Global first harmonic bifurcation diagram for odd piecewise linear control systems

The describing function method (used normally as a first approximation to study the existence and the stability of periodic orbits) is used here to analyze the dependence of periodic orbits on the parameters of autonomous systems. The method can be applied to a large class of nonlinear systems but, for simplicity, attention here is paid to a class of single-input, single-output control systems with a piecewise linear characteristic function. A first approach to the bifurcation diagram associated with the periodic orbits of such systems, called a 'first harmonic bifurcation diagram' is obtained for two and three dimensions. This diagram in two dimensions describes all the qualitative behaviour of such systems. Although this is not the case in three dimensions, the information contained in the corresponding first harmonic bifurcation diagram is of great value. It shows much of the complexity of the periodic structure that can be found in such three-dimensional systems; in fact, part of the first harmonic bifurcation diagram coincides with the actual bifurcation diagram.