一类非自治滞后-自激系统的主共振与锁模现象

研究一类受外激励作用的滞后－自激系统（van der Pol系统）的振动。对于主共振情况，用平均法求出了稳态响应方程，揭示了响应与系统参数的关系。结果表明，外激励的影响占主导地位，滞后因素对非线性共振频率有影响，而自激因素使得周期解在偏离非线性共振频率后发生Hopf分岔，转变为概周期运动。通过用映射法求得旋转数，揭示了主共振前、后的概周期运动中存在的锁模现象，其阶数分布符合Farey树规律。研究表明，锁模振动因滞后非线性因素而产生。随着滞后参数值（代表滞后程度）增大，各阶共振特别是亚谐振动的存在能力增强，对系统结构会带来较大影响。

The primary resonance and mode-locking in a certain self-excited system with hysteretic non-linearity

The nonlinear dynamic behaviours of a van der Pol system with hysteretic nonlinearity and acted by external force were studied in this paper. The results show that the first approximate analysis using the averaging method is quite adequate for determining the steady state responses of the system in primary resonant case. The external force is found playing the leading role in primary resonant responses, while the hysteretic nonlinearity plays its effect on the nonlinear resonant frequency. Under the effect of the van der Pol damping, the steady-state periodic response loses its stability through Hopf bifurcation and then the system undergoes quasi-periodic motions. The circle maps was used to get the winding numbers to reveal the super- and subharmomic resonances in various orders known as mode-locking, which take place according to the Farey number tree as having been revealed in many other systems. The study shows that as the increase of the hysteretic parameters' values, which represents increase of the level of such a nonlinearity, it's easier for occurrence of the subharmonic resonances of order 2 as well as 3, known as strong resonances and can often damage the mechanical structures.