Abstract
A controlled model for a financial system through washout-filter-aided dynamical feedback control laws is developed, the problem of anticontrol of Hopf bifurcation from the steady state is studied, and the existence, stability, and direction of bifurcated periodic solutions are discussed in detail. The obtained results show that the delay on price index has great influences on the financial system, which can be applied to suppress or avoid the chaos phenomenon appearing in the financial system.
1. Introduction
For the last two decades, there have been growing interests in studying the complex dynamics of financial systems in both micro- and macroeconomics [1, 2]. It is well known that the economic activity is a complex human behavior; it has many uncertainties, which is reflected in the nonlinear model for economic dynamics such as Goodwin’s nonlinear accelerator model [3], forced van der Pol model on business cycle [4], the dynamic IS-LM model [5], and nonlinear dynamical model on finance system [6–9]. In these models, chaotic phenomena are common. However, in economic activities, chaos is undesired sometimes, so we want to control the chaotic orbits to a stable state or a periodic orbit. For example, in [9, 10], the authors showed that the chaotic behavior of a microeconomic model can be stabilized to various periodic orbits by means of time-delayed feedback control.
On the other hand, delays are ubiquitous in life, so it is in the social and economic activities. There are at least two ways that time delays emerge in the dynamics of economic variables. One is the time lag between the time economic decisions are made and the time the decisions bear fruit [11]. The other is the behavior of economic agents known as rational expectations [12]. So, it is very meaningful to investigate the effects of time delay on economic activity [10].
The aim of this paper is to investigate the dynamics of a financial system by considering the effect of washout filters with time delay. By analyzing the characteristic equation of linearization of the system, we theoretically prove that the Hopf bifurcations occur in the model with delay. Furthermore, by using the theory of functional differential equation and Hassard’s method [13], we also give the conditions to determine the direction and stability of the bifurcating periodic solutions. Finally, numerical results are given to support the theoretical prediction.
The rest of the paper is organized as follows. In Section 2, we propose the controlled finance system through washout filters with delay. In Section 3, we study the local stability and Hopf bifurcation of the equilibria. In Section 4, using the normal form theory and the center manifold reduction, explicit formulae are derived to determine the direction of bifurcation and the stability and other properties of bifurcating periodic solutions. In Section 5, we will give some numerical simulations to support the theoretical prediction. In Section 6, a brief discussion is given.
2. The Model
In [6, 7], the authors have reported a dynamical model of financial system composed of four subblocks: production, money, stock, and labor force. By setting proper dimensions and choosing appropriate coordinates, the authors have offered the simplified financial model which describes the time variation of three variables: the interest rate , the investment demand , and the price index . The model is represented by three-dimensional ODEs: where is the saving amount, is the cost per investment, and is the elasticity of demand of commercial market. This model is well studied in [6–9]; their results show that system (1) has abundant dynamical behaviors including Hopf bifurcation and chaos; however, the effect of time delay on the dynamics of this financial system was not taken into account.
In the following, we consider the effect of washout filters with time delay. We first consider a general form of dynamical system: where is a vector and is a parameter. The washout-filter-aided controller assumes the following structure: where is a control input, is a control function, and is the washout filter time constant. The following constraints should be fulfilled: , which guarantees the stability of the washout filter; , which preserves the original equilibrium points.
In this paper, the controlled system is designed as follows: where is a control gain, is an accommodation coefficient, and is time delay. is the control input; differing from the time-delayed feedback controller (DFC) [10], the changing rate of controller is influenced by the time delay feedback on price index and adjusted by . This system has the similar character with washout filter controller, , which guarantees the stability of the controller, and the original equilibrium points were preserved [14, 15].
3. Existence of Hopf Bifurcation
In this section, we choose the gain as a constant and investigate the effect of time delay on the dynamic behavior of the controlled system (4). First, the following conclusions for the uncontrolled system (1) are needed.
Lemma 1. When , that is, , system (1) has a unique equilibrium .
Lemma 2. When , that is, , system (1) has three equilibria and .
The characteristic equation of the Jacobian matrix at the equilibria of system (1) is where , , and . Then, from Routh-Hurwitz criterion, the real parts of all the roots of the above equation are negative if and only if the conditions(H1), , and hold.
Lemma 3. (i)The equilibrium of system (1) is stable when and .(ii)The equilibria of system (1) are stable when and (H1) hold.
3.1. Hopf Bifurcation from the Stable Equilibrium
The linear equation of the controlled system (4) at (where ) is The associated characteristic equation of the linearized system is That is, where
It is well known that the equilibrium is stable if all the roots of (8) have negative real parts. Obviously, (8) always has a negative root , for all , so, we only need to investigate the third transcendental polynomial equation: Obviously, if is a pair of pure imaginary roots of (8), then satisfies Separating the real and imaginary parts, we have and it follows that where Let , and (13) becomes Denote . We have the following.
Lemma 4. For (15), one has the following results: (i)if , then (15) has at least one positive root;(ii)if and , then (15) has no positive root;(iii)if and , then (15) has positive root if and only if and .
Without loss of generality, suppose that are positive roots of (15). Then, is a root of (13). From (12), we have Denote Substituting into (10) and taking the derivative with respect to , we have From (12) and (18), through tedious computing, we get where . Since , then and have the same sign. Thus, from Lemmas 3 and 4, we have the following theorem.
Theorem 5. Suppose and ; then, one has the following: (i)if and , then the equilibrium is stable for all ;(ii)if (or and ) and , then, when , the equilibrium is stable, and system (4) undergoes Hopf bifurcation at when passes through .
3.2. Hopf Bifurcation from the Stable Equilibria
In this subsection, we assume that system (1) has two stable equilibria . Due to the symmetry of and , it is sufficient to analyze the stability of .
Let . By the linear transforms , , , and , the linear equation of the controlled system (4) at is The associated characteristic equation of system (20) is Expand (21), and we have where
Suppose is a root of (22); then, satisfies which lead to where
Denote ; then, (24) becomes Denote Clearly, if , then (27) has at least one positive root. Suppose is a positive root of (27); then, is a root of (25). From (24), we have Substituting into (22) and taking the derivative with respect to , we obtain From (24) and (30), we have where . Thus, from the above analysis, we have the following.
Theorem 6. Suppose and ; then, system (4) undergoes Hopf bifurcation at the steady state when passes through .
4. Direction and Stability of the Hopf Bifurcation
In Section 3, we obtain the conditions under which a family of periodic solutions bifurcate from the steady state at the critical value of . In this section, following the ideal of [13], we derive the explicit formulae for determining the properties of the Hopf bifurcation at the critical value of using the normal form and the center manifold theory.
In this section, we always assume that system (4) undergoes Hopf bifurcation at the steady state for , and then is the corresponding purely imaginary roots of the characteristic equation at the steady state .
Let , , , , , and and drop the bars for simplification of notations. Then, system (4) can be rewritten as a functional differential equation in : where . For ,
Obviously, is a continuous linear function mapping into . By the Riesz representation theorem, there exists a matrix function , whose elements are of bounded variation such that In fact, we can choose where denote Dirac-delta function. For , define Then, when , the system is equivalent to the system (32), where and . For , define and a bilinear inner product where ; let ; then, and are adjoint operators. By the discussion in Section 3, we know that are eigenvalues of . Thus, they are also eigenvalues of . We first need to compute the eigenvector of and corresponding to and , respectively.
Suppose that is the eigenvector of corresponding to . Then, . It follows from the definition of , , and that Thus, we can easily obtain , , , and .
Similarly, let be the eigenvector of corresponding to . By the definition of , we can compute , , and .
In order to assure that , we need to determine the value of . From (39), we have Thus, we can choose such that , .
In the following, we first compute the coordinates to describe the center manifold at . 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 , since , we have where From (43) and (44), we have In addition, ; then,
By the definition of , we have Substituting , , , and into the above equation and comparing the coefficients with (46), we get
In order to assure the value of , we need to compute and . From (37) and (43), we have where Notice that, near the origin on the center manifold , we have thus, we have Since (51), for , we have Comparing the coefficients with (51) gives that From (54), (56), and the definition of , we can get Notice that , and we have where is a constant vector. In the same way, we can also obtain where is also a constant vector.
In what follows, we will compute and . From the definition of and (54), we have
From (51) and (52), we have Substituting (58) and (62) into (60) and noticing that we obtain That is,
Similarly, substituting (59) and (63) into (61), we can get the formula of , where Thus, we can determine and . Furthermore, we can determine each . Therefore, each is determined by the parameters and delay in (4). 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 (<0), then the Hopf bifurcation is supercritical (subcritical) and the bifurcation exists for (<); determines the stability of the bifurcation periodic solutions; the bifurcating periodic solutions are stable (unstable) if (>0); and determines the period of the bifurcating periodic solutions: the period increases (decreases) if (<0).
5. Numerical Simulation
In this section, we present some numerical results to verify the analytical predictions obtained in the previous section. These numerical simulation results constitute excellent validations of our theoretical analysis; it is shown that the chaotic orbit can be controlled to a periodic orbit by using washout-filter-aided controller with time delay.
Figure 1 shows that system (1) is chaotic when , , and .
5.1. Hopf Bifurcation from the Stable Equilibrium
In this subsection, we choose , , and ; then, system (1) has only a stable equilibrium . From the algorithm of Section 3, we get that ; thus, if , then . From Lemma 4, (15) has at least one positive root. Let and . From the algorithm of Section 3, we can compute . Thus, from Theorem 5, the equilibrium is asymptotically stable when , and, as crosses , there are periodic orbits bifurcating from (Figure 2).
5.2. Hopf Bifurcation from the Stable Equilibria
In this subsection, we choose , , and ; then, from Lemma 3, system (1) has three equilibria: is unstable and are stable (Figure 3). If we choose , and , from the algorithm of Section 3, we can get that the bifurcating value of is . When pass through , a family of periodic orbits will bifurcate from equilibria , respectively (Figure 4).
5.3. Application to Control of Chaos
From Figure 1, we can see that system (1) is chaotic when , , and . If we choose , a family of periodic orbits bifurcate from the equilibria of system (4) at some critical values of . This can be verified by Figure 5.
6. Conclusion
In this paper, we have investigated a financial system with time-delayed washout-filters-aided controller. Taking the time delay as bifurcating parameter, we discussed the conditions at which periodic orbits bifurcate from the equilibria and , respectively. The stability and direction of bifurcated periodic solutions have been also investigated in detail. And the obtained results can be applied to control the chaos of this financial system.
From a financial sense, the obtained results show that the delay on price index has great influence on the financial system, which can be applied to suppress or avoid the chaos phenomenon appearing in the financial system, so as to make the economic system run well. On the other hand, the control gain is also applied to influence the dynamical behaviors of this financial system; it will be investigated in the near future.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors would like to thank the anonymous referee for the very helpful suggestions and comments which led to the improvement of the original paper. And this work is supported by Science and Technology Department of Henan Province (122300410417), Education Department of Henan Province (13A110108), 2013 Scientific Research Project of Beifang University of Nationalities (2013XYZ021), Institute of Information and System Computation Science of Beifang University (13xyb01).