On Targeted Control over Trajectories of Dynamical Systems Arising in Models of Complex Networks

Abstract

The question of targeted control over trajectories of systems of differential equations encountered in the theory of genetic and neural networks is considered. Examples are given of transferring trajectories corresponding to network states from the basin of attraction of one attractor to the basin of attraction of the target attractor. This article considers a system of ordinary differential equations that arises in the theory of gene networks. Each trajectory describes the current and future states of the network. The question of the possibility of reorienting a given trajectory from the initial state to the assigned attractor is considered. This implies an only partial control of the network. The difficulty lies in the selection of parameters, the change of which leads to the goal. Similar problems arise when modeling the response of the body’s gene networks to serious diseases (e.g., leukemia). Solving such problems is the first step in the process of applying mathematical methods in medicine and pharmacology.

Keywords

• network control
• attracting sets
• dynamical system
• phase portrait
• gene regulatory networks
• artificial neural systems