Complexity
Volume 2019 (2019), Article ID 6715036, 18 pages
https://doi.org/10.1155/2019/6715036

Research Article

Hopf Bifurcation and Chaos of a Delayed Finance System

Xuebing Zhang¹ and Honglan Zhu²

¹College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, China
²Business School, Huaiyin Institute of Technology, Huaian, China

Correspondence should be addressed to Xuebing Zhang; zxb1030@163.com

Received 11 May 2018; Revised 6 August 2018; Accepted 5 September 2018; Published 8 January 2019

Academic Editor: Avimanyu Sahoo

Copyright © 2019 Xuebing Zhang and Honglan Zhu. 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.

In this paper, a finance system with delay is considered. By analyzing the corresponding characteristic equations, the local stability of equilibrium is established. The existence of Hopf bifurcations at the equilibrium is also discussed. Furthermore, formulas for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by applying the normal form method and center manifold theorem. Finally, numerical simulation results are presented to validate the theoretical analysis. Numerical simulation results show that delay can lead a stable system into a chaotic state.

1. Introduction

Ever since economist Stutzer first revealed the chaotic phenomena in an economic system in 1980, chaotic dynamics which supports an endogenous explanation of the complexity observed in economic series has become a hot topic, and many economic models have been proposed, e.g., Goodwin’s nonlinear accelerator model [1, 2], the van der Pol model on business cycle [3–5], the IS-LM model [6, 7], and nonlinear dynamical model on finance system [8–11]. In [8, 9], Ma and Chen proposed a simplified financial model as follows:

where is the interest rate, is the investment demand, is the price index, denotes saving amount, denotes cost per investment, and denotes elasticity of demand of commercial markets. The variation of is not only influenced by the surplus between investment and saving but also structurally adjusted by the price. The changing rate of is proportional to the rate of investment and inversely proportional to the cost of investment and interest rate. The variation of is influenced by the contradiction between supply and demand in commercial markets and affected by the inflation rates. The authors studied the focus on bifurcation and topological horseshoe of chaotic financial system (1). Some delay feedback control strategies [12–15] have also been considered for system (1).

It is well known that delays are extensively encountered in many fields such as biology [16–18], chemistry [19, 20], and engineering [21–23]. Also, delay is inevitable in economic activities. For example, changes in the money supply do not cause immediate changes in the economy; there is always a lag period. The production cycle has both long and short phases. Price change always has a delay. Therefore, delay differential equations (DDEs) support a realistic economic mathematical modeling than ordinary differential equations (ODEs) [6, 7].

In [24], Wang et al. proposed a delayed fractional order financial system as follows:

where is the time delay. The authors studied its dynamic behaviors, such as single-periodic, multiple-periodic, and chaotic motions.

Based on [24], Chen et al. [25] studied the following delayed financial system:

The authors have studied the asymptotic stability and Hopf bifurcations of the unique equilibrium, and the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions were also considered.

According to the above discussions, we consider a delayed finance system as follows:

where denotes price change delay, for price change does not immediately affect the interest rate, and it often has a lag period.

The main purpose of this paper is to investigate the stability and Hopf bifurcation for system (4) with delay as the bifurcation parameter.

The structure of this paper is arranged as follows. In Section 2, we study the local stability and the existence of Hopf bifurcation. In Section 3, we give the formula determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions. Finally, to support our theoretical predictions, some numerical simulations are given which support the analysis of Sections 2–3.

2. Stability and Hopf Bifurcation

2.1. The Existence of Equilibria

In this section, we consider the stability and Hopf bifurcation of the equilibria of system (4). First, we find all possible equilibria of system (4). We make the following hypothesis:

(H1)

According to system (4), equilibria should satisfy

Obviously, system (4) has an equilibrium . For other equilibria, solving for the second and third equations of (6), we have

Substitute (7) into the first equation of (6), we obtain

So, we have following results.

Lemma 1. If (H1) holds, then system (4) has two other equilibria and , where

In the following, we consider the stability of the equilibria of system (4) by analyzing the corresponding characteristic equations. Assume that denotes an arbitrary equilibrium of system (4), then let , , and and drop the bars for the simplicity of notations. Then by linearizing system (4) around , we have

The characteristic equation associated with system (10) is

where

2.2. Stability and Hopf Bifurcation of Equilibrium

Obviously, the characteristic equation of system (4) at the equilibrium has the following form:

Clearly, is negative; we only need to consider the following equation:

For further discussion, we make following hypotheses:

(H2)

(H3)

As , (14) is equivalent to the following equation:

Obviously, is not a root of (17).

Lemma 2. If (H2) and (H3) hold, then equilibrium of system (4) is locally asymptotically stable with .

Proof. Let and be two roots of (17). Clearly, if (H2) and (H3) hold, then we have

It means that all the roots of (17) have negative real parts. So, equilibrium of system (4) with is locally asymptotically stable.

Now we discuss the effect of delay on the stability of the equilibrium of system (4). Assume that is a root of (11). Then should satisfy the following equation:

which implies that

From (20), adding the squared terms for both equations yields

Make the following assumptions:

(H4)

(H5)

Theorem 1. If (H2) and (H4) hold, then the equilibrium of system (4) is locally asymptotically stable for all .

Proof. Clearly, if (H4) holds, then we have

which means that (21) has no positive roots. That is to say, all roots of (14) have negative real parts. Combining with Lemma 2, it thus follows from the Routh-Hurwitz criterion that the equilibrium of system (4) is locally asymptotically stable for all .

Lemma 3. If (H5) holds, then (21) has a unique positive root.

Proof. (H5) holds, so we have

Hence, (21) has a unique positive root as follows:

According to Lemma 3, (21) has a unique positive root . By (20), we have

Thus, if we denote

then is a pair of purely imaginary roots of (14) with . Clearly, sequence is increasing and

Thus, we can define

Lemma 4. Let be the root of (14) near satisfying for . Then the following transversal condition holds:

Proof. Differentiating the two sides of (14) with respect to yields

Hence,

Substituting into the above equation, we obtain

Since and , then we have

By (26), we have

On the basis of Lemmas 2–4, we have the following result:

Theorem 2. If (H2), (H3), and (H5) hold, then the following statements are true:

(i)When , the equilibrium of (4) is asymptotically stable(ii)The Hopf bifurcation occurs at . That is, system (4) has a branch of periodic solutions bifurcating from near

2.3. Stability and Hopf Bifurcation of Equilibrium and

In this section, we consider stability and Hopf bifurcation of equilibria and . At the equilibria and , the characteristic (11) takes the following form:

where

As , (37) becomes

Make the following assumptions:

(H6)

(H7)

Lemma 5. Based on Lemma 1, if (H6) holds, then equilibria and are both locally asymptotically stable with .

Proof. As (H1) and (H6) hold, we have

By the Routh-Hurwitz criteria, all the roots of (39) have negative real parts. Therefore, and are both locally asymptotically stable with .

Now we discuss the effect of delay on the stability of the equilibria and of system (4). Assume that is a root of (37). Then should satisfy the following equation:

which implies that

From (44), adding up the squares of both equations, we have

Let , then (45) can be rewritten into the following form:

Denote

Lemma 6. If (H7) holds, then (46) has at least a root.

Proof. Obviously,

Therefore, (46) has at least a positive root.

According to Lemma 6, (46) has a positive root, denoted by , and thus, (45) has a positive root . By (44), we have

Thus, if we denote

then is a pair of purely imaginary roots of (21) with . Clearly, sequence is increasing and

Thus, we can define

Lemma 7. Let be the root of (37) near satisfying , . Suppose that , where is defined by (47). Then the following transversal condition holds:

and the sign of is consistent with that of .

Proof. Denote

Then (37) can be written as

and (45) can be transformed into the following form:

Thus, together with (46) and (47), we have

Differentiating both sides of (57) with respect to , we obtain

If is not simple, then must satisfy

that is, must satisfy

With (55), we have

Thus, by (56) and (57), we obtain

Since is real, i.e., , we have . We get a contradiction to the condition . This proves the first conclusion. Differentiating both sides of (55) with respect to , we obtain

which implies

It follows together with (58) that

Clearly, the sign of is determined by that of .

On the basis of Lemma 1 and Lemma 5–Lemma 7, we have the following result.

Theorem 3. If (H1), (H6), and (H7) hold, and , then the following statements are true:

(i)When , the equilibria and of system (4) are both locally asymptotically stable(ii)The Hopf bifurcation occurs at , i.e., system (4) has a branch of periodic solutions that bifurcates from and near , respectively

3. Direction and Stability of Hopf Bifurcation

In the previous section, we have shown that system (4) admits a series of periodic solutions bifurcating from the equilibrium at the critical value . In this section, we derive explicit formulae to determine the properties of the Hopf bifurcation at the critical value by using the normal form theory and center manifold reduction developed by [26].

Denote by and introduce the new parameter . Normalizing the delay by the time-scaling , (4) can then be rewritten as

where

for .

Then the linearized system of (66) at is

Based on the discussion in Section 2, we can easily know that for , the characteristic equation of (11) has a pair of simple purely imaginary eigenvalues .

Let , considering the following FDE on :

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 is the Dirac delta function.

Let denote the infinitesimal generator of the semigroup induced by the solutions of (70) and be the formal adjoint of under the bilinear pairing

for and . Then and are a pair of adjoint operators. From the discussion in Section 2, we know that has a pair of simple purely imaginary eigenvalues , and they are also eigenvalues of since and are a pair of adjoint operators. Let and be the center spaces, that is, the generalized eigenspaces of and , respectively, associated with . Then is the adjoint space of and . Direct computations give the following results.

Lemma 8. Let

Then,

is a basis of associated with and

is a basis of associated with .

Let and with

for , and

for . From (73), we can obtain and , noting that

Therefore, we have

Now, we define and construct a new basis for by

Obviously, , the second-order identity matrix. In addition, define , where

Let be defined by

for and .

Then the center space of linear Equation (69) is given by , where

and ; here denotes the complementary subspace of .

Let be defined by

where is given by

Then is the infinitesimal generator induced by the solution of (69) and (66) and can be rewritten as the following operator differential equation:

Using the decomposition and (85), the solution of (66) can be rewritten as

where

and with . In particular, the solution of (66) on the center manifold is given by

Setting and noticing that , then (91) can be rewritten as

where . Moreover, by [26], satisfies

where

Let

From (92), we have

where

Let . Then by (94), (95), and (96), we can obtain the following quantities:

Since and for appear in , we still need to compute them. It follows easily from (95) that

In addition, by [26], and satisfy

where

with . Thus, from (92), (100), (101), and (102), we can obtain that

Noticing that has only two eigenvalues with zero real parts, (102), therefore, has a unique solution in given by

From (103), we know that for

Therefore, for

By the definition of , we get from (105) that

Noting that , Hence,

Using the definition of and combining (105) and (112) we get

Notice that

Then, we have

From the above expression, we can see easily that

By the similar way, we have

Similar to the above, we can obtain that

So far, and have been expressed by the parameters of system (4). Therefore, can be expressed explicitly.

Theorem 4. System (4) has the following Poincaré normal form

where

so we can compute the following results:

which determine the properties of bifurcating periodic solutions at the critical values , i.e., 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 on the center manifold are stable (unstable), if ; and determines the period of the bifurcating periodic solutions: the periodic increase (decrease), if .

4. Numerical Simulation

In this section, we present numerical simulations of some examples to illustrate our theoretical results.

4.1. Stability of Equilibrium for All

Consider system (4) with the following parameters: , , and . By a direct calculation, we obtain that system (4) has only an equilibrium and the parameters satisfy the conditions of (H2)–(H4). According to Theorem 1, the system is locally asymptotically stable at for all ; see Figure 1. However, in this case, the interest rate and the price index are all zero; this is impractical.

(a)

(b)

Figure 1: The equilibrium is locally asymptotically stable with .

4.2. Hopf Bifurcation at Equilibrium

Consider system (4) with the following parameters: , , and . By a simple calculation, we obtain that system (4) has only an equilibrium . Obviously, (H2), (H3), and (H5) are satisfied. By (28), we obtain . According to Theorem 2, system (4) is locally asymptotically stable at for (see Figure 2) and Hopf bifurcation occurs at , as shown in Figure 3.

(a)

(b)

Figure 2: The equilibrium is locally asymptotically stable for .

(a)

(b)

Figure 3: The periodic solutions emerge from the equilibrium with .

When , we can compute , , and . Therefore, from the discussions in Section 3, we know that the bifurcated periodic solutions are orbitally asymptotically stable on the center manifold. In addition, from Theorem 4, we know that system (4) has a stable center manifold near the equilibrium for near . Therefore, the center manifold theory implies that the bifurcated periodic solutions of system (4) when in the whole phase space are orbitally asymptotically stable, and the Hopf bifurcation is supercritical for .

4.3. Hopf Bifurcation at Equilibria and

Choose the parameters of system (4) as , , and . By a simple calculation, it is easy to obtain that and . Obviously, the parameters satisfy (H6) and (H7). By (50), we obtain the critical value . According to Theorem 3, and are both stable with , and with , two limit cycles emerge from the equilibria and , as shown in Figure 4.

(a)

(b)

Figure 4: (a) and are both stable with ; (b) the periodic solutions emerge from the equilibria and with .

In addition, when , at equilibrium , we get , , and . At equilibrium , we get , , and . According to Theorem 4 in Section 3, the bifurcated periodic solutions of system (4) when in the whole phase space are both orbitally asymptotically stable, and the Hopf bifurcations are supercritical for .

However, with increasing delay , the two limit cycles emerging form the equilibria and and appear to overlap, as shown in Figure 5. Figure 5(a) shows that the maximum and minimum of varies with under two groups of different initial values. It shows that two lines about maxima and minima appear to overlap with increasing delay , which mean that two limit cycles overlap; see Figure 5(b).

(a)

(b)

Figure 5: (a) Maximum and minimum of ; (b) the two limit cycles emerging from the equilibria and appear to overlap with .

4.4. Chaos Vanishes by Delay

According to [10, 15, 27], system (4) is chaotic for appropriate parameters. Figure 6 shows the Lyapunov exponents’ spectrum of system (4) with the increasing of parameter , where and . Figure 7 shows the bifurcation diagram of system (4) in the plane. Let , and the chaotic attractor of system (4) is shown in Figure 8.

Figure 6: Lyapunov exponent spectrum of system (4) when the parameter value of is changed continuously with .

Figure 7: Bifurcation diagrams of system (4) when the parameter value of is changed continuously with .

(a)

(b)

Figure 8: (a) Time trajectories of system (4) with ; (b) phase diagram of system (4) with .

In the following, in order to investigate the effect of delay on system (4), we fix , , and , and choosing as a parameter, the Lyapunov exponent spectrum and the detailed bifurcation scenarios of system (4) are shown in Figures 9 and 10. It can be seen that chaos disappears through a cascade of inverse period-doubling; see Figure 11. This observation indicates that the delay is a sensitive factor for system bifurcation and chaos and that chaos can be suppressed by delay .

Figure 9: Lyapunov exponent spectrum of system (4) when the parameter value of is changed continuously.

Figure 10: Bifurcation diagrams of system (4) when the parameter value of is changed continuously.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 11: Phase diagrams showing chaos disappearing via inverse period-doubling: (a) ; (b) ; (c) ; (d) ; (e) ; (f) .

4.5. Chaos Induced by Delay

Consider system (4) with the following parameters , , and . Obviously, parameters satisfy condition (H6). Therefore, according to Lemma 5, are both locally stable with . However, with increasing delay , system (4) presents strong nonlinear phenomena such as periodic motion, double-periodic motion, and chaotic motion and the bifurcation diagram of system (4) with increasing delay , which can be seen from the bifurcation diagram and maximum Lyapunov exponent with the parameter value of changed continuously, as shown in Figures 12 and 13. We list dynamic behaviors of system (4) corresponding to different delays in Table 1 and Figure 14. The route to chaos in finance system (4) was shown to be via classical period-doubling bifurcations (see Figures 14(e)–14(i)).

Figure 12: Bifurcation diagram of system (4) with the parameter value of is changed continuously.

Figure 13: Maximum Lyapunov exponent of system (4) when the parameter value of is changed continuously.

Table 1: The dynamic behaviors of system (4) for different

Dynamics of system (4)
Figure

[0, 1.52)
and are both stable
Figure 14(a)

(1.52, 3.18]
System (4) is chaotic
Figure 14(b)

(3.18, 5.52]
System (4) exhibits period 1 motion
Figure 14(c)

(5.52, 5.61]
System (4) is chaotic
Figure 14(d)

(5.61, 8.32]
and are both stable
Figure 14(e)

(8.32, 9.46]
System (4) exhibits period 1 motion
Figure 14(f)

(9.46, 9.68]
System (4) exhibits period 2 motion
Figure 14(g)

(9.68, 9.71]
System (4) exhibits period 4 motion
Figure 14(h)

(9.71, 12]
System (4) is chaotic
Figure 14(i)

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Figure 14: Nonlinear phenomena of system (4) induced by delay : (a) (stable); (b) (chaos); (c) (period 1); (d) (chaos); (e) (stable); (f) (period 1); (g) (period 2); (h) (period 4); (i) (chaos).

5. Conclusions

In this study, we have investigated dynamical behaviors such as stability, Hopf bifurcation, and chaos for a delayed finance system.

Firstly, we took delay as the bifurcation parameters to study the Hopf bifurcation of system (4). We have proved theoretically that the discrete delay is responsible for the stability switch of the model and that a Hopf bifurcation occurs as the delays increase through a certain threshold.

Secondly, by the normal form method and center manifold theorem, we have derived the normal forms of Hopf bifurcation.

Finally, by numerical simulations, we have given the Hopf bifurcation (Figures 3 and 4) that was induced by delay. We have also given the bifurcation diagram (Figures 10 and 12) and the corresponding Lyapunov exponents’ spectrum (Figures 6 and 13). All these show that delay can cause the system to exhibit strong nonlinear phenomena such as periodic motion, double-periodic motion, and chaotic motion (Figure 14).

The study will help in understanding the role of financial policies and interpreting economics phenomena in theory.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

Authors would like to thank the anonymous referees for their helpful comments and suggestions. The work is supported by the Natural Science Foundation of Jiangsu Province under Grant BK20150420 and is also supported by the Startup Foundation for Introducing Talent of NUIST.

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