Nonlinear networks and invariance

Abstract

We study nonlinear networks invariant under groups (mostly cyclic) of operations. We deduce the consequences of invariance on the reduced incidence matrix A, the branch admittance operator <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{\cal Y}_{n}</tex> , and the node admittance operator <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{\cal Y}_{n}</tex> . We consider two special kinds of excitation: symmetric (for any groups) and alternating (for cyclic groups). Under the uniqueness assumption, the solutions are shown to be symmetric and alternating respectively, and the equations required for solving the network are considerably simplified. An example shows that if uniqueness does not hold, a symmetric excitation can give rise to symmetric and nonsymmetric solutions! The case of linear networks is treated as a special case of the nonlinear networks and the decoupling property is easily obtained. For the nonlinear case, we show by examples that except for some special interconnections of nonlinear elements, the function space <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L^{n}</tex> cannot be decomposed into direct sum of subspaces which are invariant under the map <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{\cal Y}_{n}</tex> ,. Finally we derive stability conditions for three kinds of periodic oscillations; symmetric oscillation, alternating oscillation, and oscillation with delay, where the last two cases are restricted to cyclic groups. It is shown how the stability conditions can be systematically simplified.