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.

We investigate the existence of periodic solutions for nonconservative superlinear second-order differential equations in the sense of rotation numbers. Specifically, we focus on equations whose solutions at infinity behave comparably to a suitable linear system. By employing a rotation number approach, spiral analysis, and fixed-point theorems, we establish the existence of periodic solutions for nonconservative superlinear second-order differential equations. Among the equations we consider, a notable subclass is partially superlinear second-order differential equations, which provide a concrete illustration of our results. Our results extend several recent results, thereby advancing to a more comprehensive understanding of periodic behavior in nonconservative systems.

We prove the existence of multiple periodic solutions for weakly coupled parametrized systems with a singularity of repulsive type at the origin and linear growth at infinity. The proof is based on a higher dimensional Poincaré-Birkhoff theorem and the phase-plane analysis of the solutions.