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.