Feedback Control of a Chaotic Finance System with Two Delays

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’ τ1,τ2 plane. Finally, we give some numerical simulations by Matlab Microsoft to show the validity of theoretical analyses.


Introduction
In the past few decades, many scholars produced the increasing interest in nonlinear dynamic economic methods [1][2][3][4][5][6][7][8][9][10][11]. In the fields of finance, because of the influence of nonlinear factors, all sorts of economy problems become more and more complicated. e misalignment of certain parameters in the economic system can lead to runaway markets and possibly even a financial crisis [12][13][14][15]. erefore, 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][17][18][19][20][21][22][23][24][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][32][33], linear feedback control [34], delay feedback approach [35][36][37][38][39][40][41][42][43][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). en, 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: x(t) denotes the interest rate, y(t) expresses the investment demand, and z(t) represents the price index: where the parameters a, b, and c represent the saving amount, the investment cost, and the elasticity of market demand, respectively, and a, b, and c are positive constants. From [48], it is known that, under the parameter values a � 0.9, b � 0.2, and c � 1.2, system (1) exists a strange attractor, as shown in Figure 1.
In this paper, our object is to control the strange attractor by using the DDFC method and study the following system: where k 1 ∈ R and k 2 ∈ R are the feedback strengths and τ 1 and τ 2 are nonnegative delays. e initial conditions of system (2) are given as e 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.
is 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 (τ 1 , τ 2 ) plane to overcome the problem that no information is given on the plane (τ 1 , τ 2 ) 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.

Stability of Equilibrium and Hopf Bifurcation
Firstly, it gives the existence of equilibria.
In the following text, it always assumes that c(1 − ab) − b > 0 is satisfied and only considers the stability of E * + and the other equilibria can be analyzed similarly. Let where Now, we use the method in [49,50] to study the root distribution of (5). When τ 1 � τ 2 � 0, (5) becomes By Routh-Hurwitz criterion, all roots of (7) have negative real parts if and only if (H1)a 2 + k 1 + k 2 > 0, holds.

2.
1. e Case τ 1 > 0 and τ 2 � 0. In this part, let τ 2 � 0, and choose τ 1 as the parameter to study the distribution of the 2 Complexity root of (5). Let iω be the root of (5), then ω must satisfy the following equations: Adding the squares of both sides of (9), it yields to where Furthermore, from (9), it can be obtained that where Let z � ω 2 , then (10) becomes Applying the results in [49], the following conclusions hold.
(iii) If all conditions in (ii) and h ′ (z k ) ≠ 0 hold, then system (2) undergoes Hopf bifurcations at E * + when From the abovementioned discussion, one can know that the stable switch may exist as τ 1 varies for system (2) with τ 2 � 0. Define I as stable interval of τ 1 .

Remark 1.
Obviously, there exists a Hopf bifurcation at τ 0 2 when τ 1 is fixed in the stable interval I. However, if we choose τ 1 in the unstable interval, then there may be no τ * 2 such that when system (2) is unstable in τ 2 ∈ [0, τ * 2 ), it is stable in τ 2 > τ * 2 . e result will be discussed in the latter section by using the stability crossing curve method in [52].
Remark 2. For some τ 1 and τ 2 , if (5) has two pairs of purely imaginary roots ± iw 1 and ± iw 2 , all the other roots have negatively real parts. Let w 1 : w 2 � l 1 : l 2 ; then, system (2) undergoes a double Hopf bifurcation (DHB) with the ratio l 1 : l 2 . If l 1 , l 2 ∈ Z + , then it is called a resonant DHB; otherwise, it is called a nonresonant DHB. Since in system (2) there are several parameters besides τ 1 and τ 2 , the co-dimension 2 bifurcation may occur. An interesting study can be found in [53].

+
In Section 3, we have obtained some sufficient conditions to guarantee that the Hopf bifurcation occurs in system (2) at E * + when τ 2 � τ 0 2 . In this section, we assume that eorem 4 (ii) is satisfied to establish the explicit formula for the property of Hopf bifurcation at τ 2 � τ 0 2 using the method proposed by Hassard et al. [54].
Substituting E 1 and E 2 into W 20 (Θ) and W 11 (Θ), respectively, furthermore, g 21 can be computed. us, it can obtain the following quantities: Hence, we have the following result.

Crossing Curve Method
e results in eorem 4 clearly show that the stability of system (2) changes depending on the parameters of system. However, the (τ 1 , τ 2 ) 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 (τ 1 , τ 2 ) 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 λ: The polynomials p 0 (λ), p 1 (λ) and p 2 (λ) do not have any common zeros e 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 (τ 1 , τ 2 ) continuously vary within R 2 + , the number of zeros (counting multiplicity) of Δ(λ, τ 1 , τ 2 ) on C + can change only if a zero appears on or across the imaginary axis.
The characteristic equation (5) has the same zeros with the zeros of where δ s (λ) � p s (λ)/p 0 (λ), s � 1, 2. erefore, in general, we may obtain all the crossing points and directions of crossing from the solutions of Δ(λ, τ 1 , τ 2 ) � 0 instead of ∇(λ, τ 1 , τ 2 ) � 0. 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 Ω of ω 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 Ω can be represented as The second step is to determine the inner angles θ 1 , θ 2 ∈ [0, π] of the triangle in Figure 2. From the cosine law, it has For any ω ∈ Ω, one can obtain (τ 1 , τ 2 ) from (54) as follows: and where u ± 0 and b ± 0 are the smallest integers so that the right sides of (58) and (59) are nonnegative. Let then which is the set of all (τ 1 , τ 2 ) such that Δ(λ, τ 1 τ 2 ) has a zero at λ � iω. T � T ω : ω ∈ Ω identifies the stability crossing curves in (τ 1 , τ 2 ) plane, and the crossing set Ω is composed by a finite number of intervals with finite length. Let these intervals be Ω k , k � 1, 2, . . . , N, arranged in such an order that the left endpoint of Ω k increases with increasing k. en, Ω � ∪ N k�1 Ω k , and the left endpoints of the intervals ω l k and the right endpoints ω r k must only satisfy one of the three equations: L 1 (ω) � 1 and L 2 (ω) � ± 1. Let Then, T � ∪ N k�1 T k . Hence, we can divide these endpoints into three types according to the conditions satisfied by the equation ω l k or ω r k . If ω l 1 � 0, then Ω 1 may have a special type. As stated by [52], the possible shapes of T k must belong to one of the following three types:  If the left endpoint of Ω k is of Type l and its right endpoint is of Type r, we call an interval Ω k is of Type lr.
ere are a total of 12 possible types, where 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.
The direction in which the ω increases is called the positive direction of the curve, which is reversed when the curve passes the point corresponding to the Ω k 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. e following results come from [52].

Lemma 5.
Let ω ∈ (ω l k , ω r k ) and (τ 1 , τ 2 ) ∈ T k so that iω is a simple root of (5) and for any ω ′ ≠ ω, Δ(iω ′ , τ 1 , τ 2 ) ≠ 0. en, as (τ 1 , τ 2 ) moves from the right-side region to the left-side region of the corresponding curve in T k , a pair of roots of (54) cross the imaginary axis to the right side if R 2 I 1 − R 1 I 2 > 0. If the inequality is reversed, the crossing direction is opposite. Theorem 6. Let ω, τ 1 , and τ 2 satisfy the conditions in Lemma 5. en, when (τ 1 , τ 2 ) crosses the curve along the direction (ℓ 1 , ℓ 2 ), 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.

Numerical Simulations
In this part, we will carry out some numerical simulations by using Matlab Microsoft to confirm the theoretical analyses.
In addition, for the following two systems: and that 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). e time-delay feedback controller ke − dτ [u(t) − u(t − τ)] 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 ke − dτ , and the function decreases   exponentially with delay τ. is 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). e systems with coefficient dependent delay increase the complexity of the analysis and are challenging, especially those with two time delays. e research is set aside for future consideration.

Conclusion
is paper analyzes a class of chaotic financial systems with two feedback delays. System (1) exists in chaos under some parameters. e 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. e 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 τ 1 changes with τ 2 � 0, system (2) exists stable switches and chaos may disappear. Furthermore, fixing τ 1 in a stability interval and taking the delay τ 2 as a parameter, proves the existence of the first critical value τ 2 . At this critical value, the equilibrium loses stability and Hopf bifurcation occurs. e 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. e abovementioned results are obtained under the condition fixed τ 1 in a stability interval; however, if we choose the τ 1 in the unstable interval, then there may exist no the critical value τ 0 2 such that τ 0 2 is the first Hopf bifurcation value. Hence, for obtaining the complete result separating the stable and unstable regions in the (τ 1 , τ 2 ), using the stability crossing curve methods in [52], it obtains the curve sets in which Hopf bifurcation occurs in (τ 1 , τ 2 ) plane for fixed a, b, and c. By numerical simulations, it can find that the different shape crossing curves, and crossing sets can produce by changing k 1 and k 2 . eoretical 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 τ 1 � τ 2 � 0. 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. e DDFC method can control the chaotic behavior of the system more effectively than the DFC method. When τ 1 cannot change the chaos behavior of system (2), system (2) can be stabilized by varying τ 2 value. ese 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
e authors declare that there are no conflicts of interest regarding the publication of this paper.