Explicit solution family for the equation of the resistively shunted Josephson junction model

We obtain and study a family of solutions of the equation $\dot\phi+\sin\phi =B+A\cos\omega t$, which is applicable to several problems in physics, mechanics, and geometry. We use polynomial solutions of double confluent Heun equations associated with this equation to construct the family. We describe the manifold $M_{\mathrm P}$ of parameters $(A,B,\omega)$ of these solutions and obtain explicit formulas for the rotation number and Poincaré map of the dynamical system on a torus corresponding to this equation with parameters $(A,B,\omega)\in M_{\mathrm rP}$.