Rotational dynamics in the system of two coupled pendulums

We consider dynamics in a pair of nonlinearly coupled pendulums.With existence of dissipation and constant torque such system can demonstrate in-phase periodical rotation in addition to the stable state. We have shown in numerical simulations that such inphase rotation becomes unstable at certain values of coupling strength. In the limit of small dissipation we have created an asymptotic theory that explains instability of the in-phase cycle. Found analytical equations for coupling strength values corresponding to the boundaries of the instability area. Numerical simulations show that there is a coupling strength interval where the system can have a pair of stable and unstable non in-phase cycles in addition to the stable in-phase motion. Therefore, we demonstrated that nonlinearly coupled pendulums have a bi-stability of the limit cycles. Analysed bifurcations which lead to originating and disappearing of non in-phase cycles.