Non-resonance with one-sided superlinear growth for indefinite planar systems via rotation numbers

We consider the non-resonance with one-sided superlinear growth conditions for the indefinite planar system $z^{\prime }=f(t,z)$ from a rotation number viewpoint, and obtain the existence of $2\pi$-periodic solutions by applying a rotation number approach together with the Poincaré-Bohl theorem. We allow that the angular velocity of solutions of $z^{\prime }=f(t,z)$ is controlled by the angular velocity of solutions of two positively homogeneous and oddly symmetric systems $z^{\prime }=L_i(t,z),i=1,2$ on the left half-plane, which have rotation numbers that satisfy $\rho (L_1)>n/2$ and $\rho (L_2)<(n+1)/2$, and allow $f(t,z)$ to grow superlinearly on the right half-plane. In order to estimate the rotation angle difference of solutions, we develop a system methodology of "tracking" the angle difference of solutions of the system $z^{\prime }=f(t,z)$ on each small interval on the given side under sign-varying conditions.