准周期响应对称破缺分岔点的一种快速计算方法

稳态响应如周期及准周期解的分岔计算, 是非线性动力学研究的难点问题之一. 与计算方法及分析理论相对完善的周期响应相比, 准周期响应的求解只是在近些年才得到较大进展, 而且其分岔分析更加棘手, 仍需要更有效的理论和方法. 目前, 稳态响应尤其是准周期响应的分岔计算, 一般需采用数值方法, 通过调节参数反复试算得到. 为此, 本文基于增量谐波平衡IHB法提出一种快速方法, 可以高效地确定准周期响应的对称破缺分岔点. 方法的理论基础是在准周期解的广义谐波级数表达基础上, 当响应发生对称破缺分岔时, 其偶次(含零次)谐波系数将逐渐由0变为小量. 基于此性质, 将零次谐波系数预先设定为小量, 同时将分岔控制参数视为可变的迭代变量, 进而通过IHB法构造迭代格式. 作为算例, 研究不可约频率作用下的双频激励Duffing系统以及Duffing-van der Pol耦合系统. 结果表明, 只要迭代格式收敛, 随着预设小量减小, 控制参数将逐渐接近分岔近似值; 同时, 通过提高谐波截断数可显著提高近似分岔值的计算精度. 所提方法无需反复试算, 只要迭代过程收敛、便可实现分岔点直接快速计算. 

A FAST CALCULATION FOR THE SYMMETRY BREAKING POINT OF QUASI-PERIODIC RESPONSES

It has been a tough task to determine the bifurcation points of steady state responses such as periodic as well as quasi-periodic solutions arising in nonlinear dynamical systems. The calculation techniques and analysis methods have been well developed for periodic responses. Compared to periodic solutions, however, the solution techniques for quasi-periodic responses have only made relative progress in recent years, and the bifurcation analysis methods are even in more urgent need. To the best of our knowledge, for example, the bifurcation values of quasi-periodic responses have so far been usually determined by numerical approaches with the help of trail and error repeated calculation. For this issue, a fast calculation approach will be proposed in this paper, based on the incremental harmonic balance (IHB) method, to determine the bifurcation point for symmetry breaking of QP responses. The method is based on the fact that, the QP response can be described by generalized Fourier series with two irreducible frequencies. As the symmetry breaking happens, the coefficients of even-order (including the zeroth-order) harmonics will change from zero to non-zero small quantities. Based on this feature, the coefficient of the zeroth-order harmonic is priorly given as a small quantity. And the controlling parameter is incorporated as a variable into the IHB iteration scheme. The bifurcation point can be approximately determined as long as the iteration scheme is convergent. As illustrative examples, the Duffing oscillator and the Duffing-van der Pol coupled system, both subjected to multiple harmonic excitations with irreducible frequencies, are investigated by the proposed method. The symmetry breaking point can be efficiently determined, without any trail and error repeated calculation, as the convergent result can directly provide the controlling parameter close to the bifurcation value. In addition, it is shown that the calculation accuracy can be significantly improved by enhancing the number of truncated harmonics in the solution expression.