Second order averaging and bifurcations to subharmonics in duffing's equation

Abstract

Periodic motions near and equilibrium solution of Duffing's equation with negative linear stiffness can evolve, lose their stability, and undergo period doubling bifurcations as excitation amplitude, frequency and damping are varied. For bifurcations to period two it is shown that these can be either sub- or supercritical, depending upon the excitation frequency. The analysis is carried out by the averaging method, and, to retain important non-linear effects, averaging must be taken to second order. Some remarks on higher order subharmonics are made. © 1981 Academic Press Inc. (London) Limited.