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

Hopf Bifurcation and Chaos of a Delayed Finance System

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; moc.361@0301bxz

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 [35], the IS-LM model [6, 7], and nonlinear dynamical model on finance system [811]. 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 [1215] have also been considered for system (1).

It is well known that delays are extensively encountered in many fields such as biology [1618], chemistry [19, 20], and engineering [2123]. 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 23.

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 24, 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. LetThen,is a basis of associated with andis 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 formwhereso 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