Synchronization, symmetry and rotating periodic solutions in oscillators with Huygens' coupling (Q2140085)

The authors consider \(n\) identical pendula, each consisting of a point mass attached to a massless rod, all connected to a rigid bar which is elastically attached to a fixed wall by means of a spring and a damper. A torque is applied to each pendulum, this torque being a nonlinear function of the pendulum's state and velocity, in order to make each pendulum a self-sustained oscillator. There is viscous damping at the point of attachment of each pendulum to the bar. The angular displacements of the pendula are assumed to be small, partially linearising the system. Rotating waves are solutions for which the oscillators follow identical periodic orbits with constant time shifts between them. The authors use the permutation symmetry of the system to classify different possible rotating waves. Various results regarding the creation of such solutions in Hopf bifurcations, based on the form of the Jacobian at the origin, are given.