Mathematical Modeling of the Evolution of Complex Networks

The purpose of the study is to describe possible behaviors of trajectories of a multi-dimensional system of ordinary differential equations that arise in the mathematical modeling of complex networks. This description is based on the combination of analytical and computational tools which allow to understand in general the behavior of trajectories. After the detailed treatment of the second-order case with multiple possible phase portraits the third order systems are considered. The emphasis is laid on the coexistence of several attracting sets. The role of knowing the attracting sets is discussed and explained. Further, the higher order systems are considered, of order four and higher. A way to obtain higher-order systems for a better understanding of them is provided. Due to the lack of results concerning modeling networks by systems of ordinary differential equations, special attention is paid to our previously obtained facts about the behavior of solutions of arbitrary order systems. The problem of control and management of such systems is discussed. Some suggestions are made.