Low-dimensional non-linear dynamical systems and generalized entropy

Low-dimensional non-linear maps are prototype models to study the emergence of complex behavior in nature. They may exhibit power-law sensitivity to initial conditions at the edge of chaos which can be naturally formulated within the generalized Tsallis statistics prescription which is characterized by the entropic index q. General scaling arguments provide a direct relation between the entropic index q and the scaling exponents associated with the extremal sets of the multifractal critical attractor. The above result comes in favor of recent conjectures that. Tsallis statistics is the natural frame for studying systems with a fractal-like structure in the phase-space. Power-law sensitivity in high-dimensional dissipative and Hamiltonian systems are also discussed within the present picture.