The devil is in the details: Self-similarity from hysteresis to mechanical trees

Abstract

Eigenvalues are usually introduced as individual quantities: natural frequencies, decay rates, or stability indicators. In large structured systems, however, the distribution of the eigenvalues can itself develop remarkable structure. In this talk two very different systems are examined, in which hierarchical organization produces a self-similar eigenvalue distribution approaching a Devil’s staircase. The first example comes from the discrete Preisach model of hysteresis. Transitions between its memory states define a recursively constructed directed graph with a block-hierarchical adjacency matrix, giving rise to a Devil’s staircase eigenvalue distribution. The second example is a binary-tree-structured mass-spring-damper system. Here the physical self-similarity of the oscillator leads to a recursively structured spectrum and, again, to a Devil’s staircase eigenvalue distribution. It will be discussed why self-similarity in the underlying system reappears as self-similarity in its spectrum, and what these two examples may tell us about spectra of hierarchical dynamical systems.

About the Speaker

Tamás Kalmár-Nagy is an Associate Professor at the Budapest University of Technology and Economics. He received his PhD in Theoretical and Applied Mechanics from Cornell University and has held research and academic positions at the United Technologies Research Center, Texas A&M University, and Mitsubishi Electric Research Laboratories. He is an associate editor of several journals, including Nonlinear Dynamics, Journal of Vibration and Control, and Robotica. His research focuses on nonlinear dynamics, complex dynamical systems, and robotics.

Host: Professor Alex Vakakis