The primary resonance and mode-locking in a certain self-excited system with hysteretic non-linearity
一类非自治滞后-自激系统的主共振与锁模现象

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.