=Paper=
{{Paper
|id=Vol-3746/Short_1.pdf
|storemode=property
|title=Example of Chaotic Behavior in Systems of Ordinary Differential Equations Arising in Modeling of Gene Regulatory Networks
|pdfUrl=https://ceur-ws.org/Vol-3746/Short_1.pdf
|volume=Vol-3746
|authors=Felix Sadyrbaev,Olga Kozlovska
|dblpUrl=https://dblp.org/rec/conf/dsmsi/SadyrbaevK23
}}
==Example of Chaotic Behavior in Systems of Ordinary Differential Equations Arising in Modeling of Gene Regulatory Networks==
https://ceur-ws.org/Vol-3746/Short_1.pdf
Example of Chaotic Behavior in Systems of Ordinary Differential
Equations Arising in Modeling of Gene Regulatory Networks
Felix Sadyrbaev 1 and Olga Kozlovska 2
1
Institute of Mathematics and Computer Science, University of Latvia, Rainis boul. 29, Riga, LV1459, Latvia
2
Department of Engineering Mathematics, Riga Technical University, Kipsalas 6a, Riga, LV1048, Latvia
Abstract
The three-dimensional system of ordinary differential equations arising in mathematical
models of genetic networks is considered. The systems under consideration first appeared in
the study of neural networks. They were later applied by several authors to the treatment of
genetic regulatory networks and telecommunication networks. In this article, we will focus on
genetic networks. The proposed equation models the evolution of the genetic regulatory
network. The system has attractors in the phase space that strongly influence the behavior of
the trajectories and other important properties of the network. Attractors are built in systems
with dimensions greater than three. These attractors are not solutions themselves, but they
attract periodic solutions of the system. A system of equations can exhibit chaotic behavior
that is not easy to detect. An information is provided about possible attracting sets in the phase
space. The role of attracting sets is discussed. An example of the system possessing the attractor
of chaotic nature is constructed.
Keywords 1
Dynamical systems, gene regulatory networks, attractors, sensitive dependence on the initial
data, chaotic behavior
1. Introduction
In this note, we consider systems of ordinary differential equations of the form
1
π₯ β²1 = β π£1 π₯1 ,
1 + π βπ1(π€11π₯1 + π€12π₯2 +π€13 π₯3 βπ1 )
1
π₯β²2 = β π£2 π₯2 , (1)
1 + π βπ2(π€21π₯1 + π€22 π₯2 + π€23 π₯3 βπ2 )
1
π₯β² = β π£3 π₯3 .
{ 3 1 + π βπ3(π€31π₯1 + π€32π₯2 + π€31π₯3βπ3)
This system appears in multiple contexts. Generally it can be interpreted as the model of three
interacting elements [1]. These elements can be of different nature. Their status at the time moment t is
described by the state vector X(t)= (x1(t),x2(t),x3(t)). Future states of a network strongly depend on the
attracting sets in the phase space (x1,x2,x3). The attracting sets must exist, as we will see later. The
system (1) is quasi-linear.
1
The nonlinearity is represented by the sigmoidal function π(π§) = , which is continuous,
1+π βππ§
bounded, and monotonically increasing from zero to unity. In the absence of the nonlinear terms, the
system (1) is linear and solutions exponentially degrade to zero, since the coefficients vi are supposed
to be positive. The slopes Β΅i of the exponential functions are positive, ο±i are adjustable parameters
interpreted as thresholds. The interrelations between elements of a network is described by the constant
matrix
Dynamical System Modeling and Stability Investigation (DSMSI-2023), December 19-21, 2023, Kyiv, Ukraine
E-MAIL: felix@latnet.lv (A. 1); olga.kozlovska@rtu.lv, (A. 2)
ORCID: 0000-0001-5074-804X (A. 1);0009-0000-6438-0602 (A. 2)
Β©οΈ 2023 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
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Workshop
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ISSN 1613-0073
85
Proceedings
π€11 π€12 π€13
π = (π€21 π€22 π€23 ). (2)
π€31 π€32 π€33
The entries can be of arbitrary sign. Zero entry wij corresponds to no-relation between the elements
xi and xj, the positivity (resp.: the negativity) of wij is associated with activation (resp.: inhibition) of the
element xj by the element xi. The self-activation and self-inhibition are allowed. Under these conditions
the dynamics of solutions of the system (1) can be rich. Our purpose here is briefly review possible
behavior of solutions. Before to go, we mention that the system (1) was used in models of neuronal
populations [2-4], in the theory of telecommunication networks [5]. The practical applications in bio-
medicine were indicated in [6-8].
2. Properties of system (1)
In the study of system (1) very efficient is the geometrical approach, based on the analyzing of the
nullclines. The nullclines are subsets of the phase space, where the trajectories change their direction.
The nullclines are defined by the system of equations
1
π£1 π₯1 = 1 + π βπ1 (π€11 π₯1 + π€12 π₯2 +π€13 π₯3βπ1 ) ,
1
π£2 π₯2 = , (3)
1 + π βπ2 (π€21 π₯1 + π€22 π₯2 + π€23 π₯3 βπ2 )
1
{π£3 π₯3 = 1 + π βπ3 (π€31 π₯1 + π€32 π₯2 + π€31 π₯3 βπ3 ) .
The common points of the nullclines are the critical points (equilibria). The local analysis of the
system (1) around the critical points is standard and can be made by linearization of system (1) at critical
points and defining their types.
The system (1) has the following properties:
1. There exists an invariant set Q3 = {0 < xi < 1/vi, i = 1, 2, 3};
2. All critical points are in Q3;
3. There exists at least one critical point;
4. The system (1) can have multiple critical points, but their number is limited;
5. The system (1) can have stable critical points, which are the simplest attractors;
6. The system (1) can have stable critical points, which are the simplest attractors;
7. The system (1) can have an attractor in the form of a stable periodic solution (limit cycle).
The property 1 can be proved by direct inspection of the vector field, defined by system (1), on the
boundary of Q3. It should be noted that the linear parts in (3) dominate over the nonlinear parts in the
right hand sides of system (3). Then attractors in that system must exist [16]. οThe property 2 is clear,
since the nullclines can intersect only in Q3. The property 3 can be proved by application of the Bohl-
Brower fixed point theorem to the system (1) and the set Q3. The property 4 follows from the analysis
1
of the system (3) and the properties of the sigmoidal function π(π§) = 1+π βππ§ . The properties 5 to 7 can
be proved by constructing the examples. All the above mentioned can be found in the references [9-
11], [17-19] and references therein.
There are multiple examples for the three-dimensional systems of ordinary differential equations to
have chaotic attractors. A plenty of examples can be found in the book [20]. The great majority of these
examples are for systems with polynomial nonlinearities. In contrast, the examples of chaotic attractors
for systems of the form (1) are few. We can refer only two results in [13] and [21].
3. Cases
It is an easy matter to obtain the two-dimensional system of the form
1
π₯ β²1 = βπ1 11 π₯1 + π€12 π₯2 βπ1 )
(π€
β π£1 π₯1 ,
{ 1 + π
1
π₯β²2 = β π£2 π₯2,
1 + π 2 21 π₯1 + π€22 π₯2 βπ2 )
βπ (π€
86
which has one, two or more stable critical points. The number of critical points cannot exceed the
number nine, for the two-dimensional systems.
The rotating vector field can be constructed by choosing the regulatory matrices of the structure
π π
π = ( ),
π π
Where k>0, a b < 0. By changing k, the limit cycles can be obtained as in [9-11, 17-19]. The two-
dimensional limit cycles can be raised to the three dimensional case by a simple trick. Let the three
dimensional matrix W be of the form
π π 0
π = (π π 0),
0 0 π
where r and is such that the equation
1
0= β π£3 π₯3
1 + π 3 ( ππ₯3 βπ3 )
βπ
has three roots. Then the third nullcline for the three-dimensional system consists of three parallel
planes, in each of them there exists the two-dimensional limit cycle, which was built earlier.
Using similar constructions, the periodic attracting sets can be obtained in systems of the form (2)
with higher dimensionalities.
4. Example
Consider system (1) with the following set (*) of parameters:
Β΅1=4.29, Β΅2 =4.7; Β΅3=3.5; v1=0.135, v2=0.1, v3=0.25; ο±1=0.64; ο±2=0.5; ο±3=0.4;
w11= 0; w12=ο0.01; w13=1.26;
w21=ο0.98; w22=0.021; w23=5.25;
w31= 0; w32=ο1; w33=2.
Let the trajectory start at x1(0)=0; x2(0)=0; x3(0)=0.2.
The phase portrait is depicted below.
Figure 1: Chaotic behavior in system (1) with the parameters (*).
The time series (x1(t), x2(t), x3(t)) are depicted in Figure 2.
87
Figure 2: Solutions of the system (1) with the parameters (*).
The plot of Lyapunov curves indicates the sensitive dependence on the initial data. The Lyapunov
numbers are (0.0119394, 0.000229502, ο0.0430462), which fit the Table 1 (below) on page 28 in the
book [20].
Table 1
Charscteristics of the attractors for a bounded three-dimensional flow
ο¬1 ο¬2 ο¬3 Attractor Dimension Dynamic
- - - Equilibrium point 0 Static
0 - - Limit cycle 1 Periodic
0 0 - Attracting 2-torus 2 Quasiperiodic
0 0 0 Invariant 1 or 2 (Quasi) periodic
+ 0 - Strange 2 to 3 Chaotic
5. Conclusion
One more (of very few) chaotic attractor was found in the three-dimensional system (1), which is
supposed to model the basic features of genetic networks. The trajectory in Figure 1 tends to a bounded
attractor, which is neither a stable equilibrium nor a limit cycle. The sensitive dependence of solutions
on the initial data is confirmed by the location of the respective Lyapunov curves [19-21]. The
Lyapunov exponents are negative, almost zero, and positive. We guess that many of the strange
attractors, listed in the book [22] (8) for polynomial systems, can be obtained also for systems of the
form (1), where the dimensionality and the number of parameters should be increased.
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