Coupled relaxation oscillators and circle maps

Abstract

We discuss the problem of two non-linear oscillators of the saw-tooth type coupled with each other at either the threshold or base in such a way that the threshold (base) of the first oscillator is equal to a constant plus a term proportional to the value of the state of the second, and similarly for the other oscillator. The solution of such problems involves the decoupling of the component subsystems through the use of geometrical techniques to reduce the problem to that of the iteration of two independent maps of the unit segment onto itself. Depending on the sign of the proportionality constant and whether the couplings takes place at the threshold or base, there are ten different systems of this type, resulting in twelve different maps. The dependence of the associated maps on the parameters of the initial system, and bifurcation spaces of a number of these maps are obtained analytically. This yields the bifurcation spaces of the original coupled non-linear oscillators.