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. data, chaotic behavior Dynamical systems, gene regulatory networks, attractors, sensitive dependence on the initial Proceedings
In this note, we consider systems of ordinary differential equations of the form
1 = ′ ′ 2 = 3 = 1 1 1 1 + − 1( 11 1 + 12 2 + 13 3− 1) − 1 1, 1 + − 2( 21 1 + 22 2 + 23 3− 2) − 2 2 1 + − 3( 31 1 + 32 2 + 31 3− 3) − 3 3
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.
The nonlinearity is represented by the sigmoidal function ( ) = 1 1+ − , which is continuous, 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 µi of the exponential functions are positive, i are adjustable parameters interpreted as thresholds. The interrelations between elements of a network is described by the constant matrix
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ceur-ws.org (1) 1 + − 3( 31 1 + 32 2 + 31 3− 3) , , 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 biomedicine were indicated in [6-8].
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
It is an easy matter to obtain the two-dimensional system of the form { ′1 = ′2 = 1 1 1 + − 1( 11 1 + 12 2 − 1) 1 + − 2( 21 1 + 22 2 − 2) − 1 1, − 2 2, (2) (3)
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. 2. 3. 4. 5. 6. 7.
There exists an invariant set Q3 = {0 < xi < 1/vi, i = 1, 2, 3}; All critical points are in Q3; There exists at least one critical point; The system (1) can have multiple critical points, but their number is limited; The system (1) can have stable critical points, which are the simplest attractors; The system (1) can have stable critical points, which are the simplest attractors;
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]. The property 2 is clear, since the nullclines can intersect only in Q3. The property 3 can be proved by application of the BohlBrower fixed point theorem to the system (1) and the set Q3. The property 4 follows from the analysis of the system (3) and the properties of the sigmoidal function ( ) = be proved by constructing the examples. All the above mentioned can be found in the references [91 1+ − . The properties 5 to 7 can 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]. = ( = ( 0 1 0 ), 0 number nine, for the two-dimensional systems.
which has one, two or more stable critical points. The number of critical points cannot exceed the The rotating vector field can be constructed by choosing the regulatory matrices of the structure Where k>0, a b < 0. By changing k, the limit cycles can be obtained as in [9-11, 17-19]. The twodimensional limit cycles can be raised to the three dimensional case by a simple trick. Let the three dimensional matrix W be of the form where r and is such that the equation 0 = 1 + − 3( 3 − 3) − 3 3 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.
Consider system (1) with the following set (*) of parameters: µ1=4.29, µ2 =4.7; µ3=3.5; v1=0.135, v2=0.1, v3=0.25; 1=0.64; 2=0.5; 3=0.4;
The phase portrait is depicted below. The time series (x1(t), x2(t), x3(t)) are depicted in Figure 2. 1 0 0 0 + 2 0 0 0 3 0
The plot of Lyapunov curves indicates the sensitive dependence on the initial data. The Lyapunov numbers are (0.0119394, 0.000229502, 0.0430462), which fit the Table 1 (below) on page 28 in the book [20].
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.