Discrete breathers in an array of self-excited oscillators: exact solutions and stability

TL;DR

This paper derives exact solutions and analyzes the stability of discrete breathers in an array of self-excited oscillators modeled after Franklin bells, revealing new types of localized oscillations and their bifurcation behaviors.

Contribution

It provides the first exact analytic solutions for discrete breathers in a self-excited oscillator array, including novel solutions with frequencies in the propagation zone.

Findings

Discrete breathers exist for all frequencies in the attenuation zone.

New localized solutions with main frequency in the propagation zone are found.

Stability regions are characterized by Neimark-Sacker and pitchfork bifurcations.

Abstract

Dynamics of array of coupled self-excited oscillators is considered. Model of Franklin bell is adopted as a mechanism for the self-excitation. The model allows derivation of exact analytic solutions for discrete breathers (DBs), and extensive exploration of their stability in the space of parameters. The DB solutions exist for all frequencies in the attenuation zone, but lose stability via Neimark-Sacker bifurcation near the boundary of propagation zone. Besides the well-known DBs with exponential localization, the considered system possesses additional and novel type of solutions - discrete breathers with main frequency in the propagation zone of the chain. The amplitude of oscillations in this solution is maximal at the localization site and then exponentially approaches constant value at infinity. We also derive these solutions in closed analytic form. They are stable in a narrow region of system parameters bounded by Neimark-Sacker and pitchfork bifurcations.