correlation Dimension Spectrum (CDS), acting as an attenuator of the noises which are more intense in lower dimensions, proving to be a valid possibility, provided that an adequate interval for r is given. Unlike GP, CDS is useful even to not self-similar attractors, as is the case for micrometeorological variables. The behaviors evaluated were from temperature, relative air humidity, net radiation and photosynthetically active radiation index data obtained at Fazenda Maracaí, 50 km from Sinop/MT, in the rainy and dry seasons, totaling 7 months. The results showed a stochastic component not detected until now and occurrences of more than one peak, reaching three in the case of net radiation. Despite this, the larger the time scale, the smaller the disparity between the algorithms.