Delocalized periodic vibrations in nonlinear electrical chains

TL;DR

This paper investigates nonlinear electrical chains, proving the existence of only five symmetry-determined nonlinear normal modes and analyzing their stability as the system size grows, using group-theoretical methods.

Contribution

It identifies the limited set of nonlinear normal modes in nonlinear electrical chains and applies a group-theoretical approach to analyze their stability.

Findings

Only five symmetry-determined NNMs exist in the system.

Stability thresholds are characterized for different system sizes.

Group-theoretical methods simplify stability analysis.

Abstract

We consider an electrical chain of N nonlinear capacitors coupled by linear inductors assuming that voltage dependence of capacitors represents an even function. We prove that only 5 symmetry determined nonlinear normal modes (NNM) can exist in the considered system. The stability of all these dynamical regimes for different N is studied with the aid of the group-theoretical method [Physical Review E 73 (2006) 36216] which allows to simplify radically the variational systems appearing in the Floquet stability analysis. The scailing of the voltage stability threshold in the thermodynamic limit (N tends to infinity) is determined for each NNM.