The saturation bifurcation in coupled oscillators

TL;DR

This paper investigates the saturation bifurcation phenomenon in coupled oscillators, where increased forcing causes energy to divert from a forced to an unforced subsystem, revealing new steady states and complex bifurcation behaviors.

Contribution

It introduces analytical predictions and numerical verification of saturation bifurcation states in weakly nonlinear coupled oscillators, expanding understanding of energy transfer and bifurcation phenomena.

Findings

Existence of saturated states where energy transfer abruptly shifts

Analytical and numerical agreement on bifurcation behavior

Identification of related phenomena like quasiperiodicity and multi-frequency bifurcations

Abstract

We examine examples of weakly nonlinear systems whose steady states undergo a bifurcation with increasing forcing, such that a forced subsystem abruptly ceases to absorb additional energy, instead diverting it into an initially quiescent, unforced subsystem. We derive and numerically verify analytical predictions for the existence and behavior of such saturated states for a class of oscillator pairs. We also examine related phenomena, including zero-frequency response to periodic forcing, Hopf bifurcations to quasiperiodicity, and bifurcations to periodic behavior with multiple frequencies.