Journal of Applied MathematicsVolume 2012 (2012), Article ID 316390, 18 pagesdoi:10.1155/2012/316390

Research Article

Bifurcation Analysis for a Kind of Nonlinear Finance System with Delayed Feedback and Its Application to Control of Chaos

Rongyan Zhang

Department of Electronic Information Engineering, Huanghe Science and Technology College, Henan, Zhengzhou 450063, China

Received 14 October 2011; Revised 15 January 2012; Accepted 23 January 2012

Academic Editor: Chuanhou Gao

Copyright © 2012 Rongyan Zhang. 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.

A kind of nonlinear finance system with time-delayed feedback is considered. Firstly, by employing the polynomial theorem to analyze the distribution of the roots to the associate characteristic equation, the conditions of ensuring the existence of Hopf bifurcation are given. Secondly, by using the normal form theory and center manifold argument, we derive the explicit formulas determining the stability, direction, and other properties of bifurcating periodic solutions. Finally, we give several numerical simulations, which indicate that when the delay passes through certain critical values, chaotic oscillation is converted into a stable steady state or a stable periodic orbit.

1. Introduction

Since the chaotic phenomenon in economics was first found in 1985, great impact has been imposed on the prominent western economics at present, because the chaotic phenomenon occurring in the economic system means that the macroeconomic operation has in itself the inherent indefiniteness. Although the government can adopt such macrocontrol measures as the financial policies or the monetary policies to interfere, the effectiveness of the interference is very limited. The instability and complexity make the precise economic prediction greatly limited, and the reasonable prediction behavior has become complicated as well. In the fields of finance, stocks, and social economics, because of the interaction between nonlinear factors, with all kinds of economic problems being more and more complicated and with the evolution process from low dimensions to high dimensions, the diversity and complexity have manifested themselves in the internal structure of the system and there exists extremely complicated phenomenon and external characteristics in such a kind of system. So it has become more and more important to study the control of the complicated continuous economic system and stabilize the instable periodic or stationary solutions, in order to make the precise economic prediction possible [1, 2].

Recent works [1, 2] have reported a dynamic model of finance, composed of three first-order differential equations. The model describes the time variations of three state variables: the interest rate , the investment demand , and the price index . By choosing an appropriate coordinate system and setting appropriate dimensions for each state variable, [1, 2] offer the simplified finance system as which is chaotic when (see Figure 1).

Figure 1: Strange attractor of finance system (1.1).

Over the last years, [3, 4] studied impulsive control and state feedback control of the finance system (1.1). In this paper, we are interesting in delayed feedback control of the finance system (1.1). The effects of the time-delayed feedback on the finance system have long been investigated [5–8].

Recently, different techniques and methods have been proposed to achieve chaos control. The existing control methods can be classified, mainly, into two categories. The first one, developed by Ott et al. [9] is based on the invariant manifold structure of unstable orbits. It is theoretically well understood but difficult to apply to fast experimental systems. The second, proposed by Pyragas [10], uses time-delayed controlling forces. In contrast to the former one, it is simple and convenient method of controlling chaos in continuous dynamical system. Thus, we adopt the second one in the present paper.

For predigesting the investigation, here we only put time delay on investment demand . By adding a time-delayed force to the second equation of finance system (1.1), we obtain the following new system Here we assume that and . The time delay is taken as the bifurcation parameter and we show that when passes through some certain critical values, the equilibrium will lose its stability and hopf bifurcation will take place; by adjusting values, we achieve the purpose of chaos control. The research of this paper is a new investigation about the hopf bifurcation and chaos control on the finance system and has important theoretical and practical value.

2. Stability of Steady States and Bifurcations of Periodic Solutions

In this section, we investigate the effect of delay on the dynamic behavior of system (1.2). Obviously, when , system (1.2) becomes the system (1.1). First, we introduce the following several lemmas in [1, 2] for T’s system(1.1).

We know that under the assumption , the system (1.1) has two equilibrium points:

The characteristic equation of the system (1.1) at is

By analyzing the characteristic equation (2.2) and the Routh-Hurwitz criteria, we get the following.

Lemma 2.1. For , the characteristic equation (2.2) has three eigenvalues with negative real parts, so two equilibrium points of the system (1.1) are asymptotic stable.

Lemma 2.2. For , the characteristic equation (2.2) has a pair of purely imaginary eigenvalues and a negative real eigenvalue , and According to the hopf bifurcation theorem [11], a hopf bifurcation of the system (1.1) occurs at .

Lemma 2.3. For , the characteristic equation (2.2) has one negative real root and one pair of conjugate complex roots with positive real parts, so two equilibrium points of the system (1.1) are unstable.

Clearly, the delayed feedback control system (1.2) has the same equilibria to the corresponding system (1.1). In this section, we analyze the effect of delay on the stability of these steady states. Due to the symmetry of and , it is sufficient to analyze the stability of . By the linear transform system (1.2) becomes It is easy to see that the origin is the equilibrium of system (2.5). The associated characteristic equation of system (2.5) at is Expanding (2.6), we have Thus, we need to study the distribution of the roots of the third-degree exponential polynomial equation: where and . We first introduce the following simple result which was proved by Ruan and Wei [12] using Rouche’s theorem.

Lemma 2.4. 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 on or crosses the imaginary axis.

Obviously, is a root of (2.8) if and only if satisfies Separating the real and imaginary parts, we have which is equivalent to Let and denote ,  , , then (2.12) becomes

In the following, we need to seek conditions under which (2.12) has at least one positive root. Denote

Therefor, applying [13], we obtain the following lemma.

Lemma 2.5. For the polynomial equation (2.13), one has the following results. (i)If , then (2.13) has at least one positive root.(ii)If and , then (2.13) has no positive roots.(iii)If and , then (2.13) has positive roots if and only if and .

Suppose that (2.13) has positive roots. Without loss of generality, we assume that it has three positive roots, defined by , and , respectively. Then (2.12) has three positive roots: From (2.11), we have Thus, if we denote where , then is a pair of purely imaginary roots of (2.8) with . Define Note that when , (2.8) becomes

Therefor, applying Lemmas 2.4 and 2.5 to (2.8), we get the following lemma.

Lemma 2.6. For (2.8), one has (i)if and , then all roots with positive real parts of (2.8) have the same sum to those of the polynomial equation (2.19) for all .(ii)if either or , , and , then all roots with positive real parts of (2.8) have the same sum to those of the polynomial equation (2.19) for .

Let be the root of (2.8) near satisfying Then by [13], we have the following transversality condition.

Lemma 2.7. Suppose that and . Then and and have the same sign.

Now, we study the characteristic equation (2.7) of the system (2.5). Comparing (2.7) with (2.8), we know that Thus, and then we can compute When , (2.7) becomes (2.2)

Applying Lemmas 2.1, 2.2, 2.6, and 2.7 to (2.7), we have the following theorems.

Theorem 2.8. Let and be defined by (2.17) and (2.18). Suppose that conditions and hold. (i)If , then (2.7) had all roots with negative real parts for all , and the equilibrium (or ) of the system (1.2) is stable.(ii)If and , (2.7) had all roots with negative real parts for , and the equilibrium (or ) of the system (1.2) is stable.(iii)If the conditions of (ii) are satisfied, and , then system (1.2) exhibits the Hopf bifurcation at the equilibrium (or ) for .

Theorem 2.9. Let and are defined by (2.17) and (2.18). Suppose that conditions and hold. (i)If , then (2.7) had two roots with positive real parts for all , and the equilibrium (or ) of the system (1.2) is unstable.(ii)If and , (2.7) has two roots with positive real parts for , and the equilibrium (or ) of the system (1.2) is unstable.(iii)If the conditions of (ii) are satisfied, and , then system (1.2) exhibits the Hopf bifurcation at the equilibrium (or ) for .

3. Direction and Stability of the Hopf Bifurcation

In the Section 2, we obtained some conditions which guarantee that the system (1.2) undergoes the Hopf bifurcation at a sequence values of . In this section, we shall study the direction and stability of the Hopf bifurcation. The method we used is based on the normal form theory and the center manifold theorem introduced by Hassard et al. [14]. Throughout this section, we always assume that system (1.2) undergoes Hopf bifurcations at the steady state for and then is corresponding purely imaginary roots of the characteristic equation at the steady state .

Letting and dropping the bars for simplification of notations, system (1.2) is transformed into an FDE in as where , and are given, respectively, by By the Riesz representation theorem, there exists a function of bounded variation for , such that for .

In fact, we can choose where is the Dirac delta function. For , define Then system (3.1) is equivalent to where for .

For , define and a bilinear inner product where . Then and are adjoins operators.

By the discussion in Section 2, we know that are eigenvalues of , thus they are also eigenvalues of .

By direct computation, we obtain that , with is the eigenvector of corresponding to , and , with is the eigenvector of corresponding to , where Using the same notation as in [14], we compute the coordinates to describe the center manifold at . Let be the solution of (3.1) when . Define On the center manifold , we have

where and are local coordinates for center manifold in the direction of and . Note that is real if is real. We consider only real solutions. For the solution of (3.1), since , we have We rewrite this equation as where Noticing we have Thus, form (3.17), we have

Comparing the coefficients of (3.17), we get Since there are and in , we need to compute them.

From (3.6) and (3.12), we have where Expanding the above series and comparing the corresponding coefficients, we obtain From (3.22), we know that for , Comparing the coefficients with (3.23) gives that From (3.24), (3.26) and the definition of , it follows that Notice that , hence where is a constant vector.

Similarly, from (3.24) and (3.27), we can obtain where is also a constant vector.

In what follows, we shall seek appropriate and. From the definition of and (3.24), we obtain where . By (3.22), we have Substituting (3.29) and (3.33) into (3.31), we obtain which leads to where

Similarly, substituting (3.30) and (3.34) into (3.32), we can get where Thus, we can determine and from (3.29) and (3.30). Furthermore, we can determine . Therefore, each in (3.21) is determined by the parameters and delay in (3.1). Thus, we can compute the following values: which determine the quantities of bifurcating periodic solutions in the center manifold at the critical value , that is, determines the directions of the Hopf bifurcation: if then the Hopf bifurcation is supercritical (subcritical) and the bifurcating periodic solutions exist for ; determines the stability of the bifurcating periodic solutions: the bifurcating periodic solutions are stable (unstable) if ; and determines the period of the bifurcating periodic solutions: the period increases (decreases) if .

4. Application to Control Chaos

In the present section, we apply the results in the previous sections to system (1.2) for the purpose of control of chaos. From Section 2, we know that under certain conditions, a family of periodic solutions bifurcate from the steady states of system (1.2) at some critical values of and the stability of the steady state maybe change along with increase of . If the bifurcating periodic solution is orbitally asymptotically stable or some steady state becomes local stable, then chaos may vanish. Following this ideal, we consider the following delayed feedback control system: which has two steady states . Clearly, when or , system (3.1) is chaotic (as depicted in Figure 1).

For the steady state or , we have the corresponding characteristic equation of system (4.1) as follows: Clearly, when , (4.2) has a negative root and a pair of complex roots with positive real parts. Following Section 2, we can obtain , , , , and for all . When or , . Thus, from Lemma 2.6 and Theorem 2.9, we know that (4.2) has roots with positive real parts. In particular, we have , that is, In this case, we can compute

Thus, from Lemma 2.7, we have and . In addition, notice that Thus, from Theorem 2.8, we have the following conclusion about the stability of the steady states of system (4.3) and Hopf bifurcation.

5. Conclusion

Suppose that is defined by (4.4).(i)When , the steady states and of the system (4.1) are unstable (see Figures 2 and 3).(ii)When , the steady states and of the system (4.1) are asymptotically (see Figures 4 and 5).(iii)When , system (4.1) undergoes a Hopf bifurcation at the steady states states and .

Figure 2: Chaos still exists for .

Figure 3: Chaos still exists for .

Figure 4: When , chaos vanishes, and becomes local stable. Here initial value is .

Figure 5: When , chaos vanishes, and becomes local stable. Here initial value is .

The above simulations indicate that when the steady state is stable or the bifurcating periodic solutions are orbitally asymptotically stable, chaos vanishes (see Figures 4–6).

Figure 6: When , chaos vanishes, and becomes a stable periodic solution. Here initial value is .

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