On a differential system arising in the network control theory

Abstract. 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 = ∑ Wijxj − 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 setΩ or to the complement ofΩ or to the boundary of Ω, where Ω is fully defined set.


Introduction
In the articles [5,6], the following problem is studied.To accommodate traffic on a wavelength-routed optical network, one can construct an optimal virtual network topology (VNT) by establishing a set of lightpaths between nodes.To treat changing in time (fluctuating) traffic on a VNT, adaptive VNT control methods should be invented, which reconfigure VNTs according to traffic conditions on VNTs.To develope such methods, one way is to observe attractor selection in biological systems that adapt to unknown changes in their surrounding environments and recover their conditions.
Mechanism and concept of Attractor Selection are described in [6]: "1) Concept of Attractor Selection: The dynamic system that is driven by attractor selection uses noise to adapt to environmental changes.In attractor selection, attractors are a part of the equilibrium points in the solution space in which the system conditions are preferable.The basic mechanism consists of two behaviors, i.e., deterministic and c Vilnius University, 2016 stochastic behaviors.When the current system conditions are suitable for the environment, i.e., the system state is close to one of the attractors, deterministic behavior drives the system to the attractor.Where the current system conditions are poor, stochastic behavior dominates over deterministic behavior.While stochastic behavior is dominant in controlling the system, the system state fluctuates randomly due to noise and the system searches for a new attractor.When the system conditions have recovered, deterministic behavior again controls the system.These two behaviors are controlled by simple feedback of the conditions of the system.In this way, attractor selection adapts to environmental changes by selecting attractors using stochastic behavior, deterministic behavior, and simple feedback." Therefore, the attractor selection represents mechanism of adaptation to unknown and rapid changes in biological systems.The overview of results in this direction can be found in [1,3,4,7].So the problem of describing the structure of attracting sets is actual and useful for understanding the above mentioned biological (and related telecommunication) networks.
We follow the above model in the following setting.We neglect the stochastic behavior in our analysis.Besides we consider the simplified model consisting of three differential equations.This system can be treated by the 3D phase plane analysis [2].Our goal is to study the structure of attracting sets in this simple model.The system involves two parameters, therefore, changes in the structure of a phase space should be described under the change of parameters.We provide full description of attracting sets depending on the choice of parameters.

System
The dynamics of the expression level of the protein on the ith gene, x i , is described by the differential system The first and second terms at the right-hand side represent the deterministic behavior of gene i; and the third term η represents stochastic behavior.The deterministic behavior controls the x i due to the effects of activation and inhibition from the other genes.Those regulations of protein expression levels on gene i by other genes are indicated by regulatory matrix W ij , the elements of which take values 1, 0, or −1, corresponding respectively to activation, no regulatory interaction, and inhibition of the ith gene by the jth gene.The rate of increase in the expression level is given by the sigmoidal regulation function Parameter θ is a regulatory parameter, which can be adjusted, and µ indicates the gain parameter of the sigmoidal function. http://www.mii.lt/NA

Three-dimensional system
We consider the simplified system (v g = 1, η = 0) The entries of W ij can take values 1, 0 or −1.We consider the specific case System (1) looks as where µ and θ are positive parameters.Our goal is to study phase portraits of this system and describe the attracting sets.

Critical points
Critical points of system (2) are to be determined from Since the right sides in (3) are positive but less than a unity, all critical points locate in the cube (0; 1) × (0; 1) × (0; 1).Lemma 1.All critical points are on the bisectrix Proof.System (3) can be written as It follows from ( 4) that We will prove that x 1 = x 2 , in the same way, it follows that x 1 = x 3 and then x 1 = x 2 = x 3 .Divide the first row in ( 5) by the second one and get Suppose that x 1 and x 2 in equation ( 6) are distinct.Consider the case x 2 > x 1 .Then the contradiction 1 + e −µ(x1+x3−θ) < 1 follows.Similarly, the case x 2 < x 1 can be considered.It follows then that x 1 = x 2 .Similarly, x 1 = x 3 , and it follows that any critical point is of the form (x; x; x), where The proof is complete.Formula ( 7) is a key relation between x, θ and µ.This is visualized in Fig. 1.
We claim that there are only three possibilities for the number of critical points of system (2).Lemma 2. There are at most three critical points of system (2).
Put values x 1 (µ) and x 2 (µ) into (8) and get The graphs of these functions θ 1 , θ 2 are depicted in Fig. 3.The upper branch is for θ 2 , and the lower one is for θ 1 .It follows from the proof of Lemma 2 and it is illustrated by Fig. 2 that there is exactly one critical point if (µ, θ) ∈ Q \ Ω.

Linearized system
In this section, we would like to study the character of possible critical points.For this, consider the linearized system By using (5) and x 1 = x 2 = x 3 = x the linearized system can by written as The matrix A of system (12) is In the new variables L = (1 + λ) and M = µx(1 − x), equation ( 13) takes a simpler form The roots of ( 14) are It is evident that λ 1 and λ 2 are negative, but λ 3 can be positive, negative and even zero.It follows from the key relation ( 7) that λ i can be written also as The above form is convenient for further analysis of characters of critical points.
It appears that the following three types of critical points are possible for system (2).
The first type of a critical point after reduction to canonical variables (v 1 , v 2 , v 3 ) is a stable node in both subspaces (v 1 , v 2 ) and v 3 : There is a degenerate stable node in the (v 1 , v 2 )-subspace attracting trajectories in v 3direction due to negativity of λ 3 .
For the purposes of this article, we will denote this type as (−, −, −).
The second type of a critical point is a stable degenerate node in the (v 1 , v 2 )-subspace with continua of copies in v 3 -direction due to zero value of λ 3 : In this article, we denote this type as (−, −, 0).

5.
Projections of the phase portrait in Fig. 4 on three coordinate planes.
Figure 6 shows the phase portrait of a linearized system in canonical variables.The respective λ 3 is zero.It demonstrates that in (v 1 , v 2 ) subspace, solutions near the critical point tend to (0; 0) point (this can be seen in the left graph of Fig. 7).This two-dimensional subspace is repeated by parallel transition along the v 3 -axis.
The third type of a critical point is a semi-stable node that is stable in the (v 1 , v 2 )subspace and unstable in v 3 -direction: The respective notation will be (−, −, +).
The graph of Fig. 8 is in canonical form.It demonstrates that in [v 1 , v 2 ] subspace, solutions tend to (0; 0) (this can be seen in the left graph of Fig. 9).In the remaining one-dimensional subspace v 3 , solutions are repelled by the critical point.http://www.mii.lt/NA

Critical points
In this section, we analyze types of critical points.For any critical point λ 1 = λ 2 < 0, it is sufficient to know the sign of λ 3 .For brevity and only for purposes of this article, let us denote the type of a critical point by symbols (−, +, 0), where −, 0 and + are respectively for λ negative, zero or positive.It is known up to now that any critical point is of the type (−, −, * ), where " * " may be −, 0 or +.
Proof.The proof follows from the fact that, in the mentioned region, the quadratic polynomial (19) and, therefore, the right-hand side in (18) are negative.Proposition 3.For any (µ, θ) ∈ Ω, there exist exactly three critical points: the middle is of the type (−, −, +), and two side points are of the type (−, −, −).
Proof.For any µ > 2, consider some θ * such that The horizontal line θ = θ * in (x, θ)-plane (Fig. 2c) has three points of intersection with the curve θ(x).The middle point x is between x 1 and x 2 .Therefore, polynomial (19) is positive, and hence, the respective λ 3 is positive.The two remaining critical points satisfy either x < x 1 or x > x 2 .In both cases, polynomial (19) is negative, and the respective λ 3 are negative also.
It follows from the above arguments that the following statement is true.
Theorem 1.The above system has attracting sets of the following structure.
All critical points are located on the bisectrix The open quadrant Q = {µ > 0, θ > 0} contains fully defined region Ω such that:

Phase portraits
Consider a number of examples illustrating (and confirming) our analysis.

Particular case I
In the first particular case, the parameter µ = 1 and θ = 0.5.System (2) has one critical point (0.71848; 0.71848; 0.71848).System (2) with parameters µ = 1 and θ = 0.5 has the following visual interpretation.In Fig. 10, we can see attractor, where all solutions from the first quadrant tend to the critical point (0.71848, 0.71848, 0.71848).The λ matrix for the critical point (0.71848, 0.71848, 0.71848) is The type of this critical point is a stable node in both subspaces according to (15).

Particular case II
In the second particular case, the parameter µ = 3 and θ = 1.The system has three critical points: (0.5, 0.5, 0.5), (0.07072, 0.07072, 0.07072), and the last one is (0.929279, 0.929279, 0.929279).In Fig. 11, one can see the visual interpretation of the system.The λ matrix for the critical point (0.5, 0.5, 0.5) is The type of this critical point is a semi-stable node that is stable in the one subspace and unstable in the second subspace according to (17).
The visual interpretation of the system is the following: The type of this critical point is a stable node in both subspaces according to (15).

Figure 3 .
Figure 3.The dependence of θ (for a critical point (x; x; x)) of µ.

Figure 7 .Figure 9 .
Figure 7. Projections of the phase portrait in Fig. 6 on three coordinate planes.

.
The type of this critical point is a stable node in one subspace and a degenerate type in the second subspace according to (16).