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

Let <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{\cal N}</tex> be an autonomous dynamic nonlinear network. Let <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{\cal N}_{RG}</tex> 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{\cal N}_{RG}</tex> has only isolated operating points. Then <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{\cal N}</tex> has only isolated equilibria if, and only if, "there are no capacitor-only cutsets and inductor-only loops." (Condition <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</tex> ). 2) If Condition <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</tex> . is violated, then there are a continuum of equilibria even if the operating points are isolated. 3) Let <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</tex> be the set of equilibria. Then each trajectory is constrained to lie on an affine submanifold <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M{\ast}</tex> , which depends on the initial state, such that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M \ca} M\ast</tex> has only isolated points. Hence each trajectory behaves as if it has only isolated equilibria. The space <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M\ast</tex> , 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.