Steady state symmetry breaking in periodically excited systems involving time delay by harmonic homotopy

Abstract

Symmetry breaking is a ubiquitous and important phenomenon arising in a wide range of physical systems. We propose the use of the harmonic balance in combination with homotopy continuation to investigate symmetry breaking occurrence in the periodically excited systems involving time delay. Two numerical examples are given to show the details. When the Hopfield neural network is subject to external excitation, we investigate the relation of the magnitude of excitation versus the amplitude of the bias term and analyze the effect of time delay on the steady state response. The second example concerns the delayed feedback control of a nonlinear beam subject to moving load. The relationship of the position feedback gain, external excitation frequency and time delay versus the amplitudes of steady state responses are studied analytically. The symmetry breaking points are accurately predicted. In addition, the Runge–Kutta numerical simulation results are used to cross-check the efficiency and accuracy.