We consider a planar system \(z'=f(t,z)\) under non-resonance or double resonance conditions and obtain the existence of \(2\uppi \)-periodic solutions by combining a rotation number approach together with Poincaré-Bohl theorem. Firstly, we allow that the angular velocity of solutions of \(z'=f(t,z)\) is controlled by the angular velocity of solutions of two positively homogeneous system \(z'=L_i(t,z),i=1,2\), whose rotation numbers satisfy \(\rho (L_1)>n\) and \(\rho (L_2)<n+1\), namely, nonresonance occurs in the sense of the rotation number. Secondly, we prove the existence of \(2\uppi \)-periodic solutions when the nonlinearity is allowed to interact with two positively homogeneous system \(z'=L_i(t,z),i=1,2\), with \(\rho (L_1)\ge n\) and \(\rho (L_2)\le n+1\), which gives rise to double resonance, and some kind of Landesman–Lazer conditions are assumed at both sides.

我们考虑非共振或双重共振条件下的平面系统\（z'= f（t，z）\），并通过结合旋转数方法和庞加莱来获得\（2 \ uppi \）周期解的存在-布尔定理。首先，我们允许\（z'= f（t，z）\）的解的角速度由两个正齐次系统\（z'= L_i（t，z），i的解的角速度控制= 1,2 \），其转数满足\（\ rho（L_1）> n \）和\（\ rho（L_2）<n + 1 \），即，在转数的意义上不发生共振。其次，我们证明\（2 \ uppi \）的存在非线性解被允许与两个正齐次系统\（z'= L_i（t，z），i = 1,2 \）相互作用的周期周期解，其中\（\ rho（L_1）\ ge n \）和\ （\ rho（L_2）\ le n + 1 \），这会引起双共振，并且在双方都假定了某种Landesman–Lazer条件。