Advances in stability theory for complex systems and networks

Abstract

Throughout natural and engineered systems, questions about stability always arise. Will a power network avoid blackouts? Will the Internet avoid traffic congestion? Will a species population survive? Engineered systems now have features comparable to living systems, namely massive scale, high degrees of nonlinearity, uncertainty and heterogeneity. Such systems are summarized as being complex. An important class of such systems has a well-defined network structure where large numbers of nodes, typically described by nonlinear dynamical systems, are connected by links. Stability theory faces many challenges in providing tools to assess stability and instability for such complex systems. Often this assessment must allow for layers of feedback control which have been implemented to regulate local and global behaviour. This paper will describe the advancement of stability theory to deal with complex systems with some emphasis on interconnected systems featuring feedback and network structures.