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Discrete Dynamics in Nature and Society

Volume 2014 (2014), Article ID 139375, 7 pages

http://dx.doi.org/10.1155/2014/139375

## Chaos and Hopf Bifurcation Analysis of the Delayed Local Lengyel-Epstein System

^{1}Department of Applied Mathematics, Kunming University of Science and Technology, Kunming, Yunnan 650500, China^{2}School of Computer Science, Beijing University of Post Telecommunication, Beijing 100876, China^{3}Department of Mathematics, Southeast University, Nanjing 210096, China^{4}Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

Received 4 January 2014; Accepted 25 February 2014; Published 30 March 2014

Academic Editor: Wenwu Yu

Copyright © 2014 Qingsong Liu et al. 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.

#### Abstract

The local reaction-diffusion Lengyel-Epstein system with delay is investigated. By choosing as bifurcating parameter, we show that Hopf bifurcations occur when time delay crosses a critical value. Moreover, we derive the equation describing the flow on the center manifold; then we give the formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions. Finally, numerical simulations are performed to support the analytical results and the chaotic behaviors are observed.

#### 1. Introduction

As is well known, in chemistry, the chlorite-iodide-malonic acid (CIMA) reaction is a typical example to indicate diffusion-driven instability mechanism. Castets et al. [1] discovered the formation of stationary three-dimensional structures of CIMA. Lengyel and Epstein [2, 3] found that although there were five variables in the reaction, in fact, three of them in the reaction process were almost unchanged. Thus, it is able to simplify the original system to a two-dimensional model, which we call Lengyel-Epstein system. We know that the local system (the ODE model) of the Lengyel-Epstein system is taking the following form: where, in the content of the CIMA reaction, and denote the chemical concentrations of the activator iodine () and the inhibitor chlorite (), respectively, at time . The positive parameters and are related to the feed concentration; similarly, the positive parameter is a rescaling parameter depending on the concentration of the starch. Yi et al. [4] gave a detailed Hopf bifurcation analysis for this ODE model (and also the associated PDE model) by choosing as the bifurcation parameter and derived conditions on the parameters for determining the direction and the stability of the bifurcating periodic solution.

In order to reflect the dynamical behaviors of models depending on the past history of the system, it is often necessary to incorporate delays into the models. The ordinary and partial differential equations models involving time delays have been widely studied in fields as diverse as biology, population dynamics, neural networks, feedback controlled mechanical systems, machine tool vibrations, lasers, and economics [5–12]. While time delay effects can also be exploited to control nonlinear systems [4, 6, 10, 12–26], in [27], Çelik and Merdan considered the following delayed system: Using the delay parameter as a bifurcation parameter, they investigated the stability and Hopf bifurcation of the above system.

Motivated by the above discussion, in the present paper, we devote our attention to the delayed local Lengyel-Epstein system taking the following form: where is the positive time delay parameter. We consider the effect of time delay on and and give the conditions of the stability and the bifurcation of the positive equilibrium. By giving numerical simulations, we find that system (3) includes chaos.

This paper is organized as follows. In Section 2, we investigate the effect of the time delay on the stability of the positive equilibrium of system (3). In Section 3, we derive the direction and stability of Hopf bifurcation by using normal form and central manifold theory. Numerical simulations are carried out to illustrate the theoretical prediction and to explore the complex dynamics including chaos in Section 4. Section 5 summarizes the main conclusions.

#### 2. Stability Analysis and Hopf Bifurcation

It is easy to see that system (3) has a unique positive equilibrium with , where .

Let , , and system (3) can be written as where and h.o.t denotes the higher order terms. Then, we obtain the linearized system The corresponding characteristic equation is where , under the condition of

For (7), we have the following Lemma.

Lemma 1. *The two roots of (7) with have always negative parts if the condition holds.*

The characteristic equation (7) can be rewritten as the following equation: Thus, is the root of (7) if and only if satisfies one of the following equations: If , then we have and it is easy to obtain or Noticing that , therefore, we have , .

If , then and are a pair of complex conjugate numbers and it is easy to get and or Clearly, we have , .

Consequently, when , (7) with has a pair of purely imaginary roots ; when , (7) with has a pair of purely imaginary roots .

Summarizing the above and combining Lemma 1, we have the following result on the distribution of roots of (7).

Lemma 2. (i)*Assume that and hold; then when , all roots of (7) have strictly negative real parts, while when , (7) has a simple pair of purely imaginary roots .*(ii)*Assume that and hold; then when , all roots of (7) have strictly negative real parts, while when , (7) has a simple pair of purely imaginary roots .*

Let be the root of (7) near satisfying , , or . It is not difficult to verify that the following result holds.

Lemma 3. *If the condition holds, the transversality conditions hold.*

From Lemma 3, we have the following result.

Lemma 4. *If the condition holds, then when (or ) and (or ), (7) has at least one root with strictly positive real part.*

By Lemmas 1–4, we have the following theorem.

Theorem 5. *For system (3), assume that the condition holds; then the following statements are true.*(i)*If (or ), then the equilibrium of system (3) is asymptotically stable for (or .*(ii)*If (or ), then the equilibrium of system (3) is unstable when (or .*(iii)*When (or ), (or ) are Hopf bifurcation values of system (3).*

#### 3. Direction and Stability of the Hopf Bifurcation

In this section, using the method based on the normal form theory and center manifold theory introduced by Hassard et al. in [11], we study the direction of bifurcations and the stability of bifurcating periodic solutions. We denote the critical values as ; let ; then is the Hopf bifurcation value of system (3). Let , , and omit “” above ; then system (3) can be rewritten as where respectively. Let , and omit “” above and ; the nonlinear terms and are

Define a family of operators as By the Riesz representation theorem, there exists a matrix whose components are bounded variation functions such that In fact, choosing where is a Dirac delta function; then (17) is satisfied.

For , define

where

Hence, (14) can be rewritten as where and . For , define and the adjoint operator of as where is the transpose of the matrix .

For , and , we define a bilinear inner product where .

Since are eigenvalues of , they will also be the eigenvalues of . The eigenvectors of and are calculated corresponding to the eigenvalues and .

Lemma 6. * is the eigenvector of corresponding to ; is the eigenvector of corresponding to and
**
where
*

Following the algorithms explained in Hassard et al. [11], we can obtain the properties of Hopf bifurcation: where

We know that and are constant vectors computed as Thus, we can compute the following quantities: These expressions give a description of the bifurcating periodic solutions in the center manifold of system (3) at critical values and when which can be stated as follows:(i) gives the direction of Hopf bifurcation; if , the Hopf bifurcation is supercritical (subcritical).(ii) determines the stability of bifurcating periodic solution; the periodic solution is stable (unstable) if .(iii) denotes the period of bifurcating period solutions; if , the period increases (decreases).

#### 4. Numerical Simulations

To demonstrate the algorithm for determining the existence of Hopf bifurcation in Section 2 and the direction and stability of Hopf bifurcation in Section 3, we carry out numerical simulations on a particular case of (3) in the following form: where , , and . It is easy to show that system (32) has unique positive equilibrium , and . From the discussion of Section 2, we have ; by calculation, we obtain .

We can see from Figure 1(a) that is asymptotically stable at , while loses stability and Hopf bifurcation occurs when ; see Figure 1(b) at . Using the algorithm derived in Section 3, we obtain that , and , and we know that the Hopf bifurcation is supercritical, bifurcating periodic solutions are stable, and periods increase, whereas with parameter increasing chaotic solution occurs; see Figure 1(c) for . In Figure 1(d), largest Lyapunov exponent diagram is plotted for variable . It is easy to know when ; the Lyapunov exponent is almost positive; then the chaotic solutions occur.

#### 5. Conclusions

In this paper, we investigate the effect of the time delay on the stability of the positive equilibrium of the delayed local Lengyel-Epstein system and derive the direction and stability of Hopf bifurcation. Numerical simulations are carried out to illustrate the theoretical prediction and to explore the complex dynamics including chaos.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The authors are grateful to the reviewers for their valuable comments and suggestions which have led to an improvement of this paper. This research is supported by the National Natural Science Foundation of China (Grant no. 11061016).

#### References

- V. Castets, E. Dulos, J. Boissonade, and P. de Kepper, “Experimental evidence of a sustained standing Turing-type nonequilibrium chemical pattern,”
*Physical Review Letters*, vol. 64, no. 24, pp. 2953–2956, 1990. View at Publisher · View at Google Scholar · View at Scopus - I. Lengyel and I. R. Epstein, “Modeling of turing structures in the chlorite-iodide-malonic acid-starch reaction system,”
*Science*, vol. 251, no. 4994, pp. 650–652, 1991. View at Scopus - I. Lengyel and I. R. Epstein, “A chemical approach to designing Turing patterns in reaction-diffusion systems,”
*Proceedings of the National Academy of Sciences of the United States of America*, vol. 89, no. 9, pp. 3977–3979, 1992. View at Scopus - F. Yi, J. Wei, and J. Shi, “Diffusion-driven instability and bifurcation in the Lengyel-Epstein system,”
*Nonlinear Analysis: Real World Applications*, vol. 9, no. 3, pp. 1038–1051, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. J. S. Allen,
*An Introduction to Mathematical Biology*, Prentice Hall, Upper Saddle River, NJ, USA, 2007. - B. Balachandran, T. Katlmar-Nagy, and D. E. Gilsinn,
*Delay Differential Equations: Recent Advances and New Directions*, Springer, New York, NY, USA, 2009. View at MathSciNet - R. Bellman and K. L. Cooke,
*Differential-Difference Equations*, Academic Press, New York, NY, USA, 1963. View at MathSciNet - I. R. Epstein and J. A. Pojman,
*An Introduction to Nonlinear Chemical Dynamics*, Oxford University Press, Oxford, UK, 1998. - J. Hale,
*Theory of Functional Differential Equations*, Springer, New York, NY, USA, 2nd edition, 1977. View at MathSciNet - Y. Kuang,
*Delay Differential Equations with Applications in Population Dynamics*, vol. 191 of*Mathematics in Science and Engineering*, Academic Press, Boston, Mass, USA, 1993. View at MathSciNet - B. D. Hassard, N. D. Kazarinoff, and Y. H. Wan,
*Theory and Applications of Hopf Bifurcation*, vol. 41 of*London Mathematical Society Lecture Note Series*, Cambridge University Press, Cambridge, Mass, USA, 1981. View at MathSciNet - J. D. Murray,
*Mathematical Biology. I*, vol. 17 of*Interdisciplinary Applied Mathematics*, Springer, New York, NY, USA, 3rd edition, 2002. View at MathSciNet - H. Akkocaoğlu, H. Merdan, and C. Çelik, “Hopf bifurcation analysis of a general non-linear differential equation with delay,”
*Journal of Computational and Applied Mathematics*, vol. 237, no. 1, pp. 565–575, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Bélair and S. A. Campbell, “Stability and bifurcations of equilibria in a multiple-delayed differential equation,”
*SIAM Journal on Applied Mathematics*, vol. 54, no. 5, pp. 1402–1424, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. L. Cooke and Z. Grossman, “Discrete delay, distributed delay and stability switches,”
*Journal of Mathematical Analysis and Applications*, vol. 86, no. 2, pp. 592–627, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Çelik, “Hopf bifurcation of a ratio-dependent predator-prey system with time delay,”
*Chaos, Solitons & Fractals*, vol. 42, no. 3, pp. 1474–1484, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. Li and J. Wei, “Stability and Hopf bifurcation analysis of a prey-predator system with two delays,”
*Chaos, Solitons & Fractals*, vol. 42, no. 5, pp. 2606–2613, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Li, S. Ruan, and J. Wei, “Stability and bifurcation in delay-differential equations with two delays,”
*Journal of Mathematical Analysis and Applications*, vol. 236, no. 2, pp. 254–280, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Ruan and J. Wei, “On the zeros of transcendental functions with applications to stability of delay differential equations with two delays,”
*Dynamics of Continuous, Discrete & Impulsive Systems. A. Mathematical Analysis*, vol. 10, no. 6, pp. 863–874, 2003. View at Zentralblatt MATH · View at MathSciNet - X.-P. Yan and Y.-D. Chu, “Stability and bifurcation analysis for a delayed Lotka-Volterra predator-prey system,”
*Journal of Computational and Applied Mathematics*, vol. 196, no. 1, pp. 198–210, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Wei and S. Ruan, “Stability and bifurcation in a neural network model with two delays,”
*Physica D*, vol. 130, no. 3-4, pp. 255–272, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Xiao, W. X. Zheng, and J. Cao, “Hopf bifurcation of an (n+1)-neuron BAM neural network model with delays,”
*IEEE Transactions on Neural Networks and Learning Systems*, vol. 24, pp. 118–132, 2013. - J. Cao and M. Xiao, “Stability and Hopf bifurcation in a simplified BAM neural network with two time delays,”
*IEEE Transactions on Neural Networks*, vol. 18, no. 2, pp. 416–430, 2007. View at Publisher · View at Google Scholar · View at Scopus - W. He and J. Cao, “Stability and bifurcation of a class of discrete-time neural networks,”
*Applied Mathematical Modelling*, vol. 31, no. 10, pp. 2111–2122, 2007. View at Publisher · View at Google Scholar · View at Scopus - J. Cao and Y. Wan, “Matrix measure strategies for stability and synchronization of inertial BAM neural network with time delays,”
*Neural Networks*, vol. 53, pp. 165–172, 2014. View at Publisher · View at Google Scholar - X. Yang, J. Cao, and Z. Yang, “Synchronization of coupled reaction-diffusion neural networks with time-varying delays via pinning-impulsive controller,”
*SIAM Journal on Control and Optimization*, vol. 51, no. 5, pp. 3486–3510, 2013. - C. Çelik and H. Merdan, “Hopf bifurcation analysis of a system of coupled delayed-differential equations,”
*Applied Mathematics and Computation*, vol. 219, no. 12, pp. 6605–6617, 2013. View at Publisher · View at Google Scholar · View at MathSciNet