Energy redistribution in hierarchical systems of oscillators

TL;DR

This paper models energy redistribution in hierarchical oscillator systems using Hamiltonian formalism, revealing localized quasi-periodic and chaotic dynamics, with spectral analysis showing strong temporal correlations across hierarchical levels.

Contribution

It introduces a mathematical model for hierarchical media and analyzes energy distribution and complex dynamics within multi-level oscillator systems.

Findings

Identification of localized quasi-periodic trajectories

Detection of chaotic trajectories in three-level models

Spectral analysis showing strong temporal correlations

Abstract

The article deals with the mathematical model for media with hierarchical structure. Using the hamiltonian formalism, the dynamical system describing the state of hierarchically connected structural elements was derived. According to the analysis of the Poincar\'e sections, we found the localized quasi-periodic and chaotic trajectories in the three-level hierarchical model. Moreover, studies of correlation functions shown that the power spectrum for three-level model possesses local maxima characterizing temporal scales with strong correlation. Using the Fourier analysis of the solution's components, we have studied the distribution of energy injected in the system over hierarchical levels. Dynamical phenomena in the multi-level system were studied as well.