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

The local reaction-diffusion Lengyel-Epstein systemwith delay is investigated. By choosing τ 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.


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 V denote the chemical concentrations of the activator iodine (I − ) and the inhibitor chlorite (ClO 2 − ), 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.

2
Discrete Dynamics in Nature and Society 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 V 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.

Stability Analysis and Hopf Bifurcation
It is easy to see that system (3) has a unique positive equilibrium * ( * , V * ) with * = , V * = 1 + 2 where = /5. Let = − * , = V − V * , and system (3) can be written aṡ= where and h.o.t denotes the higher order terms. Then, we obtain the linearized systeṁ The corresponding characteristic equation is where For (7), we have the following Lemma.
Summarizing the above and combining Lemma 1, we have the following result on the distribution of roots of (7).
Theorem 5. For system (3), assume that the condition (H) holds; then the following statements are true.

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 = 0 is the Hopf bifurcation value of system (3). Let = − * , = V − V * , = and omit " " above ; then system (3) can be rewritten aṡ( where ( ) = ( ( ) , ( )) , respectively. Let ( ) = ( ), ( ) = ( ) and omit " " above and ; the nonlinear terms 1 and 2 are 1 = Define a family of operators as By the Riesz representation theorem, there exists a matrix whose components are bounded variation functions ( , ) : In fact, choosing where is a Dirac delta function; then (17) is satisfied.
Since ± are eigenvalues of , they will also be the eigenvalues of * . The eigenvectors of and * are calculated corresponding to the eigenvalues + and − .
(ii) 2 determines the stability of bifurcating periodic solution; the periodic solution is stable (unstable) if 6 Discrete Dynamics in Nature and Society (iii) 2 denotes the period of bifurcating period solutions; if 2 > 0 ( 2 < 0), the period increases (decreases).

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 = 3, = 2, and = 0.8. It is easy to show that system (32) has unique positive equilibrium * (0.6, 1.36), + 5 − 3 2 = 4.88 > 0 and 2 − = −0.3106 < 0. From the discussion of Section 2, we have 0 = √ = 1.8787; by calculation, we obtain (3) 0 = 0.6758. We can see from Figure 1(a) that * is asymptotically stable at = 0.48 < (3) 0 = 0.6758, while * loses stability and Hopf bifurcation occurs when > (3) 0 ; see Figure 1(b) at = 0.79 > (3) 0 . Using the algorithm derived in Section 3, we obtain that 2 = 2.134, 2 = −0.857, and 2 = 2.128, 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 = 3.6 > (3) 0 . In Figure 1(d), largest Lyapunov exponent diagram is plotted for variable . It is easy to know when > 3.5; the Lyapunov exponent is almost positive; then the chaotic solutions occur.

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.