Caterpillar solutions in coupled pendula

The single pendulum is one of the fundamental model problems in the theory of dynamical systems; coupled pendula, or equivalently, two elastically coupled particles in a periodic potential on a line, are a natural extension of intrinsic interest. The system arises in various physical applications and it inherits some rudiments of the behaviour exhibited by its finite-dimensional parent, the sine-Gordon equation. Among these phenomena are the so-called caterpillar solutions, whose behaviour is reminiscent of solitons. These solutions turn out to have a transparent geometrical explanation. There is an interesting bifurcation picture associated with the system: the parameter region is broken up into the set of ‘pyramids’ parametrized by pairs of integers; these integers characterize the behaviour of the associated solutions.