On the implications of capacitor-only cutsets and inductor-only loops in nonlinear networks

Let
{\cal N}
be an autonomous dynamic nonlinear network. Let
{\cal N}_{RG}
be the associated resistive subnetwork obtained by open circuiting all capacitors and short circuiting all inductors. The following main results are proved. 1) Suppose that
{\cal N}_{RG}
has only isolated operating points. Then
{\cal N}
has only isolated equilibria if, and only if, "there are no capacitor-only cutsets and inductor-only loops." (Condition
A
). 2) If Condition
A
. is violated, then there are a continuum of equilibria even if the operating points are isolated. 3) Let
M
be the set of equilibria. Then each trajectory is constrained to lie on an affine submanifold
M{\ast}
, which depends on the initial state, such that
M \ca} M\ast
has only isolated points. Hence each trajectory behaves as if it has only isolated equilibria. The space
M\ast
, because of its nature, can be considered as the minimal dynamic space of the network. It is shown that the results can be generalized to nonautonomous networks. Finally an application of the results to eventually passive networks is given.