Hopf Bifurcation and Chaos of a Delayed Finance System

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.


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][4][5], the IS-LM model [6,7], and nonlinear dynamical model on finance system [8][9][10][11].In [8,9], Ma and Chen proposed a simplified financial model as follows: where x is the interest rate, y is the investment demand, z is the price index, a > 0 denotes saving amount, b > 0 denotes cost per investment, and c > 0 denotes elasticity of demand of commercial markets.The variation of x is not only influenced by the surplus between investment and saving but also structurally adjusted by the price.The changing rate of y is proportional to the rate of investment and inversely proportional to the cost of investment and interest rate.The variation of z 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][13][14][15] have also been considered for system (1).
It is well known that delays are extensively encountered in many fields such as biology [16][17][18], chemistry [19,20], and engineering [21][22][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 τ ≥ 0 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 τ 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 τ 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.

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: According to system (4), equilibria should satisfy Obviously, system (4) has an equilibrium P 0 = 0, 1/b, 0 .For other equilibria, solving for the second and third equations of ( 6), we have 7 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 P 1 and P 2 , where In the following, we consider the stability of the equilibria of system (4) by analyzing the corresponding characteristic equations.Assume that P * = x * , y * , z * denotes an arbitrary equilibrium of system (4), then let x = x − x * , y = y − y * , and z = z − z * and drop the bars for the simplicity of notations.Then by linearizing system (4) around P * , we have The characteristic equation associated with system (10) is where Stability and Hopf Bifurcation of Equilibrium P 0 .Obviously, the characteristic equation of system (4) at the equilibrium P 0 = 0, 1/b, 0 has the following form: Clearly, λ = −b is negative; we only need to consider the following equation: For further discussion, we make following hypotheses: (H2) As τ = 0, ( 14) is equivalent to the following equation: Obviously, λ = 0 is not a root of (17).
Proof.Let λ 1 and λ 2 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 P 0 of system (4) with τ = 0 is locally asymptotically stable.Now we discuss the effect of delay τ on the stability of the equilibrium P 0 of system (4).Assume that iω ω > 0 is a root of (11).Then ω should satisfy the following equation: From (20), adding the squared terms for both equations yields Make the following assumptions: (H4) Theorem 1.If (H2) and (H4) hold, then the equilibrium P 0 of system ( 4) is locally asymptotically stable for all τ ≥ 0.
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 P 0 of system (4) is locally asymptotically stable for all τ ≥ 0. 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 ω 0 .By (20), we have 3 Complexity Thus, if we denote then ±iω 0 is a pair of purely imaginary roots of ( 14) with τ = τ Proof.Differentiating the two sides of ( 14) with respect to τ yields Hence,
In this section, we consider stability and Hopf bifurcation of equilibria P 1 and P 2 .At the equilibria P 1 and P 2 , the characteristic (11) takes the following form: where As τ = 0, (37) becomes Make the following assumptions: (H6) Lemma 5. Based on Lemma 1, if (H6) holds, then equilibria P 1 and P 2 are both locally asymptotically stable with τ = 0.

Complexity
Proof.As (H1) and (H6) hold, we have By the Routh-Hurwitz criteria, all the roots of (39) have negative real parts.Therefore, P 1 and P 2 are both locally asymptotically stable with τ = 0. Now we discuss the effect of delay τ on the stability of the equilibria P 1 and P 2 of system (4).Assume that iω ω > 0 is a root of (37).Then ω should satisfy the following equation: From (44), adding up the squares of both equations, we have Let z = ω 2 , then (45) can be rewritten into the following form: 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 z 0 , and thus, (45) has a positive root ω 0 = z 0 .By (44), we have Thus, if we denote 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 ω, we obtain that is, ω 0 must satisfy With (55), we have Thus, by ( 56) and (57), we obtain Since τ 0 is real, i.e., Im τ 0 = 0, we have h ′ ω 2 0 = 0. We get a contradiction to the condition h ′ ω 2 0 ≠ 0. This proves the first conclusion.Differentiating both sides of (55) with respect to τ, we obtain It follows together with (58) that Clearly, the sign of d Reλ τ /dτ| τ=τ 0 is determined by that of h′ z 0 .

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 τ j 0 j ∈ N 0 .In this section, we derive explicit formulae to determine the properties of the Hopf bifurcation at the critical value τ j 0 by using the normal form theory and center manifold reduction developed by [26].

72
where δ is the Dirac delta function.
Let A τ * denote the infinitesimal generator of the semigroup induced by the solutions of (70) and A * be the formal adjoint of A τ * under the bilinear pairing 3 .Then A τ * and A * are a pair of adjoint operators.From the discussion in Section 2, we know that A τ * has a pair of simple purely imaginary eigenvalues ±iω 0 τ * , and they are also eigenvalues of A * since A τ * and A * are a pair of adjoint operators.Let P and P * be the center spaces, that is, the generalized eigenspaces of A τ * and A * , respectively, associated with Λ 0 .Then P * is the adjoint space of P and dim P = dim P * = 2. Direct computations give the following results.
75 is a basis of P associated with Λ 0 and for θ ∈ −1, 0 , and for s ∈ 0, 1 .From (73), we can obtain Ψ * 1 , Φ 1 and Ψ * 1 , Φ 2 , noting that Therefore, we have Now, we define Ψ * , Φ = Ψ * j , Φ k j, k = 1, 2 and construct a new basis ψ for Q by Obviously, Ψ, Φ = I 2×2 , the second-order identity matrix.In addition, define f 0 = ξ 1 0 , ξ 2 0 , ξ 3 0 , where Then the center space of linear Equation (69) is given by P CN C, where and C = P CN C ⊕ P S C; here P S C denotes the complementary subspace of P CN C. Let A τ * be defined by where X 0 −1, 0 → B X, X is given by Then A τ * 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 C = P CN C ⊕ P S C and (85), the solution of (66) can be rewritten as where and h x 1 , x 2 , μ ∈ P s c with h 0, 0, 0 = Dh 0, 0, 0 = 0.In particular, the solution of (66) on the center manifold is given by Setting z = x 1 − ix 2 and noticing that p 1 = Φ 1 + iΦ 2 , then (91) can be rewritten as where W z, z = h z + z /2, − z − z /2i, 0 .Moreover, by [26], z satisfies where From (92), we have

97
where Then by ( 94), (95), and (96), we can obtain the following quantities: Since W 20 θ and W 11 θ for θ ∈ −1, 0 appear in g 21 , we still need to compute them.It follows easily from (95) that In addition, by [26], W z t and z t satisfy where 9 Complexity with H ij ∈ P S C, i + j = 2. Thus, from (92), (100), (101), and (102), we can obtain that Noticing that A τ * has only two eigenvalues ±iω 0 τ * with zero real parts, (102), therefore, has a unique solution W ij i + j = 2 in P S C given by From (103), we know that for −1 ≤ θ < 0, Therefore, for −1 ≤ θ < 0, By the definition of A τ * , we get from (105) that Using the definition of A τ * and combining ( 105) and (112) we get 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, W 20 θ and W 11 θ have been expressed by the parameters of system (4).Therefore, g 21 can be expressed explicitly.

Theorem 4. System (4) has the following Poincaré normal form
where , 121 so we can compute the following results:    12 Complexity which determine the properties of bifurcating periodic solutions at the critical values τ * , i.e., σ 2 determines the directions of the Hopf bifurcation: if σ 2 > 0 σ 2 < 0 , then the Hopf bifurcation is supercritical (subcritical) and the bifurcating periodic solutions exist for τ > τ * ; β 2 determines the stability of the bifurcating periodic solutions: the bifurcating periodic solutions on the center manifold are stable (unstable), if β 2 < 0 β 2 > 0 ; and T 2 determines the period of the bifurcating periodic solutions: the periodic increase (decrease), if T 2 > 0 T 2 < 0 .
However, with increasing delay τ, the two limit cycles emerging form the equilibria P 1 and P 2 and appear to overlap, as shown in Figure 5. Figure 5(a) shows that the maximum and minimum of x varies with τ under two groups of different initial values.It shows that two lines about maxima and minima appear to overlap with increasing delay τ, which mean that two limit cycles overlap; see Figure 5(b).

Chaos Vanishes by Delay τ.
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 c, where a = 2 and b = 0 1. Figure 7 shows the bifurcation diagram of system (4) in the c − y plane.Let c = 1 1, and the chaotic attractor of system (4) is shown in Figure 8.
In the following, in order to investigate the effect of delay τ on system (4), we fix a = 2, b = 0 1, and c = 1 1, and choosing τ 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 τ. 4.5.Chaos Induced by Delay τ.Consider system (4) with the following parameters a = 3, b = 0 2, and c = 6.Obviously, parameters satisfy condition (H6).Therefore, according to Lemma 5, P 1,2 = ±0 6055,3 1667, ∓0 1009 are both locally stable with τ = 0.However, with increasing delay τ, 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 τ, which can be seen from the bifurcation diagram and maximum Lyapunov exponent with the parameter value of τ 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)).

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 τ 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 τ 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.

Figure 5 :
Figure 5: (a) Maximum and minimum of x; (b) the two limit cycles emerging from the equilibria P 1 and P 2 appear to overlap with τ = 1.

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

Figure 13 :
Figure13: Maximum Lyapunov exponent of system (4) when the parameter value of τ is changed continuously.