Dynamics of a delayed business cycle model with general investment function

Chaos, Solitons and Fractals 85 (2016) 110–119

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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

Dynamics of a delayed business cycle model with general investment function Driss Riad a, Khalid Hattaf a,b,∗, Noura Yousfi a a Department of Mathematics and Computer Science, Faculty of Sciences Ben M’sik, Hassan II University, P.O. Box 7955 Sidi Othman, Casablanca, Morocco b Centre Régional des Métiers de l’Education et de la Formation (CRMEF), Derb Ghalef, Casablanca 20340, Morocco

a r t i c l e

i n f o

Article history: Received 5 November 2015 Accepted 27 January 2016

Keywords: Business cycle Kaldor–Kalecki model Delay Hopf bifurcation

a b s t r a c t The aim of this paper is to study the dynamics of a delayed business cycle model with general investment function. The model describes the interaction of the gross product and capital stock. Furthermore, the delay represents the time between the decision of investment and implementation. Firstly, we show that the model is well posed by proving the global existence and boundedness of solutions. Secondly, we determine the economic equilibrium of the model. By analyzing the characteristic equation, we investigate the stability of the economic equilibrium and the local existence of Hopf bifurcation. Also, the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by applying the normal form method and center manifold theory. Moreover, the global existence of bifurcating periodic solutions is established by using the global Hopf bifurcation theory. Finally, our theoretical results are illustrated with some numerical simulations. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction Business cycle is an economic phenomenon also called economic cycle which is defined as the fluctuation of the macroeconomic variables caused by the instability of business systems. In the 19th century, several cyclical fluctuations were identified in economy. Therefore, many mathematical models have been developed in order to understand the dynamics of business cycle. One of the earliest of these models was introduced by Kaldor [1] which considered that the economic fluctuations are due to investment and he proposed the following nonlinear

∗ Corresponding author at: Centre Régional des Métiers de l’Education et de la Formation (CRMEF), 20340 Derb Ghalef, Casablanca, Morocco. Tel.: +212 664407825. E-mail address: [email protected], [email protected], [email protected] (K. Hattaf).

http://dx.doi.org/10.1016/j.chaos.2016.01.022 0960-0779/© 2016 Elsevier Ltd. All rights reserved.

system of differential equations

⎧ dY ⎨ = α [I (Y (t ), K (t )) − S(Y (t ), K (t ))], dt

⎩ dK = I (Y (t ), K (t )) − δ K (t ),

(1)

dt

where Y(t) and K(t) denote the gross product and capital stock at time t, respectively. The parameter α is the adjustment coefficient in the goods market and δ is the depreciation rate of the capital stock. Finally, the investment and saving functions are represented, respectively, by I(Y, K) and S(Y, K). On the other hand, Kalecki [2] introduced the idea that there is a time delay between the decision of investment and implementation. In 1999, Krawiec and Szydlowski [3] incorporated the idea of Kalecki into Kaldor model. Hence,

D. Riad et al. / Chaos, Solitons and Fractals 85 (2016) 110–119

system (1) becomes

⎧ dY ⎨ = α [I (Y (t ), K (t )) − S(Y (t ), K (t ))], dt

⎩ dK = I (Y (t − τ ), K (t )) − δ K (t ),

(2)

dt

where τ is the time delay between the decision of investment and implementation. The dynamical behavior of system (2) has been investigated by many authors. Krawiec and Szydlowski in [4,5] established the local stability and bifurcation of (2) for small time delay τ . Few years after, Zhang and Wei in [6] gave a sufficient conditions for local stability, bifurcations and global existence of periodic solutions of system (2). Furthermore, the dynamics of system (2) was studied by other researchers [7–9]. All the above studies are based on the assumptions that the investment and saving functions have the following forms: I (Y, K ) = I (Y ) + qK and S(Y, K ) = γ Y, where q < 0 and γ ∈ (0, 1) are constants. In this paper, we consider only the Keynesian saving function with the coefficient γ represents the propensity to save. Therefore, system (2) becomes

⎧ dY ⎨ = α [I (Y (t ), K (t )) − γ Y (t )], dt

⎩ dK = I (Y (t − τ ), K (t )) − δ K (t ).

111

Proof. From [10], we deduce that system (3) with initial data (φ 1 , φ 2 ) ∈ C admits a unique local solution on [0, Tmax ), where Tmax is the maximal existence time for solution of system (3). According to the second equation of system (3), we get

K (t ) = e−(δ +q¯ )t φ2 (0 ) +



¯ (ξ )]dξ . + qK

t 0

For economic considerations, we assume that the general investment function I(Y, K) is continuously differentiable in R2 with ∂∂YI > 0 and ∂∂KI < 0. The organization of our current paper is as follows. In the next section, firstly, we prove that model (3) is economically well-posed by showing the global existence and boundedness of solutions. After, we determine the economic equilibrium of the model. The local stability of economic equilibrium and the existence of Hopf bifurcation are investigated in Sections 3 and 4. The direction and stability of the Hopf bifurcation are determined in Section 5, by applying the normal form method and center manifold theory. Section 6 is devoted to global existence of bifurcating periodic solutions. The paper ends with an application and some numerical simulations to support our analytical results in Section 7. 2. Well-posedness and economic equilibrium In this section, we investigate the global existence and boundedness of solutions of system (3). Further, we determine the economic equilibrium of the model. For these reasons, we assume the general investment function I(Y, K) satisfies the following hypothesis: (H1 ) There exists two constants L > 0 and q¯ ≥ 0 such that ¯ | ≤ L for all Y, K ∈ R. |I (Y, K ) + qK Let C = C ([−τ , 0], R2 ) be the Banach space of continuous functions mapping the interval [−τ , 0] into R2 equipped with the sup-norm ϕ = sup−τ ≤θ ≤0 |ϕ (θ )| for ϕ ∈ C. Theorem 2.1. If assumption (H1 ) holds, then the system (3) with initial data (φ 1 , φ 2 ) ∈ C has a unique solution on [0, +∞ ). Moreover, this solution is uniformly bounded.

(4)

By (H1 ), we have



|K (t )| ≤ e−(δ+q¯ )t |φ2 (0 )| + L ≤ e−(δ +q¯ )t |φ2 (0 )| +

t

e−(δ +q¯ )(t−ξ ) dξ

0

L

δ + q¯

.

Since limt→+∞ e−(δ +q¯ )t |φ2 (0 )| = 0, we deduce that there exists a t0 > 0 such that e−(δ +q¯ )t |φ2 (0 )| ≤ 1 for all t ∈ [t0 , Tmax ). Hence K(t) is uniformly bounded for bound A1 = 1 + δ +L q¯ . Now, we prove the boundness of Y. From the first equation of system (3), we have

(3)

dt

e−(δ +q¯ )(t−ξ ) [I (Y (ξ − τ ), K (ξ ))

Y (t ) = Y (t0 )e−αγ (t−t0 ) + α



t

t0

e−αγ (t−ξ ) )I (Y (ξ ), K (ξ ))dξ ,

t ≥ t0 .

(5)

Then

|Y (t )| ≤ |Y (t0 )|e−αγ (t−t0 ) + α ≤ |Y (t0 )|e−αγ (t−t0 ) +



t

t0





e−αγ (t−ξ ) L + q¯ |K (ξ )| dξ

¯ 1 L + qA

γ

.

Similarly to above, we deduce that Y(t) is uniformly ¯ L+qA bounded for bound A2 = 1 + γ 1 . From the above and using the continuity of Y(t) and K(t), we conclude that Y(t) and K(t) are uniformly bounded [0, Tmax ). Therefore, Tmax = +∞. This completes the proof.  Next, we discuss the existence of equilibria of (3). In addition to (H1 ), we consider the following assumptions: (H2 ) I(0, 0) > 0; ∂I γ ∂I (Y, K ) − γ < − (Y, K ) for all (Y, K ) ∈ R2 . ∂Y δ ∂K

(H3 )

Theorem 2.2. If (H1 ) − (H3 ) hold, then system (3) has a γ unique economic equilibrium of the form E ∗ (Y ∗ , δ Y ∗ ), where Y∗ is the positive solution of the following equation:



I Y,

γ Y − γ Y = 0. δ

Proof. At any equilibrium, we have

I (Y, K ) − γ Y = 0, I (Y, K ) − δ K = 0. γ

γ

Hence, K = δ Y and I (Y, δ Y ) = γ Y . Define the function f on interval [0, +∞ ) by



f (Y ) = I Y,

γ Y − γ Y. δ

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D. Riad et al. / Chaos, Solitons and Fractals 85 (2016) 110–119

From (H1 )–(H3 ), lim f (Y ) = −∞ and

we

have

f (0 ) = I (0, 0 ) > 0,

Y →+∞

f (Y ) =

∂I γ ∂I −γ + < 0. ∂Y δ ∂K

Thus, there exists a unique economic equilibrium E ∗ = ∗ (Y , K ∗ ) with Y∗ is the positive solution of the equation γ f (Y ) = 0 and K ∗ = δ Y ∗ .  3. Stability analysis In this section, we investigate the local stability of the economic equilibrium E ∗ = (Y ∗ , K ∗ ). Let y = Y − Y ∗ and k = K − K ∗ . By substituting y and k into system (3) and linearizing we obtain the following system:

⎧ dy ⎨ = α [ay(t ) + β k(t ) − γ y(t )], dt

(6)

⎩ dk = ay(t − τ ) + β k(t ) − δ k(t ), dt

where a = ∂∂YI (Y ∗ , K ∗ ) and β = ∂∂KI (Y ∗ , K ∗ ). Then, the characteristic equation about the equilibrium E∗ is given by

λ2 − [α (a − γ ) + β − δ ]λ + α (a − γ )(β − δ ) − α aβ e−λτ = 0.

(7)

Obviously, we have the following result. Lemma 3.1. For τ = 0, the equilibrium E∗ is locally asymptotically stable if and only if



a − γ < min

 δ − β −aβ , . α δ−β

λ2 − [α (a − γ ) + β − δ ]λ + α (a − γ )(β − δ ) − α aβ = 0. (8) Thus, the roots of (8) have negative real parts if and only if

α (a − γ ) + β − δ < 0 and (a − γ )(β − δ ) − aβ > 0,

−ω2 − [α (a − γ ) + β − δ ]iω + α (a − γ )(β − δ )

α aβ e−iωτ .

(9)

Equating real parts and imaginary parts, we have

−ω[α (a − γ ) + β − δ ] = α aβ sin(ωτ ).

yields

X + [α (a − γ ) + (β − δ )2 ]X 2

2

+

2

α 2 [(a − γ )2 (β − δ )2 − a2 β 2 ] = 0,

(12)

which has no positive solution when |a − γ |(δ − β ) > −aβ . Thus, we get (i). For (ii), we consider the following function:

g( λ ) =

λ2 − [α (a − γ ) + β − δ ]λ + α (a − γ )(β − δ ) − α aβ e−λτ .

We have g(0 ) = α (a − γ )(β − δ ) − α aβ < 0 and limλ→+∞ f (λ ) = +∞. Consequently, g has a positive real root and the economic equilibrium E∗ is unstable for all τ ≥ 0. This completes the proof of the theorem.  4. Existence of Hopf bifurcation Now, we turn to the existence of local Hopf bifurcation analysis to show the existence of nontrivial periodic solutions for the system (3). We use the delay τ as a parameter of bifurcation. We view the solutions of (7) as functions of the bifurcation parameter τ . From the above analysis for the local stability of E∗ , we have the following result.

Proof. Assume that |γ − a|(δ − β ) < −aβ . It is easy to see that (11) has a unique positive root given by

ω0 =

√  12 2 − α 2 ( a − γ )2 − ( β − δ )2 , 2

where

τj =

Proof. First, we prove (i). Since (a − γ )(δ − β ) < aβ , δ > 0, a > 0 and β < 0, we have a − γ < 0. From Lemma 3.1, we deduce that E∗ is locally asymptotically stable for τ = 0. For τ = 0, we assume that iω with w > 0 is a purely imaginary root of (7). This is the case if and only if ω satisfies the following equation:

−ω2 + α (a − γ )(β − δ ) = α aβ cos(ωτ ),

(11)

(13)

= [α 2 ( a − γ )2 + ( β − δ )2 ] 2 − 4α 2 [ ( a − γ )2 ( β −

From (10), we get



(i) If (a − γ )(δ − β ) < aβ , then the economic equilibrium E∗ is locally asymptotically stable for all τ ≥ 0. (ii) If (a − γ )(δ − β ) > −aβ , then E∗ is unstable for all τ ≥ 0.



Letting X =

ω2

δ )2 − a2 β 2 ].

Theorem 3.2.

=

ω 4 + [α 2 ( a − γ ) 2 + ( β − δ ) 2 ]ω 2 + α 2 [(a − γ )2 (β − δ )2 − a2 β 2 ] = 0.

Lemma 4.1. If |γ − a|(δ − β ) < −aβ , then Eq. (7) has a unique pair of purely imaginary roots.

Proof. For τ = 0, Eq. (7) becomes

β which is equivalent to a − γ < min{ δ −αβ , δ−a }. −β

Squaring and adding both equations of (10), we obtain

(10)

1

ω0

arccos

 −ω2 + α (a − γ )(β − δ ) α aβ

+

2 jπ

ω0

,

(14)

where j ∈ N = {0, 1, 2, ...}. Therefore, ± iω0 is a pair of purely imaginary roots of (7) with τ = τ j .  Now, we consider the following assumption: β (H4 ) δa−ββ < a − γ < min{ δ −αβ , δ−a }. −β

Theorem 4.2. If (H4 ) holds, system (3) undergoes Hopf bifurcation at E∗ when τ = τ j , j ∈ N. Further, the economic equilibrium E∗ is locally asymptotically stable when τ ∈ [0, τ 0 ) and unstable when τ > τ 0 . Proof. Let λ(τ ) = μ(τ ) + iω (τ ) the root of (7) such that μ(τ j ) = 0 and ω (τ j ) = ω0 . Substituting λ(τ ) into (7) and taking the derivative with respect to τ , we have

[2λ − α (a − γ ) + δ − β + α aβτ e−λτ ]

dλ = −α aβλe−λτ . dτ

D. Riad et al. / Chaos, Solitons and Fractals 85 (2016) 110–119

1 3 (y a30 + 3y2 ka21 + 3yk2 a12 + k3 a03 ) 6 + O((y, k )4 ).

Then

+

 dλ −1 dτ

τ λ 2λ − α ( a − γ ) + δ − β = −λ3 + [α (a − γ ) + β − δ ]λ2 − α (a − γ )(β − δ )λ τ − . λ =

113

2λ − α ( a − γ ) + δ − β − −α aβλe−λτ

Hence, system (3) becomes a functional differential equation in C = C ([−1, 0], R2 ) as

u˙ (t ) = Lμ (ut ) + f (μ, ut ),

(15)

where u(t ) = (y(t ), k(t ))T ∈ R2 , Lμ : C → R2 , f : R × C → R2 , and

Hence

 dλ −1

Re

dτ

λ=iω0

= Re =



2iω0 − α (a − γ ) + δ − β [−α (a − γ ) + δ − β ]ω02 + iω0 [ω02 − α (a − γ )(β − δ )]

α 2 (a − γ )2 + (δ − β )2 + 2ω02

[−α (a − γ ) + δ − β ]ω02 + [ω02 − α (a − γ )(β − δ )]2

Therefore, the transversally condition holds. From Lemma 4.1 and applying the Hopf bifurcation theorem for functional differential equations [11, Chapter 11, Theorem 1.1], we deduce that system (3) undergoes Hopf bifurcation at E∗ when τ = τ j , j ∈ N. From Lemma 3.1, we have E∗ is locally asymptotically stable for τ = 0. So all roots of (7) have negative real part for τ = 0. While τ 0 is the minimum of τ j at which the real parts of these roots are zero. Then E∗ is locally asymptotically stable when τ ∈ [0, τ 0 ) and unstable when τ > τ 0. 

5. Direction and stability of the Hopf bifurcation In this section, we will investigate the direction and stability of the Hopf bifurcation by using the normal form method and center manifold theorem introduced by Hassard et al. [12]. We always assume that system (3) undergoes a Hopf bifurcation at the economic equilibrium E ∗ = (Y ∗ , K ∗ ) for τ = τ j , and then ± iω0 is corresponding purely imaginary roots of the characteristic equation at E∗ . Let y(t ) = Y (τ t ) − Y ∗ , k(t ) = K (τ t ) − K ∗ and τ = τ˜ + μ. Then system (3) can be rewritten as

⎧ dy    ⎪ ⎪ ⎨ dt = α (τ˜ + μ ) (a − γ )y(t ) + β k(t ) + N y(t ), k(t ) ,  dk = (τ˜ + μ ) ay(t − 1 ) + (β − δ )k(t ) ⎪ ⎪   ⎩ dt + N y(t − 1 ), k(t ) ,



> 0.

Lμ (φ ) = (τ˜ + μ )

 +

0 a



α (a − γ )



0 0

0

φ1 (−1 ) φ2 (−1 )

N (y, k ) =



1 2 ∂ 2I ∗ ∂ 2I ∂ 2I y (E ) + 2yk (E ∗ ) + k2 2 (E ∗ ) 2 ∂Y ∂ K ∂Y 2 ∂K



1 3 ∂ 3I ∗ ∂ 3I + y (E ) + 3y2 k 2 (E ∗ ) 3 6 ∂Y ∂Y ∂ K

∂ 3I ∂ 3I (E ∗ ) + k3 3 (E ∗ ) 2 ∂Y ∂ K ∂K 4 + O((y, k ) ). + 3yk2

=

def

1 2 (y a20 + 2yka11 + k2 a02 ) 2



  φ1 ( 0 ) φ2 ( 0 )

,

(16)

  αN φ1 (0 ), φ2 (0 ) f (μ, φ ) = (τ˜ + μ ) , N φ1 (−1 ), φ2 (0 ) 

(17)

with φ (θ ) = (φ1 (θ ), φ2 (θ ))T ∈ C. By the Riesz representation theorem, there exists a function η(θ , μ) of bounded variation for θ ∈ [−1, 0], such that

Lμ ( φ ) =



0 −1

d η ( θ , μ )φ ( θ ),

for

φ ∈ C.

In view of Eq. (16), we can choose

η (θ , μ ) = (τ˜ + μ )

 α (a − γ ) 

0

0 + (τ˜ + μ ) a

0 0



 αβ δ (θ ) β −δ

δ ( θ + 1 ),

where δ is the Dirac delta function. For φ ∈ C 1 ([−1, 0], R2 ), define

⎧ d φ (θ ) ⎪ ⎨ , dθ A ( μ )φ =  0 ⎪ ⎩ dη (s, μ )φ (s ), −1

where

αβ β −δ 

and

R ( μ )φ =



0, f ( μ, φ ) ,

θ ∈ [−1, 0 ), θ = 0,

θ ∈ [−1, 0 ), θ = 0.

Then system (15) can be transformed into the following operator equation:

u˙ t = A(μ )ut + R(μ )ut , where ut (θ ) = u(t + θ ) for θ ∈ [−1, 0].

(18)

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D. Riad et al. / Chaos, Solitons and Fractals 85 (2016) 110–119

For ψ ∈ C 1 ([0, 1], (R2 )∗ ), define

A

∗

ψ (s ) =

Hence,

⎧ d ψ (s ) ⎪ ⎨− ,

s ∈ (0, 1],

⎪ ⎩

s = 0,



ds

0 −1

dηT (t, 0 )ψ (−t ),

¯ θ ), W˙ = A(0 )W + H (z, z, where

¯ θ ) = H20 (θ ) H (z, z,

and a bilinear inner product

ψ , φ = ψ¯ (0 )φ (0 ) −



0 −1

 θ ξ =0

ψ¯ (ξ − θ )dη (θ )φ (ξ )dξ , (19)

where η (θ ) = η (θ , 0 ). Then A(0) and are adjoint operators. Let q(θ ) and q∗ (s) are the eigenvector for A(0) and A∗ associated to the eigenvalue iω0 τ˜ and −iω0 τ˜ , respectively. By a simple computation, we have A∗

q (s ) = D (

q∗1 , 1

)e

iω0 τ˜ s

(A(0 ) − 2iω0 τ˜ I )W20 (θ ) = −H20 (θ ), A(0 )W11 (θ ) = −H11 (θ ).

iω0 − α (a − γ )

By ut = (yt , kt = W (t, θ ) + zq(θ ) + z¯q¯ (θ ) and q(θ ) = (1, q1 )T eiω0 τ˜ θ , we get (1 ) y(t ) = z + z¯ + W20 (0 )

,

,

−αβ α (a − γ ) + (β − δ ) + 2iω0 − aαβ τ˜ eiω0 τ˜

z(t ) = q∗ , ut , W (t, θ ) = ut (θ ) − 2Re{z(t )q(θ )}.

z¯2 + ··· , 2 z2 (2 ) k(t ) = q1 z + q¯ 1 z¯ + W20 (0 ) + W11(2) (0 )zz¯ 2 2 ¯ z (2 ) (0 ) + · · · . + W02 2 g(z, z¯ ) = q¯ ∗ (0 ) f0 (z, z¯ ) = q¯ ∗ (0 ) f0 (0, ut )



   α N y(t ), k(t )  N y(t − 1 ), k(t )   a20 a02 + y(t )k(t )a11 + k2 (t ) = τ˜ D¯ α q¯∗1 y2 (t ) = τ˜ q¯ ∗ (0 )

(20)

On the center manifold C0 , we have

¯ θ) W (t, θ ) = W (z, z,

+

z2 z¯2 + W11 (θ )zz¯ + W02 (θ ) + · · · , 2 2

+

where z and z¯ are local coordinates for center manifold C0 in the direction of q∗ and q¯ ∗ . Note that W is real if ut is real. We consider only real solutions. For the solution ut ∈ C0 of (18) with μ = 0, we have



¯ 0 ) + 2Re{zq(0 )} z˙ (t ) = iω0 τ˜ z(t ) + q¯ ∗ (0 ) f 0, W (z, z,

¯ α q¯∗ (a20 + 2q1 a11 + q21 a02 ) + e−2iω0 τ˜ a20 g20 = τ˜ D[ 1

z˙ (t ) = iω0 τ˜ z(t ) + g(z, z¯ ),

(21)

where

g(z, z¯ ) = q¯ ∗ (0 ) f0 (z, z¯ ) z2 z2 z¯ z¯2 + g11 zz¯ + g02 + g21 + ··· . 2 2 2

+ 2q1 e−iω0 τ˜ a11 + q21 a02 ],



+ (q1 eiω0 τ˜ + q¯1 e−iω0 τ˜ )a11 + q1 q¯1 a02 ], ¯ α q¯∗ (a20 + 2q¯1 a11 + q¯1 2 a02 ) + e2iω0 τ˜ a20 = τ˜ D[ 1 + 2q¯1 eiω0 τ˜ a11 + q¯1 2 a02 ],

  (1 ) (0 ) a20 α q¯∗1 2W11(1) (0 ) + W20   (2 ) (2 ) + 2q1W11 (0 ) + q¯1W20 (0 ) a02 + a30  (2 ) (2 ) (1 ) + 2W11 (0 ) + W20 (0 ) + q¯1W20 (0 )  (1 ) + 2q1W11 (0 ) a11 + (q¯1 + 2q1 )a21

g21 = τ˜ D¯

˙ − z¯˙ q¯ W˙ = u˙ t − zq

θ ∈ [−1, 0 ) θ = 0,



¯ α q¯∗ a20 + (q1 + q¯1 )a11 + q1 q¯1 a02 + a20 g11 = τ˜ D[ 1 g02

(22)

From (18) and (20), we have

A(0 )W − 2Re{q¯ ∗ (0 ) f0 q(θ )}, A(0 )W − 2Re{q¯ ∗ (0 ) f0 q(θ )} + f0 ,

+

2 2 a a a 30 21 12 + y2 (t )k(t ) + y(t )k2 (t ) y3 (t ) 6 2 2  a03 a20 3 2 k (t ) + y(t − 1 )k(t )a11 + y (t − 1 ) 6 2 a02 a30 a21 + y3 (t − 1 ) + y2 (t − 1 )k(t ) k2 (t ) 2 6 2  a12 a03 2 3 y(t − 1 )k (t ) + k (t ) . 2 6

The coefficients in (22) are

We rewrite this equation as



+



= iω0 τ˜ z(t ) + q¯ ∗ (0 ) f0 (z, z¯ ).

=

z2 (1 ) + W11 (−1 )zz¯ 2

It follows from (21) and (22) that:

Next, we use the same notations as in [12], and compute the center manifold C0 at μ = 0. Let ut be the solution of Eq. (18) for μ = 0, and define

= g20

z¯2 + ··· , 2

(1 ) (−1 ) + W02

Using the normalization condition, i.e., q∗ (s ), q(θ ) = 1, we get

= W20 (θ )

z2 (1 ) + W11 (0 )zz¯ 2

(1 ) y(t − 1 ) = e−iω0 τ˜ z + eiω0 τ˜ z¯ + W20 (−1 )

αβ −i ω − (β − δ ) 0 . q∗1 = αβ

D=

(25)

)T

(1 ) (0 ) + W02

where

q1 =

z2 z¯2 + H11 (θ )zz¯ + H02 (θ ) + · · · . (24) 2 2

Substituting the expression into (23) and comparing the coefficients, we have

q(θ ) = (1, q1 )T eiω0 τ˜ θ , ∗

(23)



D. Riad et al. / Chaos, Solitons and Fractals 85 (2016) 110–119

+ (q21 + 2q1 q¯1 )a12 + q21 q¯1 a03





(1 ) (1 ) + 2W11 (−1 )e−iω0 τ˜ + W20 (−1 )e



(2 )

+ 2W11 (0 )e (2 )

+ W20 (0 )e



−iω0 τ˜

iω0 τ˜

 iω τ˜ 0

and

H11 (0 ) = −g11 q(0 ) − g¯ 11 q¯ (0 )

a20

 α a20 + α (q1 + q¯1 )a11 + α q1 q¯1 a02 + τ˜ . a20 + (q¯1 e−iω0 τ˜ + q1 eiω0 τ˜ )a11 + q1 q¯1 a02 

(1 )

+ 2q1W11 (−1 )



(1 ) + q¯1W20 (−1 ) a11

(30)



(2 ) + q¯1W20 (0 ) + 2q1W11(2) (0 ) a02 + e−iω0 τ˜ a30





+ q¯1 e−2iω0 τ˜ + 2q1 a21



+ 2q1 q¯1 e−iω0 τ˜ + q21 e

 iω τ˜ 0

 τ˜



a12 + q21 q¯1 a03 .



α (a − γ ) 0



  αβ 0 W20 (0 ) + τ˜ a β −δ

0 W20 (−1 ) 0 (31)

  αβ 0 W11 (0 ) + τ˜ a β −δ



0 W11 (−1 ) 0

= −H11 (0 ).

2

z 2 − (g11 q(θ ) + g¯ 11 q¯ (θ ))zz¯ + · · · ,



(32)

Substituting (27) into (31) and noting that

α (a − γ ) − iω0 ae−iτ˜ ω0

which on comparing the coefficients with (24) gives

τ˜

(26)

From (25), (26) and the definition of A, we get

W˙ 20 (θ ) = 2iω0 τ˜ W20 (θ ) + g20 q(θ ) + g¯ 20 q¯ (θ ).



we have

α (a − γ ) − 2iω0

ig20 ig¯ 20 q(0 )eiω0 τ˜ θ + q¯ (0 )e−iω0 τ˜ θ ω0 τ˜ 3ω0 τ˜

α (a − γ ) − 2iω0

=− (27)

(34)

Substituting (29) into this, we obtain



+ E1 e2iω0 τ˜ θ ,

(33)

 αβ E ae−2iτ˜ ω0 β − δ − 2iω0 1 = −g20 q(0 ) − g¯ 02 q¯ (0 ) − H20 (0 ).



H11 (θ ) = −g11 q(θ ) − g¯ 11 q¯ (θ ).

 αβ q(0 ) = 0, β − δ − iω0

we get

H20 (θ ) = −g20 q(θ ) − g¯ 02 q¯ (θ ),

W20 (θ ) =

0

τ˜

= −g(z, z¯ )q(θ ) − g¯ (z, z¯ )q¯ (θ )

q(0 )eiω0 τ˜ θ ,

α (a − γ )

and

¯ θ ) = −q¯ ∗ (0 ) f0 q(θ ) − q∗ (0 ) f¯0 q¯ (θ ) H (z, z, = −(g20 q(θ ) + g¯ 02 q¯ (θ ))

According to (25) and the definition of A, we obtain

= 2iω0 τ˜ W20 (0 ) − H20 (0 ),

In order to compute g21 , we need to compute W11 (θ ) and W20 (θ ). For θ ∈ [−1, 0 ), we have

Since q(θ ) =

115

ae−2iτ˜ ω0

 αβ E β − δ − 2iω0 1

α a20 + 2α q1 a11 + α q21 a02

e−2iω0 τ˜ a20 + 2q1 e−iω0 τ˜ a11 + q21 a02

 ,

Hence,

α (a20 + 2q1 a11 + q21 a02 )(β − δ − 2iω0 ) − αβ (e−2iτ˜ ω0 a20 + 2q1 e−iτ˜ ω0 a11 + q21 a02 ) , αβ ae−2iω0 τ˜ − (β − δ − 2iω0 )(α (a − γ ) − 2iω0 ) (α (a − γ ) − 2iω0 )(e−2iτ˜ ω0 a20 + 2q1 e−iτ˜ ω0 a11 + q21 a02 ) − aα e−2iτ˜ ω0 (a20 + 2q1 a11 + q21 a02 ) = . αβ ae−2iω0 τ˜ − (β − δ − 2iω0 )(α (a − γ ) − 2iω0 )

E1(1) = E1(2)

where E1 = (E1(1 ) , E1(2 ) )T ∈ R2 is a constant vector. Similarly, we have



E2(1) = E2(2) =

Using the same technique to compute E1 , we get



−δ a20 + q1 (β − β eiω0 τ˜ − δ ) + q¯1 (β − β e−iω0 τ˜ − δ ) a11 − δ q1 q¯1 a02 aβ − (β − δ )(a − γ )





−γ a20 + (a − γ )(q¯1 e−iω0 τ˜ + q1 eiω0 τ˜ ) − a(q1 + q¯1 ) a11 − γ a02 aβ − (β − δ )(a − γ )

W11 (θ ) = −

ig11 ig¯ 11 q(0 )eiω0 τ˜ θ + q¯ (0 )e−iω0 τ˜ θ + E2 , ω0 τ˜ ω0 τ˜



+ τ˜

α a20 + 2α q1 a11 + α q21 a02

e−2iω0 τ˜ a20 + 2q1 e−iω0 τ˜ a11 + q21 a02

 ,

.

Thus, we can compute the following quantities: (28)

where E2 = (E2(1 ) , E2(2 ) )T ∈ R2 is a constant vector. In the following, we will determine the values of E1 and E2 . ¯ 0 ) = −2Re{q¯ ∗ (0 ) f0 q(0 )} + f0 , we have From H (z, z,

H20 (0 ) = −g20 q(0 ) − g¯ 02 q¯ (0 )

,

c1 (0 ) =

μ2 β2



g20 g11 − 2|g11 |2 −

2ω0 τ˜ Re{c1 (0 )} =− , Re{λ (τ˜ )} = 2Re{c1 (0 )},

T2 = − (29)

i

|g02 |2 3

Im{c1 (0 )} + μ2 Im{λ (τ˜ )} . ω0 τ˜

We arrive at the following theorem.

+

g21 , 2

116

D. Riad et al. / Chaos, Solitons and Fractals 85 (2016) 110–119

Theorem 5.1. Assume that conditions of Theorem 4.2 hold. (i) The direction of the Hopf bifurcation is determined by the sign of μ2 : if μ2 > 0, then it is a supercritical bifurcation and the bifurcating periodic solutions exist for τ > τ˜ ; if μ2 < 0, then it is a subcritical bifurcation and the bifurcating periodic solutions exist for τ < τ˜ . (ii) The stability of the bifurcating periodic solution is determined by β 2 : the periodic solution is stable if β 2 < 0, and it is unstable if β 2 > 0. (iii) The period of bifurcating periodic solutions is determined by T2 : the period increases if T2 > 0, and it decreases if T2 < 0.

∂P ∂Q + < 0. ∂Y ∂ K From the Bendixson criterion [14], we deduce that system (36) has no periodic solution. This ends the proof.  Theorem 6.3. Suppose that hypotheses (H1 ), (H4 ) and (H5 ) are satisfied. Then system (3) has at least j + 1 periodic solutions for each τ > τ j ( j = 1, 2, 3, ... ). Proof. Obviously, (E∗ , τ , p) is the only stationary solution of system (35) and the corresponding characteristic function

(E ∗ ,τ ,p) (λ ) = λ2 − [α (a − γ ) + β − δ ]λ +

6. Global existence of periodic solutions It is very important to note that the periodic solutions established by Theorem 4.2 only exit in a small neighborhood of the critical value τ j . In this section, we will study the global existence of periodic solutions bifurcating from the economic equilibrium E∗ by using the global bifurcation theorem due to Wu [13]. For simplifications of the notations, setting z(t ) = (Y (t ), K (t ))T , we can rewrite the model (3) as the following functional differential equations:

dz = F (zt , τ , p), dt

(35)

where zt (θ ) = z(t + θ ) ∈ C ([−τ , 0], R2 ). Following the work of Wu [13], we need to define X = C ([−τ , 0], R2 ),  = Cl {(z, τ , p) ∈ X × R × R+ : z is a p-periodic solution of system (35)}, and let l(E ∗ ,τ , 2π ) denote the connected j

ω0

component of (E ∗ , τ j , 2ωπ ) in  , where the expressions for 0 τ j and ω0 are given by (14) and (13). From Theorem 2.1, it is easy to have the following. Lemma 6.1. If assumptions (H1 ) and (H4 ) hold, then all the periodic solutions of system (3) are uniformly bounded. Next, we assume that (H5 )

∂I (Y, K ) − γ < ∂Y

δ 1 α − α

α (a − γ )(β − δ ) − α aβ e−λτ .

¯ τ¯ , p¯ ) is called a center if F (z, ¯ τ¯ , p¯ ) = 0 We recall that (z, ¯ τ¯ , p¯ ) is said to be isoand (z,¯ τ¯ , p¯ ) ( 2pπ i ) = 0. A center (z, lated if it is the only center in some neighborhood of ¯ τ¯ , p¯ ). (z, From Theorem 4.2, we can easily verify that (E ∗ , τ j , 2ωπ ) 0 is the only isolated center and l(E ∗ ,τ , 2π ) is non-empty. j

ω0

Now, it is sufficient to prove that the projection of l(E ∗ ,τ , 2π ) onto τ -space is [τ , +∞ ) for each j > 0, where j

ω0

τ ≤ τ j . (E ∗ , τ j , 2ωπ0 ) is an isolated center and from the implicit function theorem, there exist  > 0, σ > 0 and a smooth curve λ : (τ j − σ , τ j + σ ) −→ C such that (λ(τ )) = 0, |λ(τ ) − iω0 | <  for all τ ∈ [τ j − σ , τ j + σ ] and

dReλ(τ ) > 0.

dτ τ =τ j

λ(τ j ) = iω0 , Let

  2π

 , ω2π = (u, p) : 0 < u <  , p − < .

ω0 0 It is easy to verify that on [τ j − σ , τ j + σ ] × ∂  , 2π , 

(E ∗ ,τ ,p) u + i

∂I (Y, K ) for all (Y, K ) ∈ R2 . ∂K

2π

τ = τ j, p =

p 2π

ω0

= 0 if and only if u = 0,

ω0

Lemma 6.2. If assumptions (H4 ) and (H5 ) hold, system (3) has no nonconstant periodic solution with period τ .

Therefore, the hypotheses (A1 )û(A4 ) given in [13] are satisfied. Moreover, if we define

Proof. Suppose for a contradiction that system (3) has nonconstant periodic solution with period τ . Then the following system of ordinary differential equations:

H± E∗ , τ j ,

⎧ dY ⎪ ⎨ = α [I (Y (t ), K (t )) − γ Y (t )], dt ⎪ ⎩ dK = I (Y (t ), K (t )) − δ K (t ), dt

(36)

has a non-constant periodic solution. Let (P(Y, K), Q(x, y)) denote the vector field of (36), then we have

  ∂P ∂Q ∂I ∂I + =α −γ + − δ. ∂Y ∂ K ∂Y ∂K

According to (H5 ), we get

(37)



2π



ω0

 2π (u, p) = (E ∗ ,τ j ±σ ,p) u + i , p

then we have the crossing number of the isolated center (E ∗ , τ j , 2ωπ ) as follows:



0

 E∗, τ j ,

2π

ω0







= degB H − E ∗ , τ j ,





2π

ω0

− degB H + E ∗ , τ j ,



,  , 2 π

2π

ω0





ω0

,  , 2 π



ω0

= −1, where degB denotes the classical Brouwer degree. Hence, it follows from Theorem 3.3 of Wu [13] that the connected component l(E ∗ ,τ , 2π ) through (E ∗ , τ j , 2ωπ ) in  is j

unbounded.

ω0

0

D. Riad et al. / Chaos, Solitons and Fractals 85 (2016) 110–119

117

1 delay=3 delay=10 delay=15 delay=20

0.9

0.8

0.7

Y(t)

0.6

0.5

0.4

0.3

0.2

0.1

0

0

50

100

150

200

250

300

t 2.5 delay=3 delay=10 delay=15 delay=20

2.4

2.3

K(t)

2.2

2.1

2

1.9

1.8

1.7

0

50

100

150

200

250

300

t Fig. 1. The curves of Y(t) and K(t) under different values of delay.

It follows from the definition of τ j , that τ j > 2ωπ for 0 each j ≥ 1. Now we prove that the projection of l(E ∗ ,τ , 2π ) j

ω0

onto τ -space is [τ , +∞ ), where τ ≤ τ j . It is shown that system (3) with τ = 0 has no nontrivial periodic solution under the conditions stated in Lemma 6.2. Hence, the projection l(E ∗ ,τ , 2π ) onto τ -space does not include any negj

ω0

ative number. For a contradiction, we suppose that the projection of l(E ∗ ,τ , 2π ) onto τ -space is bounded, which j ω 0

means that the projection of l(E ∗ ,τ , 2π ) onto τ -space is inj ω 0

cluded in a interval (0, τ ∗ ). Noticing 2ωπ < τ j and applying 0 Lemma 6.2, we have 0 < p < τ ∗ for (z(t), τ , p) belonging to l(E ∗ ,τ , 2π ) . By applying Lemma 6.2, we get that the conj

ω0

nected component l(E ∗ ,τ , 2π ) is bounded. This contradicts j ω0 our conclusion that l(E ∗ ,τ , 2π ) is unbounded. Consequently, j

ω0

the projection of l(E ∗ ,τ , 2π ) onto τ -space is unbounded, i.e., j ω 0

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D. Riad et al. / Chaos, Solitons and Fractals 85 (2016) 110–119 2.5

3 2.8

K(t)

Y(t)

2

1.5

2.6 2.4

1

0.5

2.2

0

200

400

600

800

2

1000

0

200

400

t Phase diagram

600

800

1000

800

1000

t

3 2.8

K

2.6 2.4 2.2 2 0.5

1

1.5

2

2.5

Y

4

3.5

3

3

2

2.5

K(t)

Y(t)

Fig. 2. Asymptotical stability of the economic equilibrium E∗ with τ = 5.3.

1

2

0

1.5

−1

0

200

400

600

800

1000

1

0

200

t Phase diagram

400

600

t

3.5 3

K

2.5 2 1.5 1 −1

0

1

2

3

4

Y Fig. 3. Asymptotical stability of the economic equilibrium E∗ with τ = 6.25.

of the form [τ , +∞ ) for each j ≥ 1, where τ ≤ τ j . This completes the proof.  7. Application The aim of this section is to apply our theoretical results to the business cycle model with specific investment function and time delay:

⎧ qK (t ) dY ⎪ ⎪ ⎨ dt = α [I (Y (t )) + 1 + ε K (t )2 − γ Y (t )], qK (t ) dK ⎪ ⎪ − δ K (t ), ⎩ dt = I (Y (t − τ )) + 1 + ε K (t )2

(38)

System (38) is a special case of system (3) with I (Y, K ) = I (Y ) + √ qK , q < 0 and ε ≥ 0. 1+ε K 2

D. Riad et al. / Chaos, Solitons and Fractals 85 (2016) 110–119

It is important to note the Kaldor–Kalecki model analyzed by many authors in [4–9] is particular case of our model (38). It suffices to take ε = 0. For simulations, we consider the following Kaldor-type investment function:

I (Y ) =

eY . 1 + eY

First, we simulate system (38) with the following parameter values: α = 3, q = −0.2, ε = 0.01, γ = 0.5, δ = 0.1. By a simple calculation, we get a = 0.2400, β = −0.1883 and (a − γ )(δ − β ) − aβ = −0.0298 < 0. By applying Theorem 3.2 (i), we see that E∗ (0.4055, 2.0273) is locally asymptotically stable for all τ ≥ 0. Fig. 1 illustrates this observation. Next, we choose γ = 0.2 and do not change the other parameter values. For this choice, the stability of the economic equilibrium E∗ (1.3570, 2.7141) depends on the investment delay τ . From (13) and (14), we obtain ω0 ≈ 0.2200 and τ 0 ≈ 6.24326.2432. By Theorem 4.2, we know that the economic equilibrium E∗ is locally asymptotically stable for τ ∈ [0, τ 0 ). When τ crosses τ 0 , a family of periodic orbits bifurcate from E∗ . Figs. 2 and 3 demonstrate this result. Acknowledgments We would like to thank the editor and the anonymous reviewers for their valuable comments and suggestions that greatly improved the quality of our study.

119

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