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Journal of Mathematics
Volume 2013 (2013), Article ID 347071, 11 pages
Stability and Andronov-Hopf Bifurcation of a System with Three Time Delays
1Department of Biomechanics, Institute of Mechanics-BAS, Acad. G. Bonchev Street, Block 4, 1113 Sofia, Bulgaria
2Department of Mechanics, University of Transport, Geo Milev Street, 158, 1574 Sofia, Bulgaria
Received 7 November 2012; Revised 25 January 2013; Accepted 26 January 2013
Academic Editor: Zindoga Mukandavire
Copyright © 2013 Svetoslav Nikolov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
A general system of three autonomous ordinary differential equations with three discrete time delays is considered. With respect to the delays, we investigate the local stability of equilibria by analyzing the corresponding characteristic equation. Using the Hopf bifurcation theorem, we predict the occurrence of a limit cycle bifurcation for the time delay parameters. Thus, some new mathematical results are obtained. Finally, the above mentioned criteria are applied to a system modelling miRNA regulation.
The future status of many systems arising from engineering, physics, mechanics, biochemistry, or systems in biology is determined not only by their current behavior but is also by their history. Such phenomena are called delay or genetic effects. Time lags in continuous systems can produce complex dynamics and instabilities. In the recent years, many mathematical models that have appeared in the literature involve either a single discrete delay to investigate the role of phosphorylation, negative or positive feedback regulation of transcription factors [1–4], and gene expression multistability . Several papers [6–13] among others consider systems with two or three discrete delays. When there is more than one delay in the equations, the local theory for stability is not fully complete.
A dynamical system which is not finite dimensional is called an infinite dimensional dynamical system. A class of infinite dimensional system can be determined by functional differential equations of the retarded type. To introduce such equations, we let denote the set with the norm defined by where denotes a norm (e.g., the Euclidean norm). Given a function defined on , let be the function determined by for . A retarded functional differential equation (with delay ) is an equation of the form where and is an open set in . The notations of the dynamical system are determined by varying over some appropriate subset of . As far as we know, the first statement similar to the Hopf theorem (concerning the bifurcation of periodic solutions from a singular point of an ordinary differential equation)  for retarded functional differential equations was given in .
In this paper, we consider stability and bifurcation behavior of a three-dimensional system of autonomous ordinary differential delay equations of the form where, are such that solutions to initial value problems exist and are continuable.
2. Derivation of the Characteristic Equation
We assume that , , and have continuous first partial derivatives with respect to their arguments and that there exists unique , , and , such that . We utilize the following notation: that represents the partial derivative of with respect to its th argument evaluated . A similar meaning is given to , and . Hence, for example, and . The variational system with respect to is We seek a solution of (3) of the form , and . For nontrivial solutions, this leads to the following characteristic equation in : where
Remark 1. We note that is a root of (4) if and only if = = = .
It is well known that the stability of the equilibrium state depends on the sign of the real parts of the roots of (4). If satisfies (4), then and are real solutions; that is, we rewrite (4) in terms of its real and imaginary parts as In the absence of delays , is locally asymptotically stable if Because of the presence of three different discrete delays in (2), the analysis of the sign of the real parts of the eigenvalues is very complicated, and a direct approach cannot be considered. Thus, in our analysis we will use a method consisting of determining the stability of the steady state when firstly two delays are equal to zero, and when secondly one delay is equal to zero. Previously, this approach is used for system with two delays [9, 12, 13, 16–18].
3. The Case and
To find the first bifurcation point, we look for purely imaginary roots , of (4) (when ); that is, we set . Then, the above two equations are reduced to or another one We note here that it is not possible for and to be both zero and can be a solution of (11) if . If the first bifurcation point is , then the other bifurcation points satisfy , .
One can notice that if is a solution of (10) (or (11)), then so is . Hence, in the following we only investigate for positive solutions of (10), or (11) respectively. By squaring the two equations into system (10) and then adding them, it follows that As is locally asymptotically stable at , it satisfies the Routh-Hurwitz conditions for stability for a cubic polynomial [19, 20]. Equation (12) is a cubic in and the left-hand side is positive for very large values of and also at . Suppose that conditions of Lemma 1(I) in  are satisfied; that is, (12) has at least one positive real simple root. Moreover, to apply the Hopf bifurcation theorem, according to , the following theorem in this situation applies.
To establish an Andronov-Hopf bifurcation at , we need to show that the following transversality condition is satisfied.
Hence, if we denote then
Evaluating the real part of this equation at and setting yield where and .
Let , then (12) reduces to Then for , we have
If is the least positive simple root of (12), then
According to the Hopf bifurcation theorem , we define the following theorem.
Theorem 3. If is the least positive root of (12), then an Andronov-Hopf bifurcation occurs as passes through .
Corollary 4. When , then the steady state of system (2) is locally asymptotically stable.
4. The Case
We return to the study of (4) which with has the form where , , , , , , , , and denotes a point in the time delay space; that is, . is the time delay space and denotes the set of nonnegative real numbers. In order to assess the stability of (2) with respect to any delay , one should know where all roots of (20) lie on the complex plane. Equation (20) has infinitely many roots on the complex plane due to the transcendental term . This makes the analytical stability assessment intractable.
In Section 2, we obtain that in the absence of delays, is locally asymptotically stable if the conditions (7) are valid. By Remark 1, this implies that is not root of (20). Further, we introduce the following simple result (which was proved by ) using Rouche’s theorem.
Lemma 5. Consider the exponential polynomial + , where and are constants. As vary, the sum of the order of the zeros of on the open right half plane can change only if a zero appears there or crosses the imaginary axis.
Obviously, is a root of (20) if and only if satisfies Separating the real and imaginary parts into (21), we obtain We square and add (22), and after simplifying, we get that and must be among the real solutions of We note that the right-hand side of (23) is always less than − + . Hence if the inequality has no real solution on , then (23) cannot be satisfied. Note that is the positive solution of first equation in (6) (when and ), which we write as Thus, for , we have where , and . It is clear that .
Rearranging terms, we write (24) as Hence, the following theorem can be formulated.
Theorem 6. Let and (27) hold. Then there is no change in stability of .
Corollary 8. If conditions of Theorem 6 are not valid and is defined as in Theorem 3, then according to Lemma 5 for any , there exists a resp. such that the steady state of system (2) is unstable when resp., and an Andronov-Hopf bifurcation takes place.
5. The General Case
Similar to Section 4, we set that is a root of (4) if and only if satisfies where , , , , , , denotes a point in the time delay space; that is, . is the time delay space and denotes the set of nonnegative real numbers.
Separating the real and imaginary parts into (29), we have where Adding up the squares of both equations into (30), we have where Clearly, the right-hand side of (32) is always less than Hence, if the inequality has no real solution on , then (32) cannot be satisfied. Similar to Section 4, we note that is the positive solution of first equation in (6) (when ), which is written as Thus, for , we have where , and . It is clear that .
Rearranging terms, we write (35) as Therefore, the following theorem can be formulated.
Theorem 9. Let and (38) hold. Then there is no change in stability of .
Remark 10. In the special case that , the characteristic equation (4) becomes
Hence, this case is a private one of Theorem 9.
Corollary 11. If conditions of Theorem 9 are not valid and is defined as in Theorem 3, then according to Lemma 5 for any , there exists a ; ; ; ; resp. such that the steady state of system (2) is unstable when ; ; ; resp., and an Andronov-Hopf bifurcation take place.
6. Application: Numerical Analysis
In this section, we apply the material and results of the previous sections to a nonlinear mathematical model with three discrete time delays, which investigate the dynamics of the gene expression regulated by the miRNA. For a general discussion of the model, we refer the reader to [22, 23]. Hence, we consider the following system: where the protein, , controls its own synthesis through the repression of mRNA, , is the concentration of miRNA, is the time delay for translation, is the time delay for transcription, is the average time delay for degradation of miRNA, is often referred as a Hill coefficient or a cooperativity coefficient, , and and are the kinetic rate constants. According to [2, 24], we consider one representative value for Hill coefficient, . Model (40) is a modified version of a model considered in ; that is, in the present paper our model contains three time delays.
The fixed points of the system, , represented by (40) can be analytically estimated and are defined by the following set of algebraic equations, including the rate constants of the model:
Let us consider a small perturbation around the fixed point of the system (40) defined as In the case when , the function can be written as a MacLaurin series where and . If we take only linear term from (43) and after substitution of (42) into differential equation (40), we have, where It is seen that notations in Section 2 here are In the next few figures we illustrate numerically the validity of the criteria proved Section 5, that is, when the three time delays are different from zero. For our simulations according to , the numerical values of the model parameters are In order to compare the predictions with numerical results, the governing equations of the model (40) were solved numerically using MATLAB . In Figure 1(a), the stable solutions for the concentration of mRNA (), the concentration of protein (), and the concentration of miRNA () are shown for absence of time delay (i.e., ). It is evident that after several physiological acceptance fluctuations, the solution of system (2) approaches a constant value (stable equilibrium state). In other words, the system possesses a stable equilibrium state which corresponds to a normal miRNA regulation process. This conclusion is in accordance with the criteria written in (7). In Figure 1(b), the unstable solutions of system (40) are shown. It is seen that for larger values of , than bifurcation one, after Andronov-Hopf bifurcation the stable limit cycle (self oscillations) occurs and sustained oscillations take place. In other words, in this case the conditions of Theorem 9 are not satisfied and the steady state of system (40) is unstable.
In this paper we have considered a system of three equations with three discrete time delays. Under the assumption that an equilibrium exists, we have estimated the length of delays for which local asymptotic stability will be preserved. We have also derived criteria for which no change in stability will occur.
If system (2) starts with a stable equilibrium, which for some delay(s) becomes unstable, it will likely destabilize by means of an Andronov-Hopf bifurcation leading to small amplitude periodic solutions. Our investigation of such a behavior is devoted to the use of the bifurcation analysis. Particularly, a Hopf bifurcation theorem was employed. The basic view that the time delays , , and