The Uniqueness of the Steady State of the Decomposed Dynamic Circuits with Nonlinear Resistors

The conditions which guarantee the nonlinear nonautonomous circuit to behave as unique steady state are very important and complex.For the nonlinear dynamical circuits with linear resistors,the problems have been solved by previously published papers.But for the network with nonlinear resistors,there have not been satisfactory results to deal with this problem.On the other hand,almost all the known results are not developed particularly for the high degree circuits,So people always find those results very difficult and inconvenient,or even impossible to utilize.The unique steady state of the dynamic circuits with nonlinear resistors in decomposition forms is studied in this paper.The adapted method is the matrix decomposition technique and Liyapunov functions.When handling the large-scale nonlinear circuit,it is very difficult or even impossible to determine its unique steady state directly.Therefore,in this paper,the large-scale circuits are decomposed into several sub-networks with comparatively lower degree,which can be relatively easy to deal with.The stability of each sub-network is determined by Liyapunov functions.And the unique steady state of the nonlinear circuits as a whole is deduce by a Hurwitz matrix,the elements of which are composed of the coefficients related to the stability of the sub networks.The main results obtained in this paper show that the unique steady state of the dynamic circuits with nonlinear resistors in decomposition form can be determined by the stability of some decomposed matrixes.Compared with the known results,the conclusions obtained in this paper make progress at least in the following two aspects.One is that this paper focuses on the nonlinear resistors'functions in the nonlinear circuit and develops an effective method to deal with them,and other is that this paper pays special attention to the handling of the high order circuits,so the results in this paper are very simple and convenient to use,when high order networks considered.From the above,the work in this paper is a great extension of the known results.