Scale Invariance, Self Similarity and Critical Behavior in Classical and Quantum Systems

Symmetry and self-affinity or scale invariance are related concepts. We explore the fractal properties of fluctuations in dynamical systems, using some of the available tools in the context of time series analysis. We carry out a power spectrum study in the Fourier domain, the method of detrended fluctuation analysis and the investigation of autocorrelation function behavior. Our study focuses on two particular examples, the logistic module-1 map, which displays properties of classical dynamical systems, and the excitation spectrum of a schematic shell-model Hamiltonian, which is a simple system exhibiting quantum chaos.