Nonsmooth bifurcations in a piecewise-linear model of the Colpitts oscillator

This paper deals with the implications of considering a first-order approximation of the circuit nonlinearities in circuit simulation and design. The Colpitts oscillator is taken as a case study and the occurrence of discontinuous bifurcations, namely, border-collision bifurcations, in a piecewise-linear model of the oscillator is discussed. In particular, we explain the mechanism responsible for the dramatic changes of dynamical behavior exhibited by this model when one or more of the circuit parameters are varied. Moreover, it is shown how an approximate one-dimensional (1-D) map for the Colpitts oscillator can be exploited for predicting border-collision bifurcations. It turns out that at a border collision bifurcation, a 1-D return map of the Colpitts oscillator exhibits a square-root-like singularity. Finally, through the 1-D map, a two-parameter bifurcation analysis is carried out and the relationships are pointed out between border-collision bifurcations and the conventional bifurcations occurring in smooth systems.