Dynamic nonlinear networks: State-of-the-art

This paper surveys the state-of-the-art of the qualitative aspects of nonlinear RLC networks. The class of networks being surveyed may contain multi-terminal and multi-port RESISTORS, INDUCTORS, AND CAPACITORS, as well as dc and time-dependent voltage and current sources. The concepts of impasse points and local solvability are introduced and shown to be of fundamental importance in modeling a physical network. Simple criteria are given which guarantee the existence of a global state equation. General theorems are presented for identifying or testing whether a dynamic nonlinear network possesses one or more of the following basic qualitative properties: 1. No finite-forward-escape-time solutions. 2. Local asymptotic stability of equilibrium points and observability of operating points. 3. Eventual uniform-boundedness of solutions. 4. Complete stability and global asymptotic stability. 5. Existence of a dc or periodic steady-state solution. 6. Unique steady-state response and spectrum conservation. The hypotheses of most of these theorems are couched in graph- and circuit-theoretic terms so they can be easily checked, often by inspection. Special efforts are made to state the concepts and results in a form that can be easily understood and used by the nonspecialist. Moreover, each concept and property is profusely illustrated with carefully conceived examples, and intuitive explanations so as to make this paper both motivating and somewhat self-contained. Extensive references are provided to facilitate researchers interested in conducting future research on the many unsolved problems in dynamic nonlinear networks.