Feedback Control of a Chaotic Finance System with Two Delays

Jiang, Zhichao; Zhang, Tongqian

doi:https://doi.org/10.1155/2020/4937569

Complexity

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Abstract Introduction Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Complexity in Economics and Business

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Research Article | Open Access

Volume 2020 | Article ID 4937569 | https://doi.org/10.1155/2020/4937569

Feedback Control of a Chaotic Finance System with Two Delays

Zhichao Jiang¹and Tongqian Zhang²

Academic Editor: Abdelalim Elsadany

Received29 Feb 2020

Revised26 Apr 2020

Accepted04 May 2020

Published26 May 2020

Abstract

In this research, we use the double-delayed feedback control (DDFC) method in order to control chaos in a finance system. Taking delays as parameters, the dynamic behavior of the system is investigated. Firstly, we study the local stability of equilibrium and the existence of local Hopf bifurcations. It can find that the delays can make chaos disappear and generate a stable equilibrium or periodic solution, which means the effectiveness of DDFC method. By using the normal form theory and center manifold argument, one derives the explicit algorithm for determining the properties of bifurcation. In addition, we also apply some mathematical methods (stability crossing curves) to show the stability changes of the financial system in two parameters’ plane. Finally, we give some numerical simulations by Matlab Microsoft to show the validity of theoretical analyses.

1. Introduction

In the past few decades, many scholars produced the increasing interest in nonlinear dynamic economic methods [1–11]. In the fields of finance, because of the influence of nonlinear factors, all sorts of economy problems become more and more complicated. The misalignment of certain parameters in the economic system can lead to runaway markets and possibly even a financial crisis [12–15]. Therefore, it is more and more important to study the internal structure characteristics of a complex financial system and uncover its causes, so as to predict and control the system.

A lot of work has been carried out in modeling nonlinear economic dynamics, such as Goodwin’s model, van der Pol model, IS-LM model, and nonlinear finance system [14, 16–25]. However, it is well known that even a simple nonlinear system can exhibit chaotic behavior. Chaos is the inherent randomness of deterministic systems. Since the first discovery of chaos in economics from 1985, a great impact has been produced on the study of western economics because chaos in the economic system means the inherent uncertainty in macroeconomic operation. Over the past two decades, many efforts had been made to control chaos, such as stability and chaos synchronization, at unstable fixed points. In recent years, many methods had been put forward to control and synchronize chaos, such as OGY method [26], PC method [27], fuzzy control [28], impulsive control method [29, 30], stochastic control [31–33], linear feedback control [34], delay feedback approach [35–44], and multiple delay feedback control (MDFC) [45]. Delayed feedback control (DFC) was first proposed by Pyragas [46] in order to stabilize unstable periodic orbits (UPO). Then, the DFC method was extended to the multidelay [47]. One of the main characteristics of the DFC method is that it does not need the knowledge of the internal dynamics of the system beyond the period nor does it require a preliminary understanding of the required UPO. At the same time of UPO control, using the DFC method to realize USS stability had become an area of concern and had been applied to some real systems. It is very successful in stabilizing UPO for the DFC method, but the control of USS is less efficient. In [45], authors put forward the MDFC method and conducted numerical simulations, which showed that the MDFC method preceded the DFC method in USS stability.

In [16], authors put forward a financial system describing the temporal changes using three variables: denotes the interest rate, expresses the investment demand, and represents the price index:where the parameters represent the saving amount, the investment cost, and the elasticity of market demand, respectively, and are positive constants. From [48], it is known that, under the parameter values , and , system (1) exists a strange attractor, as shown in Figure 1.

(a)

(b)

In this paper, our object is to control the strange attractor by using the DDFC method and study the following system:where and are the feedback strengths and and are nonnegative delays.

The initial conditions of system (2) are given aswhere and .

The purpose of this paper is to analyze and numerically study system (2). Our results show that the stability of system varies with delays. When the delay passes a certain critical value, the chaotic oscillation disappears and can be transformed into stable equilibrium or periodic orbit, which indicates that the chaotic property changes with the changes of delays.

This article is organized as follows. In Section 2, by studying the distribution of eigenvalues of exponential polynomials and using the results in [49, 50], the local stability and existence of local Hopf bifurcation are obtained. In Section 3, the properties of Hopf bifurcation are given by using central manifold theory and normal form method. In Section 4, using the crossing curve methods, it can obtain the stable changes of equilibrium in plane to overcome the problem that no information is given on the plane that comes into being stable or unstable equilibrium in Section 2. To support the analysis results, some numerical simulations are carried out in Section 5. Finally, some conclusions and discussions are given.

2. Stability of Equilibrium and Hopf Bifurcation

Firstly, it gives the existence of equilibria.

Lemma 1. (i)If holds, then system (2) has only a boundary equilibrium (ii)If holds, then system (2) has two interior equilibria besides , where .In the following text, it always assumes that is satisfied and only considers the stability of and the other equilibria can be analyzed similarly.
Let , and , then system (2) becomeswhose characteristic equation iswhereNow, we use the method in [49, 50] to study the root distribution of (5). When , (5) becomesBy Routh–Hurwitz criterion, all roots of (7) have negative real parts if and only ifholds.

2.1. The Case and

In this part, let , and choose as the parameter to study the distribution of the root of (5). Let be the root of (5), then must satisfy the following equations:

Adding the squares of both sides of (9), it yields towhere

Furthermore, from (9), it can be obtained thatwhere and .

Let , then (10) becomes

Applying the results in [49], the following conclusions hold.

Lemma 2. (i)If and hold, then (13) has no positive root(ii)If holds, then (13) has at least a positive root(iii)If and hold, then (13) has a positive roots iff and , where and Without loss of generality, it supposes that (13) has three positive roots, denoted by , respectively. Then, (10) has three positive roots . Substituting into (9) giveswhere .
Define Let be the root of (5) with satisfying and .

Lemma 3. Suppose that , then and .

Proof. Let , and differentiate both sides of (5) about , and it hasHence,where . Since and , then we haveBy 3 and applying the Hopf bifurcation theorem in [51], for system (2) it has the following theorem.

Theorem 1. It assumes that holds:(i)If and hold, then, for all , is locally asymptotically stable (LAS)(ii)If either or and , , hold, then for , is LAS(iii)If all conditions in (ii) and hold, then system (2) undergoes Hopf bifurcations at when , We know that the condition (H1) guarantees that all roots of (7) have negative real parts. If (H1) is violated, we defineLet , then (7) becomeswhere and .
DefineThen, from Cardano’s formula, it has the following Theorem.

Theorem 2. (i)If , then (19) has three real roots(ii)If , then (19) has a real root and a pair of conjugate complex roots Furthermore, we assume that

Theorem 3. It assumes that holds. For system (2), it has the following results.(i)If and , then, for all , of system (2) is unstable.(ii)If either or and , , hold, then for , of system (2) is unstable. In addition, if , then is LAS when , where is the second critical value.(iii)If all conditions in (ii) and hold, then system (2) undergoes Hopf bifurcations at when , .From the abovementioned discussion, one can know that the stable switch may exist as varies for system (2) with . Define as stable interval of .

2.2. The Case and

In this part, let , , and be the root of (5), and it haswhich yields towhereand

Hence, we havewith and .

It can be easily known that equation (26) has at most positive roots, denoted by . From (22), we have

Denote

Let be the root of (5) satisfying and . By computation, we obtainwhere

To sum up, we have the following theorem.

Theorem 4. Suppose that either or is satisfied, and for system (2).(i)If (26) has no positive roots, then for all , is LAS.(ii)If (26) has positive roots, then of is LAS when . In addition, if , then system (2) undergoes Hopf bifurcation at when .

Remark 1. Obviously, there exists a Hopf bifurcation at when is fixed in the stable interval . However, if we choose in the unstable interval, then there may be no such that when system (2) is unstable in , it is stable in . The result will be discussed in the latter section by using the stability crossing curve method in [52].

Remark 2. For some and , if (5) has two pairs of purely imaginary roots and , all the other roots have negatively real parts. Let ; then, system (2) undergoes a double Hopf bifurcation (DHB) with the ratio . If , then it is called a resonant DHB; otherwise, it is called a nonresonant DHB. Since in system (2) there are several parameters besides and , the co-dimension 2 bifurcation may occur. An interesting study can be found in [53].

3. Property of Hopf Bifurcation at

In Section 3, we have obtained some sufficient conditions to guarantee that the Hopf bifurcation occurs in system (2) at when . In this section, we assume that Theorem 4 (ii) is satisfied to establish the explicit formula for the property of Hopf bifurcation at using the method proposed by Hassard et al. [54].

For convenience, we assume and the phase space . Let , and dropping “–”. Then, system (2) occurs Hopf bifurcation at . System (2) can be transformed into the following system:where , and are given, respectively, bywherewhere .

By the Riesz representation theorem, for , there exists a bounded variation function such that

In fact, one may choose

For , define

For , it has . Then, system (31) can be rewritten aswhere .

For and , defineand the inner productwhere . By direct computations, we obtain that is an eigenvector of corresponding to the eigenvalue , and is an eigenvector of corresponding to the eigenvalue . Furthermore, it has thatwhere

Let be the solution of system (31) when . Define ; then,where

Rewriting (42) aswhere

Furthermore,where

Notice that

Hence, we can obtain the following important quantities:wherewhere .

Substituting and into and , respectively, furthermore, can be computed. Thus, it can obtain the following quantities:

Hence, we have the following result.

Theorem 5. Hopf bifurcation is supercritical (subcritical) if . The bifurcation periodic solutions are orbitally stable (unstable) if . The period increase (decrease) if .

4. Crossing Curve Method

The results in Theorem 4 clearly show that the stability of system (2) changes depending on the parameters of system. However, the plane analysis results for bifurcation generation are not obtained by this method in Section 2. Gu et al. [52] gave an effective approach to separate the stable and unstable regions in the plane by using the stability crossing curves. In this part, we carry out the method. on the basis of equation (5), and it can define the following polynomials about :satisfying

The following discussions will follow the continuity of the zeros with respect to the delay parameters as stated in the following lemma [52].

Lemma 4. As the delays continuously vary within , the number of zeros (counting multiplicity) of on can change only if a zero appears on or across the imaginary axis.
The characteristic equation (5) has the same zeros with the zeros ofwhere , . Therefore, in general, we may obtain all the crossing points and directions of crossing from the solutions of instead of . Now, based on the procedure proposed by [52], the procedure is comprised of the following steps.
The first step is to determine the crossing set of that satisfies the feasibility condition so that the purely imaginary root exists, and geometrically, the vectors that satisfy (54) form a triangle (see Figure 2).
From Figure 2, the crossing set can be represented asThe second step is to determine the inner angles of the triangle in Figure 2. From the cosine law, it hasFor any , one can obtain from (54) as follows:andwhere and are the smallest integers so that the right sides of (58) and (59) are nonnegative.
Letthenwhich is the set of all such that has a zero at .
identifies the stability crossing curves in plane, and the crossing set is composed by a finite number of intervals with finite length. Let these intervals be , , arranged in such an order that the left endpoint of increases with increasing . Then, , and the left endpoints of the intervals and the right endpoints must only satisfy one of the three equations: and .
LetThen, .
Hence, we can divide these endpoints into three types according to the conditions satisfied by the equation or . If , then may have a special type. As stated by [52], the possible shapes of must belong to one of the following three types:(i)A series of closed curves.(ii)A series of spiral-like curves oriented along horizontally, vertically, or diagonally.(iii)A series of open-ended curves whose ends approach .If the left endpoint of is of Type and its right endpoint is of Type , we call an interval is of Type . There are a total of 12 possible types, where Type 1: is satisfied. links at the end. Type 2: is satisfied. links at the end. Type 3: is satisfied. links at the end. Type 0: . As , and approach .In 12 possible types, Type 11, Type 22, and Type 33 form a series of closed curves. Type 12 and Type 21, Type 13 and Type 31, and Type 23, and Type 32 form series spiral-like curves oriented along diagonally, vertically, and horizontally, respectively. Type 01, Type 02, and Type 03 form a series of open-ended curves.
Next, to determine the existence of Hopf bifurcation, we consider the direction of the root of (5) through the imaginary axis by the method given in [52]. By (54) and the implicit function theorem, and can be expressed as the function of . As moves along the imaginary axis, moves along . For a fixed , letwhere .
The direction in which the increases is called the positive direction of the curve, which is reversed when the curve passes the point corresponding to the endpoint. When we move in the positive direction of the curve, we also call the region on the left-hand side the region on the left. The following results come from [52].

Lemma 5. Let and so that is a simple root of (5) and for any , . Then, as moves from the right-side region to the left-side region of the corresponding curve in , a pair of roots of (54) cross the imaginary axis to the right side if . If the inequality is reversed, the crossing direction is opposite.

Theorem 6. Let satisfy the conditions in Lemma 5. Then, when crosses the curve along the direction , a pair of roots of (54) cross the imaginary axis to the right side ifIf the inequality is reversed, the crossing direction is opposite.

5. Numerical Simulations

In this part, we will carry out some numerical simulations by using Matlab Microsoft to confirm the theoretical analyses.

Firstly, as an example, we investigate the following system:and the initial functions are , and . With these parameters, condition (H2) holds. When , (10) has two positive roots and . Substituting them into (14) gives, respectively,

Furthermore, . By Theorem 4, is unstable when , and LAS when . The numerical simulation results are shown in Figures 3–5.

(a)

(b)

(a)

(b)

(a)

(b)

Fix , and it computes . By Theorem 4, we know that is LAS for . Choosing , is stable (see Figure 6). Furthermore, by Section 3, it has and when , and the bifurcating periodic solution is stable, which is illustrated in Figure 7.

(a)

(b)

(a)

(b)

Next, one gives some examples for the crossing curve method using Matlab Microsoft. Firstly, it still chooses the parameters in system (65). It can obtain the crossing set based on the equation of and . In equations (58) and (59), we regard and as the function of , by drawing the parametric equation curves in plane, and it obtains the crossing curves. The crossing set has only a interval and , satisfying with and (see Figure 8(a)). So, the interval is Type 33 and the crossing curves form a series of closed curves (see Figures 8(b) and 8(c)). Firstly, it chooses and , and it can obtain that is unstable (see Figure 9). In the following, it chooses, respectively, for fixed , and it can find that is stable when (see Figure 10) and (see Figure 11), unstable when , and there exists a stable periodic solution (see Figure 12). Furthermore, it can fix , and let change, and the crossing curves can produce different shapes (see Figures 13–15). When and , the crossing sets have two intervals and . Here, and satisfy while and satisfy (see Figure 13(a)). So, the interval belongs to Type 32 and the interval belongs to Type 23. The crossing curves are spatial-like curves, as shown in Figures 13(b) and 13(c). Choosing and , the crossing sets include two intervals and (see Figure 14(a)). The crossing curves belongs to Type 31 and Type 13 with the spiral-like shape (see Figure 14(b)). Here, the stability crossing curves in belonging to type 31 are not drawn. If and , the crossing set includes a interval . Since and (see Figure 15(a)), the stability crossing curves belongs to Type 03 with the open-ended shapes (see Figure 15(b)). These show that the changes of the feedback strengths and can alter the stable region of the system in plane, and it has an important effect on the stability of the financial system.

(a)

(b)

(c)

(a)

(b)

(a)

(b)

(a)

(b)

(a)

(b)

(a)

(b)

(c)

(a)

(b)

(a)

(b)

In addition, for the following two systems:andthat is, delayed feedback terms appear on the investment demand or the price index, respectively. Systems (67) and (68) can be investigated as system (2) and can also obtain similar results to system (2).

The time-delay feedback controller with delay correlation coefficients can also be designed to control system (1) which can modify the bifurcation characteristics of a nonlinear system to obtain some specific dynamical behaviors. Note that the strength of feedback control is in the form of , and the function decreases exponentially with delay . This means that the feedback effects of past states diminish over time. Hence, it can carry out the feedback with time-delay correlation coefficients in system (1). The systems with coefficient dependent delay increase the complexity of the analysis and are challenging, especially those with two time delays. The research is set aside for future consideration.

6. Conclusion

This paper analyzes a class of chaotic financial systems with two feedback delays. System (1) exists in chaos under some parameters. The purpose of this study is to control the chaos of the system. For controlling chaos, we improve the DFC method and introduce the double-delay feedback control method in system (1). We introduce the control term in the equation of the interest rate. The system may exist in three equilibria, and we choose one of these equilibria as the research target. It finds that the single delay feedback control can make the system stable and produce the stable switches, i.e., when changes with , system (2) exists stable switches and chaos may disappear. Furthermore, fixing in a stability interval and taking the delay as a parameter, proves the existence of the first critical value . At this critical value, the equilibrium loses stability and Hopf bifurcation occurs. The properties of Hopf bifurcation are also studied by using central manifold theory and normal form method for determining the direction of Hopf bifurcation and the stability of bifurcating periodic solution. The abovementioned results are obtained under the condition fixed in a stability interval; however, if we choose the in the unstable interval, then there may exist no the critical value such that is the first Hopf bifurcation value. Hence, for obtaining the complete result separating the stable and unstable regions in the , using the stability crossing curve methods in [52], it obtains the curve sets in which Hopf bifurcation occurs in plane for fixed . By numerical simulations, it can find that the different shape crossing curves, and crossing sets can produce by changing and . Theoretical analysis and numerical simulation results show that, for chaotic financial systems, chaotic oscillation can be controlled by delays. In other words, the multiple delay financial system we study has chaotic oscillations when . When the delay increases, the chaos disappears, the equilibrium point gains stability or the system appears periodic oscillation, and the periodic solution is generated by the Hopf bifurcation. The DDFC method can control the chaotic behavior of the system more effectively than the DFC method. When cannot change the chaos behavior of system (2), system (2) can be stabilized by varying value. These show that the effectiveness of the DDFC method.

Data Availability

Data sharing is not applicable to this article as all datasets are hypothetical during the current study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

Z. Jiang was supported by National Natural Science Foundation of China (11801014), Natural Science Foundation of Hebei Province (A2018409004), University Discipline Top Talent Selection and Training Program of Hebei Province (SLRC2019020) and Graduate Student Demonstration Course Construction of Hebei Province (KCJSX2020093), 2020 Talent Training Project Funding of Hebei Province. T. Zhang was supported by Shandong Provincial Natural Science Foundation from China (ZR2019MA003).

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Copyright © 2020 Zhichao Jiang and Tongqian 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.

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