In this paper, periodic orbits in the nonlinear dynamical system with a fixed point vortex in a periodic flow are investigated. Under the influence of periodic perturbations in the phase space, an infinite number of nonlinear resonances with elliptic and hyperbolic periodic orbits arise. It is shown that these orbits exist even with completely destroyed resonant islands. In the perturbed system, all periodic orbits with periods up to $T=4T_0,$ where T₀ is period of perturbation, are found. The existence of nonlinear resonances of the KAM and non-KAM nature is shown, and a genetic relationship between different orbits is established. All elliptic orbits are destroyed with an increase of perturbation by the universal cascade of period doubling. It is shown that the rates of cascades for different orbits have close values and are consistent with the value of the Feigenbaum constant for two-dimensional conservative mappings. The complex interaction of hyperbolic orbits of the secondary resonances with the elliptic orbit of the primary resonance is demonstrated. It is shown that in addition to the universal cascade of period doubling, other bifurcation scenarios common to different orbits are possible.

本文研究了非线性动力系统中具有定点涡旋的周期性流动中的周期性轨道。在相空间中的周期扰动的影响下，出现了无数个具有椭圆和双曲周期轨道的非线性共振。结果表明，即使完全破坏了共振岛，这些轨道也存在。在扰动系统中，周期最大为$Ť=4{Ť}_0，$找到T ₀是扰动周期。证明了KAM和非KAM性质的非线性共振的存在，并建立了不同轨道之间的遗传关系。周期倍增的通用级联会随着扰动的增加而破坏所有椭圆轨道。结果表明，不同轨道的级联速率具有接近的值，并且与二维保守映射的费根鲍姆常数的值一致。演示了次要共振的双曲轨道与主要共振的椭圆轨道的复杂相互作用。结果表明，除了周期倍增的通用级联之外，不同轨道共有的其他分叉情况也是可能的。