On the phase space decomposition for weakly connected oscillatory networks with 2nd order cells

Oscillatory nonlinear networks represent a circuit architecture for image and information processing. It has been shown that they can be exploited to implement associative and dynamic memories. It has also been shown that phase noise play an important role as a limiting key factor for the performances of oscillatory cells. A tool of paramount importance for the design of oscillatory networks and the analysis of phase noise are phase models. These models require to treat the noise and the couplings among the cells as perturbations, and to identify the proper directions along which project the perturbations. In this paper we discuss the proper decomposition of the phase space for second order cells of oscillatory nonlinear networks, and we derive analytical formulas for the vectors spanning the directions for the proper phase space decomposition. We also discuss the implications of this decomposition in control theory and to what extent a simple orthogonal projection is correct.