Global Stability Analysis of Coupled-Oscillator Systems

A methodology for the prediction and in-depth understanding of the stability properties of coupled-oscillator systems is presented. Unlike former investigations, all possible combinations of phase-shift values between the oscillator elements are considered. This provides greater insight into the mechanisms leading to instability. The analysis is based on the determination of the bifurcation loci in the space defined by the inter-stage phase shifts, enabling the detection of both ordinary and co-dimension-two bifurcations. The new methodology for bifurcation detection is applicable to any number N of oscillator elements. For illustration, the case of three oscillator elements is considered. This is representative of the behavior for any number N and admits a planar representation of the bifurcation loci. The loci facilitate the comprehension of stability changes commonly observed during the system tuning and enable the evaluation and increase of the stability margins. Using these loci, it is possible to predict and synthesize the stable phase-shift regions, which will have interest in the case of nonconstant distributions used in null formation and other applications. Good agreement has been found between simulations and measurements of a practical coupled-oscillator system at 3.85 GHz.