Application of graph theory in stability of nonlinear complex dynamic systems

In this paper we follow a graph theoretic approach to develop a decomposition tool which exploits the structure of the directed graphs associated with a nonlinear dynamical system. Through this structural exploitation a new stability results for a nonlinear complex systems described by time varying ordinary differential equations are established. The present results make use of directed graph to transform complex systems into an interconnection of strongly connected subsystems (SCS). The stability is then accomplished in terms of the subsystems and in terms of the interconnection structure of the complex systems. To demonstrate the applicability of these results to physical systems, a damped transiently driven pendulum is considered as a specific example.