Bifurcation Analysis for a Kind of Nonlinear Finance System with Delayed Feedback and Its Application to Control of Chaos

Abstract

A kind of nonlinear finance system with time-delayed feedback is considered. Firstly, by employing the polynomial theorem to analyze the distribution of the roots to the associate characteristic equation, the conditions of ensuring the existence of Hopf bifurcation are given. Secondly, by using the normal form theory and center manifold argument, we derive the explicit formulas determining the stability, direction, and other properties of bifurcating periodic solutions. Finally, we give several numerical simulations, which indicate that when the delay passes through certain critical values, chaotic oscillation is converted into a stable steady state or a stable periodic orbit.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Recommended by Chuanhou Gao

Department of Electronic Information Engineering, Huanghe Science and Technology College, Henan, Zhengzhou 450063, China

Received 14 October 2011; Revised 15 January 2012; Accepted 23 January 2012

1. Introduction

Since the chaotic phenomenon in economics was first found in 1985, great impact has been imposed on the prominent western economics at present, because the chaotic phenomenon occurring in the economic system means that the macroeconomic operation has in itself the inherent indefiniteness. Although the government can adopt such macrocontrol measures as the financial policies or the monetary policies to interfere, the effectiveness of the interference is very limited. The instability and complexity make the precise economic prediction greatly limited, and the reasonable prediction behavior has become complicated as well. In the fields of finance, stocks, and social economics, because of the interaction between nonlinear factors, with all kinds of economic problems being more and more complicated and with the evolution process from low dimensions to high dimensions, the diversity and complexity have manifested themselves in the internal structure of the system and there exists extremely complicated phenomenon and external characteristics in such a kind of system. So it has become more and more important to study the control of the complicated continuous economic system and stabilize the instable periodic or stationary solutions, in order to make the precise economic prediction possible [1, 2].

Recent works [1, 2] have reported a dynamic model of finance, composed of three first-order differential equations. The model describes the time variations of three state variables: the interest rate x , the investment demand y , and the price index z . By choosing an appropriate coordinate system and setting appropriate dimensions for each state variable, [1, 2] offer the simplified finance system as [figure omitted; refer to PDF] which is chaotic when a=1.69, b=4 (see Figure 1).

Figure 1: Strange attractor of finance system (1.1).

[figure omitted; refer to PDF]

Over the last years, [3, 4] studied impulsive control and state feedback control of the finance system (1.1). In this paper, we are interesting in delayed feedback control of the finance system (1.1). The effects of the time-delayed feedback on the finance system have long been investigated [5-8].

Recently, different techniques and methods have been proposed to achieve chaos control. The existing control methods can be classified, mainly, into two categories. The first one, developed by Ott et al. [9] is based on the invariant manifold structure of unstable orbits. It is theoretically well understood but difficult to apply to fast experimental systems. The second, proposed by Pyragas [10], uses time-delayed controlling forces. In contrast to the former one, it is simple and convenient method of controlling chaos in continuous dynamical system. Thus, we adopt the second one in the present paper.

For predigesting the investigation, here we only put time delay on investment demand y . By adding a time-delayed force K(y(t)-y(t-τ)) to the second equation of finance system (1.1), we obtain the following new system [figure omitted; refer to PDF] Here we assume that (_C1 ) a,b,τ∈(0,∞) and K∈R . The time delay τ is taken as the bifurcation parameter and we show that when τ passes through some certain critical values, the equilibrium will lose its stability and hopf bifurcation will take place; by adjusting K values, we achieve the purpose of chaos control. The research of this paper is a new investigation about the hopf bifurcation and chaos control on the finance system and has important theoretical and practical value.

2. Stability of Steady States and Bifurcations of Periodic Solutions

In this section, we investigate the effect of delay on the dynamic behavior of system (1.2). Obviously, when τ=0 , system (1.2) becomes the system (1.1). First, we introduce the following several lemmas in [1, 2] for T's system(1.1).

We know that under the assumption (_C1 ) , the system (1.1) has two equilibrium points: [figure omitted; refer to PDF]

The characteristic equation of the system (1.1) at _S1 (_S2 ) is [figure omitted; refer to PDF]

By analyzing the characteristic equation (2.2) and the Routh-Hurwitz criteria, we get the following.

Lemma 2.1.

For a<1 , the characteristic equation (2.2) has three eigenvalues with negative real parts, so two equilibrium points _S1 , _S2 of the system (1.1) are asymptotic stable.

Lemma 2.2.

For a=1 , the characteristic equation (2.2) has a pair of purely imaginary eigenvalues _λ1,2 =±i_ω0 (_ω0 =b) and a negative real eigenvalue _λ3 =-2 , and [figure omitted; refer to PDF] According to the hopf bifurcation theorem [11], a hopf bifurcation of the system (1.1) occurs at a=1 .

Lemma 2.3.

For a>1 , the characteristic equation (2.2) has one negative real root and one pair of conjugate complex roots with positive real parts, so two equilibrium points _S1 , _S2 of the system (1.1) are unstable.

Clearly, the delayed feedback control system (1.2) has the same equilibria to the corresponding system (1.1). In this section, we analyze the effect of delay on the stability of these steady states. Due to the symmetry of _S1 and _S2 , it is sufficient to analyze the stability of _S1 . By the linear transform [figure omitted; refer to PDF] system (1.2) becomes [figure omitted; refer to PDF] It is easy to see that the origin _S0 (0,0,0) is the equilibrium of system (2.5). The associated characteristic equation of system (2.5) at _S0 (0,0,0) is [figure omitted; refer to PDF] Expanding (2.6), we have [figure omitted; refer to PDF] Thus, we need to study the distribution of the roots of the third-degree exponential polynomial equation: [figure omitted; refer to PDF] where _ai ,_bi ∈R (i=0,1,2) and ^∑i=02bi2 ...0;0 . We first introduce the following simple result which was proved by Ruan and Wei [12] using Rouche's theorem.

Lemma 2.4.

Consider the exponential polynomial [figure omitted; refer to PDF] where _τi ...5;0 (i=1,2,...,m) and ^pj(i) (i=0,1,...,m, j=1,2,...,n) are constants. As (_τ1 ,_τ2 ,...,_τm ) vary, the sum of the order of the zeros of P(λ,^e-λ_τ1 ,...,^e-λ_τm ) on the open right half plane can change only if a zero appears on or crosses the imaginary axis.

Obviously, iω(ω>0) is a root of (2.8) if and only if ω satisfies [figure omitted; refer to PDF] Separating the real and imaginary parts, we have [figure omitted; refer to PDF] which is equivalent to [figure omitted; refer to PDF] Let z=^ω2 and denote p=^a22 -^b22 -2_a1 , q=^a12 -2_a0a2 -^b12 +2_b0b2 , r=^a02 -^b02 , then (2.12) becomes [figure omitted; refer to PDF]

In the following, we need to seek conditions under which (2.12) has at least one positive root. Denote [figure omitted; refer to PDF]

Therefor, applying [13], we obtain the following lemma.

Lemma 2.5.

For the polynomial equation (2.13), one has the following results.

(i) If r<0 , then (2.13) has at least one positive root.

(ii) If r...5;0 and Δ=^p2 -3q...4;0 , then (2.13) has no positive roots.

(iii): If r...5;0 and Δ=^p2 -3q>0 , then (2.13) has positive roots if and only if ^z1* =(1/3)(-p+Δ)>0 and h(^z1* )...4;0 .

Suppose that (2.13) has positive roots. Without loss of generality, we assume that it has three positive roots, defined by _z1 , _z2 , and _z3 , respectively. Then (2.12) has three positive roots: [figure omitted; refer to PDF] From (2.11), we have [figure omitted; refer to PDF] Thus, if we denote [figure omitted; refer to PDF] where k=1,2,3; j=0,1,2,... , then ±i_ωk is a pair of purely imaginary roots of (2.8) with ^τk(j) . Define [figure omitted; refer to PDF] Note that when τ=0 , (2.8) becomes [figure omitted; refer to PDF]

Therefor, applying Lemmas 2.4 and 2.5 to (2.8), we get the following lemma.

Lemma 2.6.

For (2.8), one has

(i) if r...5;0 and Δ=^p2 -3q...4;0 , then all roots with positive real parts of (2.8) have the same sum to those of the polynomial equation (2.19) for all τ...5;0 .

(ii) if either r<0 or r...5;0 , Δ=^p2 -3q>0 , ^z1* =(1/3)(-p+Δ)>0 and h(^z1* )...4;0 , then all roots with positive real parts of (2.8) have the same sum to those of the polynomial equation (2.19) for τ∈[0,_τ0 ) .

Let [figure omitted; refer to PDF] be the root of (2.8) near τ=^τk(j) satisfying [figure omitted; refer to PDF] Then by [13], we have the following transversality condition.

Lemma 2.7.

Suppose that _zk =^ωk2 and ^{h[variant prime]} (_zk )...0;0 . Then [figure omitted; refer to PDF] and Rλ(^τk(j) )/dτ and ^{h[variant prime]} (_zk ) have the same sign.

Now, we study the characteristic equation (2.7) of the system (2.5). Comparing (2.7) with (2.8), we know that [figure omitted; refer to PDF] Thus, [figure omitted; refer to PDF] and then we can compute [figure omitted; refer to PDF] When τ=0 , (2.7) becomes (2.2) [figure omitted; refer to PDF]

Applying Lemmas 2.1, 2.2, 2.6, and 2.7 to (2.7), we have the following theorems.

Theorem 2.8.

Let ^τk(j) and _τ0 be defined by (2.17) and (2.18). Suppose that conditions (_C1 ) and a<1 hold.

(i) If Δ...4;0 , then (2.7) had all roots with negative real parts for all τ...5;0 , and the equilibrium _S1 (or _S2 ) of the system (1.2) is stable.

(ii) If Δ>0, ^z1* >0 and h(^z1* )...4;0 , (2.7) had all roots with negative real parts for τ∈[0,_τ0 ) , and the equilibrium _S1 (or _S2 ) of the system (1.2) is stable.

(iii): If the conditions of (ii) are satisfied, and ^{h[variant prime]} (_zk )...0;0 , then system (1.2) exhibits the Hopf bifurcation at the equilibrium _S1 (or _S2 ) for τ=^τk(j) .

Theorem 2.9.

Let ^τk(j) and _τ0 are defined by (2.17) and (2.18). Suppose that conditions (_C1 ) and a>1 hold.

(i) If Δ...4;0 , then (2.7) had two roots with positive real parts for all τ...5;0 , and the equilibrium _S1 (or _S2 ) of the system (1.2) is unstable.

(ii) If Δ>0, ^z1* >0 and h(^z1* )...4;0 , (2.7) has two roots with positive real parts for τ∈[0,_τ0 ) , and the equilibrium _S1 (or _S2 ) of the system (1.2) is unstable.

(iii): If the conditions of (ii) are satisfied, and ^{h[variant prime]} (_zk )...0;0 , then system (1.2) exhibits the Hopf bifurcation at the equilibrium _S1 (or _S2 ) for τ=^τk(j) .

3. Direction and Stability of the Hopf Bifurcation

In the Section 2, we obtained some conditions which guarantee that the system (1.2) undergoes the Hopf bifurcation at a sequence values of τ . In this section, we shall study the direction and stability of the Hopf bifurcation. The method we used is based on the normal form theory and the center manifold theorem introduced by Hassard et al. [14]. Throughout this section, we always assume that system (1.2) undergoes Hopf bifurcations at the steady state (_x* ,_y* ,_z* ) for τ=_τk and then ±i_ωk is corresponding purely imaginary roots of the characteristic equation at the steady state (_x* ,_y* ,_z* ) .

Letting _x1 =x-_x* , _x2 =y-_y* , _x3 =z-_z* , _x¯i (t)=_xi (τt), τ=_τk +μ and dropping the bars for simplification of notations, system (1.2) is transformed into an FDE in C=C([-1,0],^R3 ) as [figure omitted; refer to PDF] where x(t)=^{(_x1 (t),_x2 (t),_x3 (t))T} ∈^R3 , and _Lμ :C[arrow right]R, f:R×C[arrow right]R are given, respectively, by [figure omitted; refer to PDF] By the Riesz representation theorem, there exists a function η(θ,μ) of bounded variation for θ∈[-1,0] , such that [figure omitted; refer to PDF] for [varphi]∈C[-1,0] .

In fact, we can choose [figure omitted; refer to PDF] where δ is the Dirac delta function. For [varphi]∈^C1 ([-1,0],^{(R3 )*} ) , define [figure omitted; refer to PDF] Then system (3.1) is equivalent to [figure omitted; refer to PDF] where _xt (θ)=x(t+θ) for θ∈[-1,0] .

For ψ∈^C1 ([0,1],^R3 ) , define [figure omitted; refer to PDF] and a bilinear inner product [figure omitted; refer to PDF] where η(θ)=η(θ,0) . Then A=A(0) and ^A* =^A* (0) are adjoins operators.

By the discussion in Section 2, we know that ±i_ωkτk are eigenvalues of A , thus they are also eigenvalues of ^A* .

By direct computation, we obtain that q(θ)=_q0^eiθ_ωkτk , with [figure omitted; refer to PDF] is the eigenvector of A corresponding to i_ωkτk , and ^q* (s)=D^{q0*eis_ωkτk} , with [figure omitted; refer to PDF] is the eigenvector of ^A* corresponding to -i_ωkτk , where [figure omitted; refer to PDF] Using the same notation as in [14], we compute the coordinates to describe the center manifold _C0 at μ=0 . Let _xt be the solution of (3.1) when μ=0 . Define [figure omitted; refer to PDF] On the center manifold _C0 , we have [figure omitted; refer to PDF]

where [figure omitted; refer to PDF] z and z¯ are local coordinates for center manifold _C0 in the direction of ^q* and ^q¯* . Note that W is real if _xt is real. We consider only real solutions. For the solution _xt ∈_C0 of (3.1), since μ=0 , we have [figure omitted; refer to PDF] We rewrite this equation as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Noticing [figure omitted; refer to PDF] we have [figure omitted; refer to PDF] Thus, form (3.17), we have [figure omitted; refer to PDF]

Comparing the coefficients of (3.17), we get [figure omitted; refer to PDF] Since there are _W20 (θ) and _W11 (θ) in _g21 , we need to compute them.

From (3.6) and (3.12), we have [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Expanding the above series and comparing the corresponding coefficients, we obtain [figure omitted; refer to PDF] From (3.22), we know that for θ∈[-1,0) , [figure omitted; refer to PDF] Comparing the coefficients with (3.23) gives that [figure omitted; refer to PDF] [figure omitted; refer to PDF] From (3.24), (3.26) and the definition of A , it follows that [figure omitted; refer to PDF] Notice that q(θ)=^{(1,α,β)Teiθ_ωkτk} , hence [figure omitted; refer to PDF] where _E1 =^{(E1(1) ,E1(2) ,E1(3) )T} ∈^R3 is a constant vector.

Similarly, from (3.24) and (3.27), we can obtain [figure omitted; refer to PDF] where _E2 =^{(E2(1) ,E2(2) ,E2(3) )T} ∈^R3 is also a constant vector.

In what follows, we shall seek appropriate _E1 and _E2 . From the definition of A and (3.24), we obtain [figure omitted; refer to PDF] [figure omitted; refer to PDF] where η(θ)=η(θ,0) . By (3.22), we have [figure omitted; refer to PDF] [figure omitted; refer to PDF] Substituting (3.29) and (3.33) into (3.31), we obtain [figure omitted; refer to PDF] which leads to [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Similarly, substituting (3.30) and (3.34) into (3.32), we can get [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Thus, we can determine _W20 (0) and _W11 (0) from (3.29) and (3.30). Furthermore, we can determine _g21 . Therefore, each _gij in (3.21) is determined by the parameters and delay in (3.1). Thus, we can compute the following values: [figure omitted; refer to PDF] which determine the quantities of bifurcating periodic solutions in the center manifold at the critical value _τk , that is, _μ2 determines the directions of the Hopf bifurcation: if _μ2 >0(_μ2 <0) then the Hopf bifurcation is supercritical (subcritical) and the bifurcating periodic solutions exist for τ>_τk (τ<_τk ) ; _β2 determines the stability of the bifurcating periodic solutions: the bifurcating periodic solutions are stable (unstable) if _β2 <0(_β2 >0) ; and _T2 determines the period of the bifurcating periodic solutions: the period increases (decreases) if _T2 >0(_T2 <0) .

4. Application to Control Chaos

In the present section, we apply the results in the previous sections to system (1.2) for the purpose of control of chaos. From Section 2, we know that under certain conditions, a family of periodic solutions bifurcate from the steady states of system (1.2) at some critical values of τ and the stability of the steady state maybe change along with increase of τ . If the bifurcating periodic solution is orbitally asymptotically stable or some steady state becomes local stable, then chaos may vanish. Following this ideal, we consider the following delayed feedback control system: [figure omitted; refer to PDF] which has two steady states _S+ =...(0.65,-0.65,0.5917), _S- =...(-0.65,0.65,0.5917) . Clearly, when τ=0 or K=0 , system (3.1) is chaotic (as depicted in Figure 1).

For the steady state _S+ or _S- , we have the corresponding characteristic equation of system (4.1) as follows: [figure omitted; refer to PDF] Clearly, when τ=0 , (4.2) has a negative root and a pair of complex roots with positive real parts. Following Section 2, we can obtain p=-2K-6.2839 , q=22.8488K-77.2289 , r=522.0677>0 , Δ=^p2 -3q>0 , and ^z1* =(1/3)(-p+Δ)>0 for all K∈R . When K<-0.1907 or K>12.107 , h(^z* )<0 . Thus, from Lemma 2.6 and Theorem 2.9, we know that (4.2) has roots with positive real parts. In particular, we have K=-1 , that is, [figure omitted; refer to PDF] In this case, we can compute [figure omitted; refer to PDF]

Thus, from Lemma 2.7, we have Reλ(^τ1(j) )/dτ>0 and Reλ(^τ2(j) )/dτ<0 . In addition, notice that [figure omitted; refer to PDF] Thus, from Theorem 2.8, we have the following conclusion about the stability of the steady states of system (4.3) and Hopf bifurcation.

5. Conclusion

Suppose that ^τk(j) , k=1,2; j=0,1,2,... is defined by (4.4).

(i) When τ∈[0,^τ2(0) )∪(^τ1(1) ,∞) , the steady states _S1 and _S2 of the system (4.1) are unstable (see Figures 2 and 3).

(ii) When τ∈(^τ2(0) ,^τ1(1) ) , the steady states _S1 and _S2 of the system (4.1) are asymptotically (see Figures 4 and 5).

(iii): When τ=^τk(j) , system (4.1) undergoes a Hopf bifurcation at the steady states states _S1 and _S2 .

Figure 2: Chaos still exists for K=-1, τ=0.2 .

[figure omitted; refer to PDF]

Figure 3: Chaos still exists for K=-1, τ=2.5 .

[figure omitted; refer to PDF]

Figure 4: When K=-1, τ=0.8 , chaos vanishes, and _S1 becomes local stable. Here initial value is (0.5, -0.5, 0.6) .

[figure omitted; refer to PDF]

Figure 5: When K=-1, τ=0.8 , chaos vanishes, and _S2 becomes local stable. Here initial value is (-0.5, 0.5, 0.6) .

[figure omitted; refer to PDF]

The above simulations indicate that when the steady state is stable or the bifurcating periodic solutions are orbitally asymptotically stable, chaos vanishes (see Figures 4-6).

Figure 6: When K=-1, τ=2 , chaos vanishes, and _S2 becomes a stable periodic solution. Here initial value is (5, -5, 5) .

[figure omitted; refer to PDF]

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