Attraction in n ‐dimensional differential systems from network regulation theory

Attracting sets in network regulatory theory

Modern telecommunication networks are very complex and they should be able to deal with rapid and unpredictable changes in traffic flows. Virtual Network Topology is used to carry IP traffic over Wavelength-division multiplexing optical network. To use network resources in the most optimal way, there is a need for an algorithm, which will dynamically re-share resources among all devices in the particular network segment, based on links utilization between routers. Attractor selection mechanism could be used to dynamically control such Virtual Network Topology. The advantage of this algorithm is that it can adopt to very rapid, unknown and unpredictable changes in traffic flows. This mechani…

Networks Describing Dynamical Systems

Abstract We consider systems of ordinary differential equations that arise in the theory of gene regulatory networks. These systems can be of arbitrary size but of definite structure that depends on the choice of regulatory matrices. Attractors play the decisive role in behaviour of elements of such systems. We study the structure of simple attractors that consist of a number of critical points for several choices of regulatory matrices.

On a differential system arising in the network control theory

We investigate the three-dimensional dynamical system occurring in the network regulatory systems theory for specific choices of regulatory matrix { { 0, 1, 1 } { 1, 0, 1 } { 1, 1, 0 } } and sigmoidal regulatory function f(z) = 1 / (1 + e-μz), where z = ∑ Wij xj - θ. The description of attracting sets is provided. The attracting sets consist of respectively one, two or three critical points. This depends on whether the parameters (μ,θ) belong to a set Ω or to the complement of Ω or to the boundary of Ω, where Ω is fully defined set.

Remarks on GRN-type systems

Systems of ordinary differential equations that appear in gene regulatory networks theory are considered. We are focused on asymptotical behavior of solutions. There are stable critical points as well as attractive periodic solutions in two-dimensional and three-dimensional systems. Instead of considering multiple parameters (10 in a two-dimensional system) we focus on typical behaviors of nullclines. Conclusions about possible attractors are made.

Dynamical Models of Interrelation in a Class of Artificial Networks

The system of ordinary differential equations that models a type of artificial networks is considered. The system consists of a sigmoidal function that depends on linear combinations of the arguments minus the linear part. The linear combinations of the arguments are described by the regulatory matrix W. For the three-dimensional cases, several types of matrices W are considered and the behavior of solutions of the system is analyzed. The attractive sets are constructed for most cases. The illustrative examples are provided. The list of references consists of 12 items.