Bifurcation analysis and chaotic behavior of a discrete-time delayed genetic oscillator model
- Feng Liu^{1}Email author,
- Xiang Yin^{1},
- Fenglan Sun^{2},
- Xinmei Wang^{1} and
- Hua O Wang^{3}
https://doi.org/10.1186/s13662-016-1053-6
© The Author(s) 2017
Received: 22 May 2016
Accepted: 2 December 2016
Published: 3 January 2017
Abstract
In this paper, a genetic oscillator model with time delay is discretized by the Euler method. The discrete oscillator model is discussed by using Neimark-Sacker bifurcation theory. The direction and the stability of the Neimark-Sacker bifurcation has been studied using the center manifold theorem and normal form theory. Numerical simulations illustrate the theoretical results.
Keywords
1 Introduction
In recent years, the dynamics of genetic oscillators has been investigated in the past decades [1–12]. Goldbeter shows that the genetic oscillators have a crucial impact on many aspects of cell physiology activities [3, 4]. Mathematical modeling has been playing key roles for understanding the dynamics of genetic oscillators [1–4]. The evolution of genetic oscillators can be described by ordinary differential equations or difference equations. Oscillation phenomena are common in nature, especially in those systems that have rhythm behavior. It has been demonstrated that one of the most common causes of the oscillation phenomenon is the existence of certain bifurcations [5–16]. In continuous-time systems, oscillations appear mainly due to Hopf bifurcations [8–11]. In discrete-time systems, however, the essential cause is Neimark-Sacker bifurcation or period-doubling bifurcation [13, 14].
Recently, increasing attention has been paid to discrete time models. The reasons are as follows: The numerical simulations of continuous-time models are obtained by the discretizing models. It is common practice to discretize the continuous-time model for experimental or computational purposes. The discrete-time model inherits the dynamic characteristics of the continuous-time model, and it also retains functional similarity to the continuous-time system and any physical or biological reality that the continuous-time model has [17]. At last, the discrete-time models have rich dynamical behaviors as compared to continuous-time models. We can get more accurate numerical simulations results from discrete time models. Herein, we will consider an oscillator model described by difference equations.
This paper is organized as follows. In Section 2, the stability of the positive equilibrium and the existence of a Neimark-Sacker bifurcation are discussed. In Section 3, the direction and the stability of bifurcating periodic solutions can be determined by using the center manifold theorem and normal form theory. In Section 4, a simulation example is applied to verify the theoretical results. The last section contains conclusions.
2 Stability analysis
In the following sections, we will study the dynamics behavior of the system (2.3) including stability and the bifurcation phenomenon. We will use some lemmas to illuminate existence conditions of a unique positive root of equation (2.4).
Let \(f(y) = y^{3} + My - N\).
Lemma 2.1
- (i)
\(0 < n_{1}n_{4} / n_{3}n_{5} < 1\);
- (ii)
\(n_{1}n_{4} / n_{3}n_{5} > 1\) and \(f(r_{1}) < 0\);
- (iii)
\(n_{1}n_{4} / n_{3}n_{5} > 1\), \(f(r_{1}) > 0\), and \(f(r_{2}) > 0\),
Proof
As \(f(y) = y^{3} + My - N\). Since \(f(0) = - N <0\), \(f( + \infty ) = + \infty \), it is clear that equation (2.5) has at least one positive solution.
We will consider three cases as follows:
If (i) holds, then \(\Delta = - 12M \le 0\).
This means that \(f'(y) \ge 0\). Hence \(f(y)\) is monotonically increasing on \([0, + \infty )\). Based on the above analysis, we know that the positive root is unique.
Similarly, if (iii) holds\(,f(y)\) has a unique positive root.
It means that under one of the assumptions (i)-(iii), there is a unique positive equilibrium point \(( x^{ *},y^{ *} ) \) in the system (2.3). This completes the proof. □
According to the knowledge of the dynamics, we know that the equilibrium stability of equation (2.6) is determined by the roots distribution of equation (2.9). We will utilize a lemma of Zhang et al. [18] to analyze the roots distribution of equation (2.9).
Lemma 2.2
There exists a \(\bar{\tau } > 0\) such that, for \(0 < \tau < \bar{\tau }\), all roots of equation (2.9) have modulus less than one.
Proof
When \(\tau = 0\), we can obtain \(a_{1} = - 2\), \(a_{2} = 1\), \(a_{3} = 0\).
Consider the root \(\lambda (\tau )\) such that \(\vert \lambda (0) \vert =1\). This root depends continuously on τ and is a differentiable function of τ.
Lemma 2.3
Assume the step size h is sufficiently small. If \(\Delta _{1} \ge 0\), and \(0 < 16a_{2}^{*}( - a_{1}^{*}a_{2}^{*} - a_{1}^{*} \pm \sqrt{\Delta _{1}} ) < 1\), then equation (2.9) has no root with modulus one for all \(\tau > 0\).
Proof
Assume that the step size h is sufficiently small.
For sufficiently small \(h > 0\), if \(\Delta _{1} \ge 0\), and \(0 < 16a_{2}^{*}( - a_{1}^{*}a_{2}^{*} - a_{1}^{*} \pm \sqrt{\Delta _{1}} ) < 1\), then we obtain \(\cos \omega ^{*} > 1\). This is a contradiction. So we complete proof. □
Lemma 2.4
The inequality \(d_{h} = \frac{d\vert \lambda \vert ^{2}}{d\tau }|_{\lambda = \lambda ^{ *},\tau = \tau ^{ *}} > 0\) holds for sufficiently small h.
Proof
From Lemmas 2.1-2.4, we obtain the following theorem.
Theorem 2.1
- (i)
When \(\tau \in [0,\tau _{0})\), the equilibrium \((x^{*},y^{*}) \) of system (2.3) is asymptotically stable.
When \(\tau > \tau _{0}\), the equilibrium \(( x^{ *},y^{ *} ) \) of system (2.3) is unstable.
- (ii)
When \(\tau = \tau _{0}\), for the equilibrium \(( x^{ *},y^{ *} ) \) of system (2.3) there will exist a Neimark-Sacker bifurcation. This is to say, for system (2.3) there exists a cluster of periodic solutions bifurcating near the equilibrium at \(\tau = \tau _{0}\).
Remark
According to the above discussions and applying the Neimark-Sacker bifurcation theory presented in Kuznetsov [19], we obtain Theorem 2.1. We can see that the stability of the equilibrium \(( x^{ *},y^{ *} ) \) varies as the parameter τ varies. It is shown that the equilibrium is asymptotically stable for \(\tau \in [0,\tau _{0})\), and unstable for \(\tau > \tau _{0}\). We can observe that the Neimark-Sacker bifurcation occurs when the time delay crosses the critical value\(\tau _{0}\). According to the results of Theorem 2.1, we can see that the conclusions of the discrete system (2.3) are consistent with those of the continuous models (see [8, 10, 11]).
3 Direction and stability of the Neimark-Sacker bifurcation
In this section, we will use bifurcation theory [19–21] to discuss the direction and stability of the Neimark-Sacker bifurcation of system (2.3) for considering the delay time as a bifurcation parameter.
Let \(\tau ^{ *}\) is the critical value \(\tau _{j}\) (\(j = 0,1,2, \ldots\)) of the origin, at which system (2.6) undergoes a Neimark-Sacker bifurcation.
Lemma 3.1
Proof
It is noted that \(q(\tau ^{ *} )\), \(p(\tau ^{ *} )\) should satisfy \(\langle p,q \rangle = 1\), where \(\langle p,q \rangle = \sum_{j = 0}^{d} \bar{p}_{j} q_{j}\).
Let \(q = ( q_{1},q_{2}, \ldots,q_{d + 2} ) ^{T}\) and \(p = ( p_{1},p_{2}, \ldots,p_{d + 2} ) ^{T}\).
Let \(q_{1} = 1\), then we can get the eigenvector q.
Let a real two dimensional eigenspace of \(e^{ \pm \omega ^{ *} j}\) be \(T^{c}\), and \(T^{s}\) be a d dimensional eigenspace which is other than \(T^{c}\).
Theorem 3.1
When \(l > 0\) (<0), where \(l = - \operatorname{Re} [ e^{i\omega ^{ *}} c_{1} ( \tau ^{ *} ) ] / d_{h}\), we know that if the curve exists for \(\tau > \tau ^{ *}\), the bifurcation of equation (2.4) is supercritical (subcritical); when \(\operatorname{Re} [e^{ - i\omega ^{ *}} c_{1}(\tau ^{ *} )] < 0\), or \(\operatorname{Re} [e^{ - i\omega ^{ *}} c_{1}(\tau ^{ *} )] > 0\) the bifurcation is orbitally stable or unstable.
4 Numerical simulation
In Figure 1, we show that the waveform plot and phase diagram for equation (2.5) when \(\tau = 1 < \tau _{0}\) and for the equilibrium \(( x^{ *},y^{*} ) \) are asymptotically stable.
From Figures 1-5, we can observe that a Neimark-Sacker bifurcation occurs when the time delay crosses the critical value\(\tau _{0}\), It is shown that the positive equilibrium is asymptotically stable for \(\tau \in [0,\tau _{0})\), and unstable for \(\tau > \tau _{0}\) which is entirely consistent with the results in Theorem 2.1. It can be found that there are different kinds of Neimark-Sacker bifurcations for different time delays and when the system undergoes a variety of bifurcations, for the system can be found the chaos phenomenon. Chaos has important significance to the life system. It shows that the parameter time delay has important influence on the dynamic behaviors of the system, which may easily generate complex oscillations and chaos.
5 Conclusions
A discrete-time genetic network with delay was considered in this paper. Some sufficient conditions of local stability of equilibrium points were given. By choosing the time delay as a bifurcation parameter, we show that the Neimark-Sacker bifurcation would occur when the bifurcation parameter crosses some critical values. We obtained a formula for determining the direction and stability of a Neimark-Sacker bifurcation.
The theoretical studies on discrete-time genetic network model may not only contribute to the understanding of dynamic relation among different elements.
Declarations
Acknowledgements
The authors are very grateful to the editor and anonymous referees for their valuable comments and helpful suggestions, which have led to a great improvement of the original manuscript. This work was partially supported by National Natural Science Foundation (NNSF) of China under Grant 61472374, 61503053, 61672112, 11401110, 61603358, the fund of research center for advanced control of complex systems & intelligent geosciences instrument, China University of Geosciences (Wuhan) (AU2016CJ021).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Hidde, DJ: Modeling and simulation of genetic regulatory systems: a literature review. J. Comput. Biol. 9, 67-103 (2002) View ArticleGoogle Scholar
- Kobayashi, T, Chen, L, Aihara, K: Modeling genetic switches with positive feedback loops. J. Theor. Biol. 221, 379-399 (2003) MathSciNetView ArticleGoogle Scholar
- Goldbeter, A: Computational approaches to cellular rhythms. Nature 420, 238-245 (2002) View ArticleGoogle Scholar
- Tyson, JJ: Design principles of biochemical oscillators. Nat. Rev. Mol. Cell Biol. 9, 981-982 (2008) View ArticleGoogle Scholar
- Chen, L, Aihara, K: Stability of genetic regulatory networks with time delay. IEEE Trans. Biomed. Circuits Syst. 49, 602-608 (2002) MathSciNetView ArticleGoogle Scholar
- Li, C, Chen, L, Aihara, K: Stability of genetic networks with sum regulatory logic: Lur’e system and LMI approach. IEEE Trans. Biomed. Circuits Syst. 53(11), 2451-2458 (2006) MathSciNetView ArticleGoogle Scholar
- He, W, Cao, J: Robust stability of genetic regulatory networks with distributed delay. Cogn. Neurodyn. 2, 355-361 (2008) View ArticleGoogle Scholar
- Wan, A, Zou, X: Hopf bifurcation analysis for a model of genetic regulatory system with delay. J. Math. Anal. Appl. 356, 464-476 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Shen, J, Liu, Z, Zheng, W, Xu, F, Chen, L: Oscillatory dynamics in a simple gene regulatory network mediated by small RNAs. Physica A 388, 2995-3000 (2009) MathSciNetView ArticleGoogle Scholar
- Wang, K, Wang, L, Jiang, H, Teng, Z: Stability and bifurcation of genetic regulatory networks with delays. Neurocomputing 73, 2882-2892 (2010) View ArticleGoogle Scholar
- Xiao, M, Zheng, W, Cao, J: Stability and bifurcation of genetic regulatory networks with small RNAs and multiple delays. Int. J. Comput. Math. 91, 5907-5927 (2014) View ArticleMATHGoogle Scholar
- Wan, X, Xu, L, Fang, H, Yang, F, Li, X: Exponential synchronization of switched genetic oscillators with time-varying delays. J. Franklin Inst. 351, 4395-4414 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Jiang, XW, Zhan, XS, Guan, ZH, Zhang, XH, Yu, L: Neimark-Sacker bifurcation analysis on a numerical discretization of Gause-type predator-prey model with delay. J. Franklin Inst. 352(1), 1-15 (2015) MathSciNetView ArticleMATHGoogle Scholar
- He, X, Li, CD, Huang, TW, Yu, JZ: Bifurcation behaviors of an Euler discretized inertial delayed neuron model. Sci. China, Technol. Sci. 59, 418-427 (2016) View ArticleGoogle Scholar
- Wang, B, Jian, J: Stability and Hopf bifurcation analysis on a four-neuron BAM neural network with distributed delays. Commun. Nonlinear Sci. Numer. Simul. 15, 189-204 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Yu, P, Lin, W: Complex dynamics in biological systems arising from multiple limit cycle bifurcation. J. Biol. Dyn. 10, 263-285 (2016) MathSciNetView ArticleGoogle Scholar
- Mohamad, S, Naim, A: Discrete-time analogues of integro-differential equations modelling bidirectional neural networks. J. Comput. Appl. Math. 138, 1-20 (2002) MathSciNetView ArticleMATHGoogle Scholar
- Zhang, C, Liu, M, Zheng, B: Hopf bifurcation in numerical approximation of a class delay differential equations. Appl. Math. Comput. 146, 335-349 (2003) MathSciNetMATHGoogle Scholar
- Kuznetsov, YA: Elements of Applied Bifurcation Theory. Springer, New York (1998) MATHGoogle Scholar
- Hassard, BD, Kazarinoff, ND, Wan, YH: Theory and Applications of Hopf Bifurcation. Cambridge University Press, Cambridge (1981) MATHGoogle Scholar
- Han, M, Yu, P: Normal Forms, Melnikov Functions, and Bifurcations of Limit Cycles. Springer, New York (2012) View ArticleMATHGoogle Scholar