Periodic solutions of indefinite planar systems with asymmetric nonlinearities via rotation numbers.

We consider an indefinite planar system z′=f(t,z)$$ {z}^{\prime }=f\left(t,z\right) $$ under asymmetric non‐resonance conditions and obtain the existence of 2π$$ 2\pi $$‐periodic solutions by combining a rotation number approach together with the Poincaré‐Bohl theorem. We allow that the angular velocity of solutions of z′=f(t,z)$$ {z}^{\prime }=f\left(t,z\right) $$ is controlled by the angular velocity of solutions of two positively homogeneous systems z′=Li(t,z),i=1,2$$ {z}^{\prime }={L}_i\left(t,z\right),i=1,2 $$ and then characterize the rotation behavior of solutions of the planar system by the rotation numbers of the positively homogeneous systems. Because the system under consideration is strongly asymmetric and controlled by two positively homogeneous systems on the left half‐plane, the time that solutions stay in the left half‐plane in 2π$$ 2\pi $$‐period is intermittent; therefore, we estimate the angle difference of the dwell time of solution orbits in the half phase plane and develop a system method of "tracking" the angle difference of solutions of system z′=f(t,z)$$ {z}^{\prime }=f\left(t,z\right) $$ on each small interval on the given side under the sign‐varying conditions. [ABSTRACT FROM AUTHOR]