Linear and nonlinear vibrational properties of a hierarchical continuous system

Abstract

Abstract We report some results on the vibrational properties of a one-dimensional continuous hierarchical system that was investigated both theoretically and experimentally, and that shows a very varied behaviour, both in the linear and in the nonlinear regime. As well as reviewing some previous work, new results are presented. In particular, a direct node-counting technique has been used for computing the density of state of very large systems in order to measure spectral dimension, band-width scaling and the multifractal spectrum. The results, together with previous evidence, strongly suggest that the spectrum is composed of frequency intervals where a singular continuous spectrum exists, separated by regions where it coexists with an absolutely continuous spectrum of extended modes. The review part of the paper concerns experimental investigation of the structure, discusses some properties of the eigenstates, and gives an analytical expression for the spectral dimension. Finally it deals with the nonlinear behaviour of the system, which has been found to be particularly effective in coupling extended and non-extended states. We argue that this feature, extremely relevant for transport phenomena, is not restricted to self-similar structures alone, but can probably be shared by a wider class of aperiodic and disordered systems.