  In this paper, the existence and number of non-contractible limit cycles of
the Josephson equation $\beta \frac{d^{2}\Phi}{dt^{2}}+(1+\gamma \cos
\Phi)\frac{d\Phi}{dt}+\sin \Phi=\alpha$ are studied, where $\phi\in \mathbb
S^{1}$ and $(\alpha,\beta,\gamma)\in \mathbb R^{3}$. Concretely, by using some
appropriate transformations, we prove that such type of limit cycles are
changed to limit cycles of some Abel equation. By developing the methods on
limit cycles of Abel equation, we prove that there are at most two
non-contractible limit cycles, and the upper bound is sharp. At last, combining
with the results of the paper (Chen and Tang, J. Differential Equations, 2020),
we show that the sum of the number of contractible and non-contractible limit
cycles of the Josephson equation is also at most two, and give the possible
configurations of limit cycles when two limit cycles appear.
