On global dynamic behavior of weakly connected oscillatory networks

The global dynamics of weakly connected oscillatory networks is investigated: as a case study one-dimensional arrays of third order oscillators are considered. Through the joint application of the describing function technique and of Malkin’s Theorem a very accurate analytical expression of the phase deviation equation (i.e. the equation that describes the phase deviation due to the weak coupling) is derived. The total number of limit cycles and their stability properties are estimated via the analytical study of the phase deviation equation. The proposed technique significantly extends the results available in the literature and can be applied to almost all complex networks of oscillators. In particular two-dimensional, space variant and fully connected networks can be dealt with.