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