Oscillations of Interconnected Systems withC⁰Nonlinearities

Abstract

In this paper we establish conditions which ensure the existence of self-excited oscillations in complex dynamical systems with nondifferentiable nonlinearities, by considering those types of systems which can be viewed as an interconnection of several simpler subsystems. We find that the nonlinear terms of the system in which we are interested do not need to satisfy the Lipschitz condition. IntroductionIn recent years, many researchers have concerned themselves with the qualitative analysis of large-scale dynamical systems. The analysis is in terms of the qualitative properties of the free subsystems and of the structure of the interconnecting system. Examples of this method can be found in [2,6,7,8,10,11]. However these results are not applicable to some systems, for example, when the nonlinearity does not satisfy the Lipschitz condition. In this paper, we improve upon the old results and present new results, by providing conditions for the existence of limit cycles in interconnected systems with continuous nonlinearities which do not necessarily satisfy the Lipschitz condition. Using the method described in this paper, we are able to improve the oscillation result in [3] and discuss the existence of periodic solutions of second order difference equations.Of particular interest to the present discussion are some results in [1] and [9]. In this paper, we extend their results to a large class of interconnected systems.