On a Planar Dynamical System Arising in the Network Control Theory

Abstract

Abstract. We study the structure of attractors in the two-dimensional dynamical− x2 that appears in the network control theory. We provide description of the attracting set and follow changes this set suffers under the changes of positive parameters µ and Θ.

Attraction in n‐dimensional differential systems from network regulation theory

“…Theorems 1 and 2 are the main results of the paper along with the examples in Section 5. For systems (9) with the regulatory matrices of the form (8), there are at most 3 critical points, all of them of symmetric form (x, … , x). The character of a critical point can be detected by analyzing the respective parameters and .…”

“…It follows from properties of f ( z ) function ( S ‐shaped graph, monotonicity, and boundedness) that Equation has at most 3 roots for any positive θ , w . For the specific sigmoidal function , the case n = 2 was studied in Atslega et al, and the general case of arbitrary n was investigated in Brokan and Sadyrbaev . It was proved that there exists decomposition of the first quadrant of the parameter ( μ , θ )‐plane in 2 disjoint sets separated by a simple curve so that for parameters in the first set, there is exactly 1 critical point of the type stable node, and for parameters in the second set, there are exactly 3 critical points of the type stable node (2 side points), and the remaining (middle) point is a saddle.…”

Control in Inhibitory Genetic Regulatory Network Models

“…Some of these facts were known and some were proved in [8][9][11][12][13][14]. Similar technique was used in the works [15][16][17].…”

“…We can obtain, following the arguments in [14] and using (10) and (11), the figure depicted in Figure 5.…”

On attracting sets in artificial networks: cross activation

“…Even in the simplest cases the analysis of such systems provide nontrivial results. For instance, in the work [8] the simplified system (for a network of two elements) of the form…”