Phase retrapping in a pointlike $\varphi$ Josephson junction: the Butterfly effect

We consider a $\ensuremath{\varphi}$ Josephson junction, which has a bistable zero-voltage state with the stationary phases $\ensuremath{\psi}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\varphi}$. In the nonzero voltage state the phase ``moves'' viscously along a tilted periodic double-well potential. When the tilting is reduced quasistatically, the phase is retrapped in one of the potential wells. We study the viscous phase dynamics to determine in which well ($\ensuremath{-}\ensuremath{\varphi}$ or $+\ensuremath{\varphi}$) the phase is retrapped for a given damping, when the junction returns from the finite-voltage state back to the zero-voltage state. In the limit of low damping, the $\ensuremath{\varphi}$ Josephson junction exhibits a butterfly effect---extreme sensitivity of the destination well on damping. This leads to an impossibility to predict the destination well.