Substituting an exponential solution pδ(t)=eλtu into the linearized Equation (2) yields the corresponding characteristic equation (3)λ−α−β1e−λτ1−β2e−λτ2+β3e−λτ3=0

Dividing the left hand side of Equation (3) by λ − α and denote the result by a(λ), (4)a(λ)=1+a1(λ)e−λτ1+a2(λ)e−λτ2+a3(λ)e−λτ3=0 with a1(λ)=−β1λ−α, a2(λ)=−β2λ−α,a3(λ)=β3λ−α.

Since λ = 0 is not a solution of Equation (3), a pair of pure imaginary roots must exist if stability switch occurs. We thus assume λ = iω, ω > 0 is a solution of Equation (3). For λ = iω, a1(iω)=αβ1+iβ1ωα2+ω2 and |a1(iω)|=|β1|α2+ω2, a2(iω)=αβ2+iβ2ωα2+ω2 and |a2(iω)|=|β2|α2+ω2 and a3(iω)=−αβ3+iβ3ωα2+ω2 and |a1(iω)|=β3α2+ω2.

Each term of a(iω) may be viewed as a vector in the complex plane. Hence solving a(iω) = 0 analytically is equivalent to constructing a quadrangle geometrically as shown in Figure 2 in which these four vectors are arranged from head to tail.

The determination of the stability switching surface in the (τ₁, τ₂, τ₃) plane is challenging². In order to make the problem manageable, we reduce the three-delay equation to a two-delay equation by fixing the value of τ₃ at a certain positive level and then construct the stability switching curve in the (τ₁, τ₂) plane in which we can apply the method used earlier in Matsumoto and Szidarovszky [11], and Gu et al. [14]. Notice that vector 1 is connected to vector a3(iω)e-iωτ3 in Figure 2. If the sum of these vectors is denoted by a_d(iω, τ₃), ad(iω,τ3)=1+a3(iω)e−iωτ3, then a(λ) with λ = iω can be rewritten as a(iω)=ad(iω,τ3)+a1(iω)e−iωτ1+a2(iω)e−iωτ2.

As already shown in Figure 2, a(iω) = 0 means that these three vectors in a(iω) must form a triangle having the dotted base with interior angles, θ₁ and θ₂. Sufficient and necessary conditions for forming a triangle are given by (5)(i) fd(ω)=|a1(iω)|+|a2(iω)|−|ad(iω,τ3)|≥0,(ii) f1(ω)=|a2(iω)|+|ad(iω,τ3)|−|a1(iω)|≥0,(iii) f2(ω)=|ad(iω,τ3)|+|a1(iω)|−|a2(iω)|≥0.

Each inequality condition implies that the length of any segment of the triangle is not greater than the sum of the lengths of the remaining two segments. Define the crossing frequency set Ω of all ω > 0 such that a(iω) = 0 holds for at least one delay combination (τ₁, τ₂, τ₃) ≥ 0. For ω = 0 in a(iω) leads to a(iω)|ω = 0=1α(α+β1+β2−β3)<0.

This inequality implies 0 ∉ Ω.

By the law of cosine, we can determine the values of θ₁ and θ₂ of the triangle in Figure 2, θ1(ω)=cos−1[|ad(iω,τ3)|2+|a1(iω)|2−|a2(iω)|22|ad(iω,τ3)||a1(iω)|] and θ2(ω)=cos−1[|ad(iω,τ3)|2+|a2(iω)|2−|a1(iω)|22|ad(iω,τ3)||a2(iω)|].

Vertices A and B maybe located above or below the horizontal axis and the slope of segment AB can be positive or negative. So we have eight possibilities to construct a quadrangle. However, a simple geometric consideration shows that there are only two different possibilities: arg(a1(iω)e−iωτ1)+(2k−1)π−arg(ad(iω,τ1))±θ1(ω)=0 and arg(a2(iω)e−iωτ2)+(2n−1)π−arg(ad(iω,τ1))∓θ2(ω)=0.

Solving these equations for τ₁ and τ₂ yields the threshold values of the delays, (6)τ1±(ω,k)=1ω[arg(a1(iω))−arg(ad(iω,τ3))+(2k−1)π                       ± θ1(ω)] and (7)τ2∓(ω,n)=1ω[arg(a2(iω))−arg(ad(iω,τ3))+(2n−1)π                        ∓ θ2(ω)].

The stability switching curves consist of two sets of parametric segments, (8)L1(k,n)={τ1+(ω,k),τ2−(ω,n)} for k,n=0,1,2,... and (9)L2(k,n)={τ1−(ω,k),τ2+(ω,n)} for k,n=0,1,2,... where the parameter ω runs through the crossing frequency set Ω.

Substituting τ₁ = τ₂ = τ₃ = 0 into Equation (3) presents λ=α+β1+β2−β3=−fm<0.

This inequality implies that the steady state with no delays is stable. Next, assuming that τ₁ = τ₂ = 0, we examine the existence of a threshold value of τ₃ at which the system loses stability. Without loss of generality, assuming λ = iω, ω > 0 and then substituting it into Equation (3) with τ₁ = τ₂ = 0 reduce the characteristic equation to iω−(α+β1+β2)+β3e−iωτ3=0.

We denote α + β₁ + β₂ by Δ where Δ=f(1−m)(1−α1−α2−m1−m) and break down the characteristic equation into the real and imaginary parts, (10)−Δ+β3cosωτ3=0ω−β3sinωτ3=0.

Moving Δ and ω to the right hand side and adding the squared equations give ω2=β32−Δ2 where β3−Δ=fm>0 and β3+Δ=2f(1−m)(φ(α1)−α2) with φ(α1)=2−3m2(1−m)−α1.

Notice that the dotted curve in Figure 1 is described by α₂ = φ(α₁). It is clear first that β3+Δ⋛0 according to φ(α1)⋛α2 and second that β3+Δ<0 if m≥2/3.

This leads to the following result.

Proposition 1. Given τ₁ = τ₂ = 0, the solution of delay system (1) is stable for any τ₃ ≥ 0 if the fundamentalists dominate over the chartists in the sense that m ≥ 2/3.

If β₃ + Δ > 0, then there is a ω such as ω-=β32-Δ2>0. In this case, loss of stability for λ=iω- can be shown in the following way. The characteristic equation with τ₁ = τ₂ = 0 can be written as (11)β3e−λτ3=β1+β2+α−λ.

Taking λ as a function of τ₃ and differentiating it with respect to τ₃ yield dλdτ3=λβ3e−λτ31−τ3β3e−λτ3         =λ(β1+β2+α−λ)1−τ3(β1+β2+α−λ) where the right hand side of Equation (11) are used. Substituting λ = iω and taking the real part of the resulting expression give d[Reλ]dτ3|λ = iω=Re[λ(β1 + β2 + α − λ)1−τ3(β1 + β2 + α − λ)]=Re[ω2+iω(β1+β2)1−τ3(β1+β2+α)−iω1+τ3(β1+β2+α)−iω1+τ3(β1+β2+α)−iω]=ω2[1−τ3(β1+β2+α)]2+ω2>0.

This inequality implies that all roots cross the imaginary axis at iω from left to right as τ₃ increases so stability is lost. Correspondingly, solving the first equation of Equation (10) for τ₃ presents the threshold value (12)τ¯3(α1,α2,m)=1ω¯cos−1(Δβ3)>0 and the system is stable for τ3<τ-3 and unstable otherwise³. We summarize the results obtained so far.

Proposition 2. Given τ₁ = τ₂ = 0, the solution of delay system (1) is stable for τ3<τ-3(α1,α2,m), loses stability for τ3=τ-3(α1,α2,m) and becomes unstable for all τ3>τ-3(α1,α2,m) if the following condition holds, α2<φ(α1) whereas it is always stable for any τ₃ > 0 otherwise.

It is to be noticed that with τ₁ = τ₂ = 0, the three delay model is reduced to a one delay model and thus this stability result is essentially the same as the one of Qu and Wei [10]. In the next section we will proceed to the case of τ_i > 0 for i = 1, 2, 3.

We start with the τ₁-effect and present some numerical examples to see how different values of τ₁ affect dynamics of Equation (1). We first check the triangle conditions. Subtracting the third equation from the second equation Equation in (5) yields (13)f1(ω)−f2(ω)=6f(1−m)α2+ω2(α2−13) implying that the relative location of the curves of f₁(ω) and f₂(ω) depends on whether the value of α₂ is larger or less than 1/3. To simplify the analysis, we assume that the value of α₂ is kept constant at 1/3;

Assumption 4. α₂ = 1/3

Assumption 2 implies η_i > 0 for all i, that is, the weights of the delay prices to calculate the moving average are positive. Their relative magnitudes depend on the selected values of α_i and thus there are many possibilities. Among them, we will limit analysis to only two cases, η₁ = η₂ > η₃ if α₁ < 1/2 and η₃ > η₁ = η₂ if α₁ < 1/2 since Assumption 4 makes η₁ = η₂ always⁴. With Assumption 4, f₁(ω) = f₂(ω) holds regardless of the value of α₁. Further, f_i(ω) > 0, i = 1, 2, for ω > 0 which will be numerically confirmed below. Consequently, it is enough to verify only the location and the shapes of the f_d(ω) curve. We consider dynamics over an interval [α10,2/3] of α₁. The left hand side extreme value, α10≃0.356 is numerically determined so as to make the maximum of f_d(ω) equal to zero. Thus for α1<α10, f_d(ω) < 0 for all ω > 0, implying that one of the triangle conditions is always violated. The other extreme value, α₁ = 2/3, together with α₂ = 1/3 makes α₃ = 0 via α₂ = 1 − α₁. For α₁ > 2/3, the nonnegative condition α₃ ≥ 0 is violated. In the following numerical analysis, α₁ is increased from 6/15 to 9/15 with an increment of 1/15. The resultant four f_d(ω) curves are colored in black in Figure 3 where the red curve with α₁ = 1/2 and the blue curves with those extreme values are also depicted. The corresponding f₁(ω) = f₂(ω) curves are illustrated in the dotted curves with the same color and are seen to be positive for all ω ≥ 0. In the gray region below the lower blue curve, no stability switch occurs and thus the steady state is stable. We do not consider the τ₁-effect there. On the other hand, the gray region above the upper blue curve is not feasible and thus eliminated from consideration. Figure 3 also illustrates various shapes of the f_d(ω) curves and shows that increasing α₁ shifts the curve upward and removes its unevenness. Indeed, the lower blue curve with α1=α10 located at the bottom of the yellow region is high-wave shaped while the higher blue curve with α₁ = 2/3 at the top is monotonically downward-sloping. In addition, as the red curve passes through the origin, we then see that the curves below the red curve have two intersections with the horizontal axis and the curves above have only one intersection.

To be more specific, we pick up α₁ = 6/15 and construct a stability switching curve. The corresponding f_d(ω) curve is the black one just above the lower blue curve in Figure 3 and intersects the horizontal axis twice at ω¯1≃0.438 and ω¯2≃0.582.

The triangle conditions in Equation (5) hold for ω∈Ω=[ω-1,ω-2]. We now ready to derive the stability switching curves, L₁(k, n) and L₂(k, n) defined in (8) and (9). Taking k = n = 1, we obtain the points of τ1±(ω,1) and τ2∓(ω,1) by changing values of ω with increment 0.0002 inside the interval [ω-1,ω-2] and then plot these points in the (τ₁, τ₂) plane. In Figure 4A, segment L₂(1, 1) is illustrated as a real curve in the region above the diagonal and segment L₁(1, 1) is a dotted curve in the region below⁵. Two segments form an elliptical-shaped closed curve having the starting point S=(τ1+(ω-1,1),τ2-(ω-1,1)) and the ending point E=(τ1-(ω-2,1),τ2+(ω-2,1)) on the diagonal where, for i = 1, 2, τi±(ω¯1,1)=τs≃8.478 and τi±(ω¯2,1)=τe≃2.789.

Since the delays have two constraints, τ_i ≤ 5 for i = 1, 2 and τ₁ < τ₂ due to Assumptions 1 and 3, the feasible region should be above the diagonal line and subject to τ₂ ≤ 5. It is colored in yellow and further sub-divided into two sub-regions by the L₂(1, 1) segment that intersects two lines, one is the diagonal at (τ^e, τ^e) and the other is the horizontal line at τ₂ = 5⁶.

It is clear from the yellow region that stability is preserved for τ₁ and τ₂ such as 0<τ1<τe and τ1<τ2<τe. When the varying pair (τ₁, τ₂) with τ2>τe crosses the L₂(1, 1) segment, we have the stability switching as the real part of at least one eigenvalue turns to be positive and then the stationary point loses stability. To see this, we perform a simulation by increasing the value of τ₁ along the dotted horizontal line at τ₂ = 4 that crosses the L₂(1, 1) curve at point (τ1c,4) with τ1c≃1.640⁷. The numerical results are summarized in Figure 4B in which a bifurcation diagram with respect to τ₁ is depicted. For τ1≤τ1c, a pair of (τ₁, 4) stays within the stability region. So the stationary point p^* = f is stable and the corresponding part of the bifurcation diagram is a horizontal line at p(t) = f. Once τ1>τ1c, the pair is in the instability region above the L₂(1, 1) segment. The stationary point, therefore, loses stability and bifurcates to a limit cycle having two extreme values (i.e., maximum and minimum). The existence of the limit cycle is confirmed by the Hopf bifurcation theorem. Figure 4B further implies that the limit cycle becomes larger as τ₁ gets larger and the stability is never regained for τ₁ ≤ 4⁸. The τ₁-effect is summarized as follows:

Proposition 3. Stability of dynamic Equation (1) with respect to τ₁ depends on the selected value τ-2 of τ₂; If τ¯2≤τe,then the steady state is stable for τ1<τ¯2;If τ¯2>τe,then there is the threshold value τ1c such that thesteady state is stable for τ1<τ1c and loses stability forτ1≥τ1c. where τ1c is the τ₁-point of the intersection of the switching curve and the horizontal line at τ-2.

Taking various values of α₁ from the interval [α10,2/3] and constructing the stability switching curves, we detect the τ₁-effect more. In Figure 5A, the switching curves with various values of α₁ are illustrated and their end-points on the diagonal are denoted by the green dots. The upper most blue curve has α1=α10 and the lower most blue curve has α₁ = 2/3 while the curve shifts downward as α₁ increases from α10 to 2/3. As far as the curve is single-valued in τ₂, we have essentially the same result on the τ₁-effect as in the case of τ₁ = 6/15: fixing the value of τ₂ at some level and increasing the value of τ₁, the stability is preserved for τ₁ below the threshold value τ1c and it is lost for τ1>τ1c. As seen in Figure 5A, this threshold value becomes smaller as the value of α₁ becomes larger⁹, a larger value of α₁ reinforces the destabilizing τ₁-effect.

Taking τ₁ fixed, we are next concerned with the τ₂-effect, how changes in the value of τ₂ affect dynamics. We first return to Figure 4A in which both values of α₁ and α₂ are fixed at 6/15 and 1/3, respectively. The downward-sloping shape of the switching curve in the yellow region indicates that changing the value of τ₂ with constant τ₁ may generate similar effect to the τ₁-effect. Let τ1m denote the τ₁-value of the intersection of the stability switching curve and the horizontal line at τ₂ = 5¹⁰. If τ₁ is selected to be less than τ1m, then the steady state is stable for any τ₂ that should be above the diagonal and ≤5. If τ1m<τ1<τe, then the steady state is stable for τ2<τ2c and bifurcates to a limit cycle for τ2>τ2c where τ2c is defined in the same way as τ1c and depends on the selected value of τ₁. One more case exists and is specific to the τ₂-effect, namely, if τ1>τe, any feasible pair of τ₁ and τ₂ is in the unstable region so the steady state is unstable.

We now turn attention again to Figure 5A and consider the τ₂-effect when α₁ is increased but α₂ is still fixed at 1/3. It is already seen that the stationary point is stable in the region left to the stability switching curve and increasing the value of α₁ shifts the switching curve leftward. The stability switching curve with α = α₀ is not depicted as it is located outside Figure 5A. The shift of the curve implies that a larger value of α₁ reinforces the destabilizing τ₂-effect in the sense that it makes the stability region smaller. In addition, we have interesting dynamic phenomenon when the switching curve with α₁ being close to 2/3 has forward- and backward-bending segments. This shape suggests the stability loss and gain with respect to τ₂. If the initial point of (τ₁, τ₂) is selected on the diagonal such that the stationary point is stable, then the vertical line standing at this value of τ₁ denoted as τ10=0.85 crosses the stability switching curve three times at τ2a≃1.014, τ2b≃1.687 and τ2c≃3.359 as illustrated in Figure 5A. That is, the stability is lost at the first intersection and a limit cycle emerges. The cycle is getting larger and then smaller to merge the steady state at the value of the second intersection at which the stability is regained. Further increasing τ₂ crosses the stability switching curve again at which the stability is lost again. The corresponding bifurcation diagram is given in Figure 5B in which the stability losses and gains are shown. This τ₂-effect is summarized as follows:

Proposition 4. With a value of τ₁ being close to 2/3, the increasing delay τ₂ has a double edge effect in a sense that it can destabilize the market price to generate a limit cycle, then stabilize it and later destabilize it again.

We now look into the τ₃-effect, the effect caused by changing the value of τ₃. To this end, α₁ = (6 − 0.1)/12 is taken and dynamics with three different values of τ₃, τ₃ = 5, τ₃ = 6, and τ₃ = 7, are considered. The corresponding switching curves are illustrated in different colors in Figure 6A (that is, a blue curve with τ₃ = 5, red curves with τ₃ = 6, and black curves with τ₃ = 7). As τ₃ increases, it is observed first that the feasible regions of τ₁ and τ₂ increase since the upper bound of τ₂ becomes larger and second that the downward-sloping switching curve shifts upward, which implies an enlargement of the stability region. It is further observed that island-shaped switching curves emerge for τ₃ = 6 and τ₃ = 7 and the steady state is unstable inside them. To detect the reason why the island-shaped region is born under larger values of τ₃, we draw the green f₁(ω) = f₂(ω) curve and the red f_d(ω) curve for τ₃ = 7 in Figure 6B. It is seen that f₁(ω) = f₂(ω) > 0 for ω > 0 and the larger τ₃ value makes the second hump of the f_d(ω) curve high enough to cross over the horizontal line. As a result, the red curve intersects the horizontal line four times at ω1≃0.104,ω2≃0.562,ω3≃1.059 and ω4≃1.256.

In consequence, the triangle conditions (5) hold in the two intervals, [ω₁, ω₂] and [ω₃, ω₄]. It is verified that the switching curve defined on [ω₁, ω₂] is downward-sloping and the one on [ω₃, ω₄] forms an island shape. Both curves are colored in black. The existence of the smaller red islands indicates that the f_d(ω) curve also intersects the horizontal axis four times even when τ₃ = 6. A critical value of τ₃ for the birth of the island is somewhere between τ₃ = 5 and τ₃ = 6, for which the maximum of the second hump of f_d(ω) becomes zero.

Proposition 5. As far as τ₃ is relatively small, increasing the value of τ₃ stabilizes the steady state by shifting the downward-sloping switching curve upward whereas for a relatively larger value, increasing value of τ₃ additionally generates an island-shaped switching curve within which the steady state is unstable.

To see the τ₃-effect on dynamics, we perform two simulations. In the first simulation, we increase the value of τ₁ along the horizontal line at τ2=τ20(≃4.356). This line successively crosses the stability switching curves, the blue curve at τ1a(≃0.972), the red curve at τ1b(≃1.372) and the black curve at τ1c(=2). Although, these bifurcating values are denoted as black dots, they are not labeled in Figure 6A to avoid confusion. The corresponding bifurcation diagrams are illustrated in Figure 7A in which the black curve is depicted first, the red curve is then put on it and finally the blue curve is further placed upon. Each diagram has a qualitatively similar shape as the one shown in Figure 4B, that is, the steady state loses stability at the bifurcation value and a limit cycle emerges for a larger value of τ₁. Thus larger τ₃ value does not alter qualitative aspects of the τ₁-effect, however, does affect its quantitative aspects because the bifurcation value of τ₁ becomes larger as τ₃ increases. Increasing value of τ₃ enhances the stabilizing τ₁-effect by delaying the loss of stability.

In the second simulation, we shift the emphasis away from the τ₁-effect to the τ₂-effect. Fixing τ₃ = 7 and starting at point (2, 2) on the diagonal, we increase the value of τ₂ from 2 to 7 along the vertical real line at τ1=τ10(=2) in Figure 6A. The line crosses the lower black island-shaped curve at τ2a(≃2.563), the black downward sloping curve at τ2b(≃4.356) and then upper black circle-shaped curve at τ2c(≃6.283). These threshold values are not labelled in Figure 6A by the same reason as before. The simulation results are given in Figure 7B in which stability gain and loss occur in the following way:

Proposition 6. Increasing the value of τ₃ does not affect essentially dynamics with respect to τ₁ and can affect the dynamics with respect to τ₂ as it can generate multiple stability switching.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.