Neimark-Sacker Bifurcation in a Discrete-Time Financial System

Abstract

A discrete-time financial system is proposed by using forward Euler scheme. Based on explicit Neimark-Sacker bifurcation (also called Hopf bifurcation for map) criterion, normal form method and center manifold theory, the system's existence, stability and direction of Neimark-Sacker bifurcation are studied. Numerical simulations are employed to validate the main results of this work. Some comparison of bifurcation between the discrete-time financial system and its continuous-time system is given.

1. Introduction

Economic dynamics have recently become more prominent in mainstream economics [1]. The real financial and economic systems show a lot of complex dynamical phenomena, such as, business cycle, financial crisis, irregular growth, and bullwhip effect. Many nonlinear dynamical models of economics and finance [2–9] present various complex dynamical behaviors such as, chaos, fractals, and bifurcation.

Bifurcation refers to a class of phenomena in dynamic systems such that the dynamic properties of the system cause a sudden “qualitative” or topological change when the parameter values (the bifurcation parameters) cross a boundary. Bifurcation boundaries, for example, Hopf bifurcations [10–13], have been discovered in many macroeconomic systems [14]. Hopf bifurcations occur at points where the system has a nonhyperbolic equilibrium with a pair of purely imaginary eigenvalues, but without zero eigenvalues. For a financial or economic system, there can be disequilibrium thresholds where society decides it cannot afford the increasing cost of misallocated resources as disequilibrium increases. Such a threshold then forces a restructuring of the market system. This concept of restructuring to maintain the survival of the system is known as bifurcation theory. A bifurcation in a financial or economic system is a point (or threshold) where the system is restructured to operate at a more acceptable or stable level of disequilibrium. Bifurcations do not usually lead to equilibrium conditions, only to a stable or comfortable disequilibrium condition under which the system can continue to survive [15].

Huang and Li [16] proposed a nonlinear financial model as follows:̇̇𝑥=𝑧+(𝑦−𝑎)𝑥,𝑦=1−𝑏𝑦−𝑥²,𝑧̇=−𝑥−𝑐𝑧, (1.1) where 𝑥 denotes the interest rate, 𝑦 denotes the investment demand, 𝑧 denotes the price index, 𝑎 is the saving amount, 𝑏 is the cost per investment, 𝑐 is the demand elasticity of commercial markets, and all three constants 𝑎,𝑏,𝑐≥0.

Chen [1] and Ma et al. [11–13] studied some complex dynamics in system (1.1), such as, a steady state, a periodic oscillation, a quasiperiodic motion and a chaotic motion. In this paper, we will apply the forward Euler scheme to system (1.1) in order to obtain an autonomous discrete-time financial system as follows:𝑥_𝑛+1=𝑥_𝑛+𝛿(𝑧_𝑛+(𝑦_𝑛−𝑎)𝑥_𝑛𝑦),_𝑛+1=𝑦_𝑛+𝛿(1−𝑏𝑦_𝑛−𝑥²_𝑛𝑧),_𝑛+1=𝑧_𝑛+𝛿(−𝑥_𝑛−𝑐𝑧_𝑛), (1.2) where 0<𝛿<1 is the step size.

An arduous task in the study of nonlinear dynamical systems like system (1.2) is to identify different types of complex nonlinear behaviors and to present how the behavior evolves as a system parameter varies [17, 18]. Thereinto, bifurcation is a very important nonlinear behavior which can indicate a qualitative change of system properties as a system parameter changes. Neimark-Sacker bifurcations give rise to closed invariant curves which present more interesting complex behaviors. The criterion of Hopf bifurcation in continuous-time system can be stated in terms of eigenvalues or the coefficients of characteristic polynomial [19, 20]. The later method, called Schur-Cohn stability criterion, which is more convenient and efficient for detecting the existence of Hopf bifurcation in high-order and multiparameters systems was also demonstrated in discrete dynamical systems [21–23].

The remainder of this paper is organized as follows. In Section 2, we present some preliminaries. In Section 3, we prove stabilities of the fixed points in system (1.2). In Section 4, we analyze the existence of Neimark-Sacker bifurcation in system (1.2) by means of Wen’s Neimark-Sacker bifurcation criterion. In Section 5, we study the stability and direction of Neimark-Sacker bifurcation in system (1.2) by utilizing Kuznetsov’s normal form method and center manifold theory. In Section 6, we illustrate the Neimark-Sacker bifurcation in system (1.2). In Section 7, we give some comparison of bifurcation between the continuous-time system (1.1) and the discrete-time system (1.2). Finally conclusions in Section 8 close the paper.

2. Preliminaries

Lemma 2.1 (see [24]). Let 𝐹 from ℝ^𝑛 to ℝ^𝑛 be 𝐶². Assume 𝑝₀ is a period-𝑘 point. Let 𝜆₁,𝜆₂,…,𝜆_𝑛 be the eigenvalues of 𝐷(𝐹^𝑘)_(𝑝0).(i)If all the eigenvalues 𝜆_𝑖 of 𝐷(𝐹^𝑘)_(𝑝0) have |𝜆_𝑖|<1, then the periodic orbit 𝒪⁺_𝐹(𝑝₀) is attracting. (ii)If one eigenvalue 𝜆_𝑖0 of 𝐷(𝐹^𝑘)_(𝑝0) has |𝜆_𝑖0|>1, then the periodic orbit 𝒪⁺_𝐹(𝑝₀) is unstable. (iii)If all the eigenvalues 𝜆_𝑖 of 𝐷(𝐹^𝑘)_(𝑝0) have |𝜆_𝑖|>1, then the periodic orbit 𝒪⁺_𝐹(𝑝₀) is repelling.

Next, we will study the stability of a nonlinear discrete dynamical system which can be described as follows:𝑋_𝑡+1=𝐹(𝑋_𝑡),𝑋(0)=𝑋₀=(𝑥₁₀,𝑥₂₀,…,x_𝑛0)^𝑇,where (2.1)𝐹(𝑋_𝑡𝑓)=¹(𝑥_1𝑡,𝑥_2𝑡,…,𝑥_𝑛𝑡)𝑓₂(𝑥_1𝑡,𝑥_2𝑡,…,𝑥_𝑛𝑡)⋮𝑓_𝑛(𝑥_1𝑡,𝑥_2𝑡,…,𝑥_𝑛𝑡), (2.2)𝑋_𝑡=(𝑥_1𝑡,𝑥_2𝑡,…,𝑥_𝑛𝑡)^𝑇∈𝑅^𝑛.

Theorem 2.2. Let 𝑋=(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇 be a fixed point of system (2.1), that is, 𝑋=𝐹(𝑋) and 𝐴=(𝜕𝐹/𝜕𝑋)∣_𝑋=𝑋 is the Jacobian matrix at the point 𝑋; then the necessary condition for asymptotically stability of the point 𝑋 is that (i)|tr(𝐴^𝑡)|<𝑛 for all 𝑡>0,(ii)|det(𝐴^𝑡)|<1 for all 𝑡>0,where tr(𝐴) denotes the trace of A and det(𝐴) the determinant of A.

Proof. Assume that the point 𝑋 is asymptotically stable and let 𝜆₁,…,𝜆₁,𝜆_𝑛 be the eigenvalues of the Jacobian matrix A at the point 𝑋. Then it follows from Lemma 2.1 that all the eigenvalues satisfy |𝜆_𝑖|<1,𝑖=1,2,…,𝑛. (2.3) Thus |tr(𝐴^𝑡)|=|^𝑛_𝑖=1𝜆^𝑡_𝑖|≤^𝑛_𝑖=1|𝜆_𝑖|^𝑡<𝑛,∀𝑡>0,|det(𝐴^𝑡)|=|^𝑛_𝑖=1𝜆^𝑡_𝑖|=^𝑛_𝑖=1|𝜆_𝑖|^𝑡<1,∀𝑡>0. (2.4) The theorem is proved.

Theorem 2.3. Let 𝑋=(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇 be a fixed point of system (2.1), that is, 𝑋=𝐹(𝑋) and 𝐴=(𝜕𝐹/𝜕𝑋)∣_𝑋=𝑋 is the Jacobian matrix at the point 𝑋; then the necessary condition for repellent of the point 𝑋 is that (i)|tr(𝐴^𝑡)|>𝑛 for all 𝑡>0,(ii)|det(𝐴^𝑡)|>1 for all 𝑡>0,where tr(𝐴) denotes the trace of A and det(𝐴) the determinant of A.

Proof. Assume that the point 𝑋 is repelling and let 𝜆₁,…,𝜆₁,𝜆_𝑛 be the eigenvalues of the Jacobian matrix A at the point 𝑋. Then it follows from Lemma 2.1 that all the eigenvalues satisfy |𝜆_𝑖|>1,𝑖=1,2,…,𝑛. (2.5) Thus |tr(𝐴^𝑡)|=|^𝑛_𝑖=1𝜆^𝑡_𝑖|≤^𝑛_𝑖=1|𝜆_𝑖|^𝑡>𝑛,∀𝑡>0,|det(𝐴^𝑡)|=|^𝑛_𝑖=1𝜆^𝑡_𝑖|=^𝑛_𝑖=1|𝜆_𝑖|^𝑡>1,∀𝑡>0. (2.6) The theorem is proved.

Lemma 2.4 (An explicit criterion of Neimark-Sacker bifurcation [22]). For an 𝑛th-order discrete-time dynamical system like system (1.2), assume first that at the fixed point 𝑥₀, its characteristic polynomial of Jacobian matrix 𝐴=(𝑎_𝑖𝑗)_𝑛×𝑛 takes the following form: 𝑝_𝜇(𝜆)=𝜆^𝑛+𝑎₁𝜆^𝑛−1+⋯+𝑎_𝑛−1𝜆+𝑎_𝑛, (2.7) where 𝑎_𝑗=𝑎_𝑗(𝜇,𝑘),𝑗=1,…,𝑛,𝜇, is the bifurcation parameter, and 𝑘 is the control parameter or the other to be determined. Consider the sequence of determinants Δ^±₀(𝜇,𝑘)=1,Δ^±₁(𝜇,𝑘),…,Δ^±_𝑛(𝜇,𝑘), where Δ^±_𝑗(𝜇,𝑘)=|1𝑎₁𝑎₂⋯𝑎_𝑗−101𝑎₁⋯𝑎_𝑗−2001⋯𝑎_𝑗−3±𝑎⋯⋯⋯⋯⋯000⋯1^{𝑛−𝑗+1}𝑎_{𝑛−𝑗+2}⋯𝑎_𝑛−1𝑎_𝑛𝑎_{𝑛−𝑗+2}𝑎_{𝑛−𝑗+3}⋯𝑎_𝑛0𝑎⋯⋯⋯⋯⋯_𝑛−1𝑎_𝑛𝑎⋯00_𝑛0⋯00|,𝑗=1,…,𝑛. (2.8) If the following conditions hold,(H1)eigenvalue assignment Δ⁻_𝑛−1(𝜇₀,𝑘)=0,  𝑝_𝜇0(1)>0,(−1)^𝑛𝑝_𝜇0(−1)>0,Δ⁺_𝑛−1(𝜇₀,𝑘)>0,Δ^±_𝑗(𝜇₀,𝑘)>0,𝑗=𝑛−3,𝑛−5,…,1(𝑜𝑟2) when 𝑛 is even (or odd, resp.),(H2)transversality condition 𝑑Δ⁻_𝑛−1(𝜇₀,𝑘)/𝑑𝜇≠0,(H3)nonresonance condition cos(2𝜋/𝑚)≠𝜓 or resonance condition cos(2𝜋/𝑚)=𝜓, where 𝑚=3,4,5,… and 𝜓=1−0.5𝑝_𝜇0(1)Δ⁻_𝑛−3(𝜇₀,𝑘)/Δ⁺_𝑛−2(𝜇₀,𝑘),then a Neimark-Sacker bifurcation occurs at 𝜇₀.

3. Stability of the Fixed Points

The fixed points of system (1.2) satisfy the following equations:𝑥=𝑥+𝛿(𝑧+(𝑦−𝑎)𝑥),𝑦=𝑦+𝛿(1−𝑏𝑦−𝑥²),𝑧=𝑧+𝛿(−𝑥−𝑐𝑧). (3.1) By the analysis of roots for (3.1), one obtains the following proposition.

Proposition 3.1. (1) If 𝑐−𝑏−𝑎𝑏𝑐≤0, system (1.2) has only one fixed point 𝑃₀=(0,1/𝑏,0).
(2) If 𝑐−𝑏−𝑎𝑏𝑐≥0, system (1.2) has three fixed points: 𝑃₁1=(0,𝑏,0),𝑃_2,3=(±𝑐−𝑏−𝑎𝑏𝑐𝑐,1+𝑎𝑐𝑐1,∓𝑐𝑐−𝑏−𝑎𝑏𝑐𝑐). (3.2) The Jacobian matrix 𝐽(𝑃) of system (1.2) evaluated at the fixed point 𝑃(𝑥^∗,𝑦^∗,𝑧^∗) is given by 𝐽(𝑃)=1+𝛿(𝑦^∗−𝑎)𝛿𝑥^∗𝛿−2𝛿𝑥^∗1−𝛿𝑏0−𝛿01−𝛿𝑐. (3.3)

Following from Theorem 2.2, it is easy to obtain the following propositions.

Proposition 3.2. When 𝑐−𝑏−𝑎𝑏𝑐≤0, the fixed point 𝑃₀ is not asymptotically stable if 1−𝑎𝑏−𝑏²−𝑏𝑐>0𝑜𝑟(1−𝑎𝑏−𝑏²−𝑏𝑐)ℎ+6𝑏<0. (3.4)

Proposition 3.3. When 𝑐−𝑏−𝑎𝑏𝑐≥0, (1)the fixed point 𝑃₁ is not asymptotically stable if 1−𝑎𝑏−𝑏²−𝑏𝑐>0𝑜𝑟(1−𝑎𝑏−𝑏²−𝑏𝑐)ℎ+6𝑏<0; (3.5)(2)the fixed points 𝑃_2,3 are not asymptotically stable if 1−𝑏𝑐−𝑐²>0𝑜𝑟(1−𝑏𝑐−𝑐²)ℎ+6𝑐<0. (3.6) That is, if one of Propositions 3.2 and 3.3 holds, it is possible that bifurcation occurs in system (1.2).
The Jacobian matrix 𝐽(𝑃₀) of the system (1.2) evaluated at the fixed point 𝑃₀=(0,1/𝑏,0) is given by 𝐽(𝑃₀1)=1+𝛿(𝑏−𝑎)0𝛿01−𝛿𝑏0−𝛿01−𝛿𝑐. (3.7) Its eigenvalues can be written as 𝜆_1,2=12𝑏(𝛿+2𝑏−𝑎𝑏𝛿−𝑏𝑐𝛿±𝑖𝛿4𝑏²−(𝑎𝑏−b𝑐−1)²),𝜆₃=1−𝑏𝛿. (3.8)

Following from Theorem 2.2, it is easy to obtain the following propositions.

Proposition 3.4. When 𝑐−𝑏−𝑎𝑏𝑐≤0,(1)the fixed point 𝑃₀ is asymptotically stable if 𝑎>(𝑐𝛿−𝑏𝛿+𝑏𝑐−1)/𝑏(𝛿−1);(2)the fixed point 𝑃₀ is unstable if 𝑎<(𝑐𝛿−𝑏𝛿+𝑏𝑐−1)/𝑏(𝛿−1).(3)a bifurcation occurs at the fixed point 𝑃₀ if 𝑎=(𝑐𝛿−𝑏𝛿+𝑏𝑐−1)/𝑏(𝛿−1).

4. Existence of Neimark-Sacker Bifurcation

The main task of this paper is to determine the value of bifurcation parameter when the system (1.2) has only one fixed point 𝑃₀=(0,1/𝑏,0) with 𝑐−𝑏−𝑎𝑏𝑐<0.

The characteristic polynomial of the Jacobian matrix (3.7) is𝑝(𝜆)=𝜆³+𝑝₂𝜆²+𝑝₁𝜆+𝑝₀=0, (4.1) where 𝑝₁1=𝛿(𝑎+𝑏+𝑐−𝑏𝑝)−3,₂=𝛿²𝑐(𝑎𝑏+𝑎𝑐+𝑏𝑐−𝑏1)−2𝛿(𝑎+𝑏+𝑐−𝑏𝑝)+3,₃=𝛿³(𝑎𝑏𝑐+𝑏−𝑐)−𝛿²𝑐(𝑎𝑏+𝑎𝑐+𝑏𝑐−𝑏1)+𝛿(𝑎+𝑏+𝑐−𝑏)−1. (4.2) According to Lemma 2.4, for 𝑛=3, we can get the following equalities and inequalities:Δ⁻₂(𝑎)=|1𝑝₁−𝑝01²𝑝₃𝑝₃01|=1+𝑏²(𝛿𝑏−1)²(𝛿²𝑏−𝛿𝑏𝑐−𝛿𝑎𝑏+𝛿+𝑏+𝛿²𝑎𝑏𝑐−𝛿²𝑐)²𝑝>0,_𝑎(1)=𝛿³(𝑎𝑏𝑐+𝑏−𝑐)>0,(−1)³𝑝_𝑎1(−1)=𝑏(2−𝑏𝛿)(4𝑏−2𝑏𝑐𝛿+2𝛿−𝑐𝛿²−2𝑎𝑏𝛿+𝑎𝑏𝑐𝛿²+𝑏𝛿²Δ)>0,₊₂(𝑎)=|1𝑝₁+𝑝01²𝑝₃𝑝₃01|=1−𝑏²(𝛿𝑏−1)²(𝛿²𝑏−𝛿𝑏𝑐−𝛿𝑎𝑏+𝛿+𝑏+𝛿²𝑎𝑏𝑐−𝛿²𝑐)²>0. (4.3) According to (4.3), the critical value of Neimark-Sacker bifurcation of system (1.2) can be obtained as𝑎^∗=𝑐𝛿−𝑏𝛿+𝑏𝑐−1.𝑏(𝛿−1) (4.4) Thus, it follows (3.8) that the eigenvalues modules |𝜆_1,2|=1,|𝜆₃|=1−𝑏𝛿 satisfy the condition (H1) in Lemma 2.4, that is, Neimark-Sacker bifurcation occurs at the fixed point 𝑃₀=(0,1/𝑏,0).

5. Direction and Stability of the Neimark-Sacker Bifurcations

In this section, we will use Kuznetsov’s normal form method and center manifold theory [25] to investigate the direction and stability of the Neimark-Sacker bifurcations in system (1.2).

Since the fixed point 𝑃₀=(0,1/𝑏,0) is not the origin 𝑂(0,0,0), the 𝑃₀ needs to be transformed to the origin by the change of variables 1𝑥=𝑢,𝑦=𝑏+𝑣,𝑧=𝑤. (5.1) This transforms system (1.2) into the following equivalent system:𝑢_𝑛+1=𝑢_𝑛+𝛿(𝑤_𝑛+(𝑣_𝑛1−𝑎+𝑏)𝑢_𝑛𝑣),_𝑛+1=𝑣_𝑛+1𝑏−𝛿(𝑏𝑣_𝑛+𝑢²_𝑛𝑤),_𝑛+1=𝑤_𝑛−𝛿(𝑢_𝑛+𝑐𝑤_𝑛), (5.2) This system can be written as 𝑋_𝑛+1=𝐽𝑋_𝑛+12𝐵(𝑋_𝑛,𝑋_𝑛1)+6𝐶(𝑋_𝑛,𝑋_𝑛,𝑋_𝑛)+𝑂(𝑋⁴_𝑛), (5.3) where 𝐽 is the Jacobin matrix of system (5.2) evaluated at the origin 𝑂(0,0,0) as follows.𝐽(𝑂)=2𝛿²+2𝑐𝛿−𝑐²𝛿²+10𝛿01−𝛿𝑏0−𝑑01−𝛿𝑐. (5.4) And the multilinear functions 𝐵∶ℝ²×ℝ²→ℝ² and 𝐶∶ℝ³×ℝ³→ℝ³ are defined, respectively, by 𝐵_𝑖(𝑥,𝑦)=^𝑛_𝑗,𝑘=1𝜕²𝑋_𝑖(𝜉,0)𝜕𝜉_𝑗𝜕𝜉_𝑘|_𝜉=0𝑥_𝑗𝑦_𝑘𝐶,𝑖=1,2,3,_𝑖(𝑥,𝑦,𝑧)=^𝑛_{𝑗,𝑘,𝑙=1}𝜕³𝑋_𝑖(𝜉,0)𝜕𝜉_𝑗𝜕𝜉_𝑘𝜕𝜉_𝑙|_𝜉=0𝑥_𝑗𝑦_𝑘𝑧_𝑙,𝑖=1,2,3. (5.5) For the system (5.2), 𝐵(𝜉,𝜂)=𝛿𝜉₁𝜂₂−𝛿𝜉₁𝜂₁0000,𝐶(𝜉,𝜂,𝜁)=. (5.6) The eigenvalues of the matrix 𝐽(𝑂) are𝜆_1,2=12𝑏(𝛿+2𝑏−𝑎^∗𝑏𝛿−𝑏𝑐𝛿±𝑖ℎ4𝑏²−(𝑎^∗𝑏−𝑏𝑐−1)²)=𝑒^±𝑖𝜃0,where0<𝜃₀<𝜋. (5.7)

Let 𝑞∈𝐶³ be a complex eigenvector of the matrix 𝐽 corresponding to 𝜆₁ given by (5.7), and satisfy 𝐽𝑞=𝑒^𝑖𝜃0𝑞. (5.8)

Let 𝑝∈𝐶³ be a complex eigenvector of the transposed matrix 𝐽 corresponding to 𝜆₂ given by (5.7), and satisfy 𝐽^𝑇𝑝=𝑒^−𝑖𝜃0𝑝. (5.9) Then we can obtain 1𝑞∼(ℎ(1−ℎ𝑐−𝜆₁),0,1)^𝑇1,𝑝∼(ℎ(1−ℎ𝑐−𝜆₂),0,1)^𝑇. (5.10) For the eigenvector 𝑞=(1/ℎ(1−ℎ𝑐−𝜆₁),0,1)^𝑇, to normalize 𝑝, let 𝑝=(−2𝐾²|𝐾²|(𝑐−𝑎+(1/𝑏)+(𝐾_𝐴/𝑏𝛿))−𝑏𝐾𝛿,0,−4𝑏𝛿𝐾(𝑐−𝑎+(1/𝑏)+(𝐾_𝐴)/𝑏𝛿))−4𝑏𝛿^𝑇, (5.11) where 𝐾_𝐴=𝑖ℎ4𝑏²−(𝑎𝑏−𝑏𝑐−1)²,𝐾=𝑎𝑏𝛿−𝛿−𝑏𝑐𝛿+𝐾_𝐴. (5.12) We have ⟨𝑝,𝑞⟩=1, where ⟨⋅,⋅⟩ means the standard scalar product in ℂ²∶⟨𝑝,𝑞⟩=𝑝₁𝑞₁+𝑝₂𝑞₂

So the coefficients of the normal of the system (5.2) can be computed by the formulas as follows:𝑔₂₀𝑔=⟨𝑝,𝐵(𝑞,𝑞)⟩,₁₁=⟨𝑝,𝐵(𝑞,𝑔𝑞)⟩,₀₂=⟨𝑝,𝐵(𝑞,𝑔𝑞)⟩,₂₁=⟨𝑝,𝐶(𝑞,𝑞,𝑞)⟩+2⟨𝑝,𝐵(𝑞,(𝐼_𝑛−𝐽)⁻¹𝐵(𝑞,𝑞))⟩+⟨𝑝,𝐵(𝑞,(𝑒^2𝑖𝜃0𝐼_𝑛−𝐽)⁻¹+𝑒𝐵(𝑞,𝑞))⟩_−𝑖𝜃0(1−2𝑒^𝑖𝜃0)1−𝑒^𝑖𝜃0𝑔₂₀𝑔₁₁+21−𝑒^−𝑖𝜃0|𝑔₁₁|²+𝑒^𝑖𝜃0𝑒^3𝑖𝜃0−1|𝑔₀₂|². (5.13) The direction coefficient of bifurcation of a closed invariant curve can be obtained by following formula𝑒𝑑=Re(^−𝑖𝜃0𝑔₂₁2𝑒)−Re(^{−2𝑖𝜃0}(1−2𝑒^𝑖𝜃0)2(1−𝑒^𝑖𝜃0)𝑔₂₀𝑔₁₁1)−2|𝑔₁₁|²−14|𝑔₀₂|². (5.14) Thus we can obtain the theorem as follows.

Proposition 5.1. The direction and stability of Neimark-Sacker bifurcation of system (1.2) can be determined by the sign of 𝑑. If 𝑑<0(>0), then the Neimark-Sacker bifurcation of system (1.2) at 𝑎^∗=(𝑐𝛿−𝑏𝛿+𝑏𝑐−1)/𝑏(𝛿−1) is supercritical (subcritical), and the unique closed invariant curve bifurcating from 𝑃₀=(0,1/𝑏,0) is asymptotically stable (unstable).

6. Numerical Simulations

In this section, we will give an example to illustrate above analytic results.

Let ℎ=0.3,𝑏=0.6,and𝑐=0.2 with an initial state (𝑥₀,𝑦₀,𝑧₀)=(0.4,0.6,0.8); we can obtain the critical saving amount 𝑎^∗=(𝑐𝛿−𝑏𝛿+𝑏𝑐−1)/𝑏(𝛿−1)=1.773. By substituting this into (5.14), we have 𝑑=−1.83<0. It follows from Proposition 5.1 that system (1.2) undergoes a supercritical Neimark-Sacker bifurcation at 𝑎=1.773. When we give a small perturbation Δ𝑎=0.0001, a sufficiently small positive real number, that is, 𝑎=𝑎^∗+Δ𝑎=1.773+0.0001=1.7731, system (1.2) has a stable closed invariant curve around the equilibrium (quasiperiodic solution), as shown in Figures 1 and 2.

Figure 1 

Phase portrait with 𝑎=1.7731

Figure 2 

Time histories with 𝑎=1.7731

7. Comparison

For system (1.1) at the fixed point 𝑃₀=(0,1/𝑏,0) with 𝑐−𝑏−𝑎𝑏𝑐<0, Ma and Chen [11] gave the critical value of Hopf bifurcation as follows. 𝑎^∗=1𝑏−𝑐. (7.1) By simple calculation, we can get the following conclusions.

Proposition 7.1. Hopf bifurcations of continuous-time system (1.1) and discrete-time system (1.2) occur simultaneously at 𝑎=1/𝑏−1 when 𝑐=1.

Proposition 7.2. The continuous-time system (1.1) undergoes Hopf bifurcation earlier than the discrete-time system (1.2) when (I)𝑐<1 and (𝑏+𝛿)/𝑏(1−𝛿)<−1or(II)𝑐>1 and (𝑏+𝛿)/𝑏(1−𝛿)<1.

Proposition 7.3. The discrete-time system (1.2) undergoes Hopf bifurcation earlier than the continuous-time system (1.1) when (I)𝑐<1 and (𝑏+𝛿)/𝑏(1−𝛿)<1or(II)𝑐>1 and (𝑏+𝛿)/𝑏(1−𝛿)>1.

8. Conclusion

In this paper, we introduce a discrete-time financial system obtained by Euler method. The existence of Neimark-Sacker bifurcation is studied by means of Wen’s Neimark-Sacker bifurcation criterion. The stability and direction of Neimark-Sacker bifurcation are proved by utilizing Kuznetsov’s normal form method and center manifold theory. Numerical simulations are used to illustrate the above main results. We give Some comparison of bifurcation between the discrete-time financial system and its continuous-time system.

Acknowledgments

The authors are very grateful to the referees for their valuable suggestions, which help to improve the paper significantly. This work is supported partly by the China Postdoctoral Science Foundation (Grant no. 20100470783 ) and the Specialized Research Fund for the Doctoral Program of Higher Education from Ministry of Education of China (Grant no. 20090032110031).