Simple-Zero and Double-Zero Singularities of a Kaldor-Kalecki Model of Business Cycles with Delay

We study the Kaldor-Kalecki model of business cycles with delay in both the gross product and the capital stock. Simple-zero and double-zero singularities are investigated when bifurcation parameters change near certain critical values. By performing center manifold reduction, the normal forms on the center manifold are derived to obtain the bifurcation diagrams of the model such as Hopf, homoclinic and double limit cycle bifurcations. Some examples are given to confirm the theoretical results.


Introduction
In the last decade, the study of delayed differential equations that arose in business cycles has received much attention.The first model of business cycles can be traced back to Kaldor 1 who used a system of ordinary differential equations to study business cycles in 1940 by proposing nonlinear investment and saving functions so that the system may have cyclic behaviors or limit cycles, which are important from the point of view of economics.Kalecki 2 introduced the idea that there is a time delay for investment before a business decision.Krawiec  where Y is the gross product, K is the capital stock, α > 0 is the adjustment coefficient in the goods market, q ∈ 0, 1 is the depreciation rate of capital stock, I Y, K and S Y, K are investment and saving functions, and τ ≥ 0 is a time lag representing delay for the investment due to the past investment decision.This model has been studied extensively by many authors; see 6-11 .Several authors also discussed similar models 12-14 and established the existence of limit cycles.
Considering that past investment decisions 6 also influence the change in the capital stock, Kaddar and Talibi Alaoui 15 extended the model 1.1 by imposing delays in both the gross product and capital stock.Thus adding the same delay to the capital stock K in the investment function I Y, K of the second equation of Sys. 1 1. 4 Kaddar and Talibi Alaoui 15 studied the characteristic equation of the linear part of Sys.
1.4 at an equilibrium point and used the delay τ as a bifurcation parameter to show that the Hopf bifurcation may occur under some conditions as τ passes some critical values.However, they did not obtain the stability of the bifurcating limit cycles and the direction of the Hopf bifurcation.Wang and Wu 18 further studied Sys.1.4 and gave a more detailed discussion of the distribution of the eigenvalues of the characteristic equation which has a pair of purely imaginary roots.They derived the normal forms on the center manifold for sys.1.4 to give the direction of the Hopf bifurcation and the stability of the bifurcating limit cycles for some critical values of τ.However, under certain conditions, the characteristic equation of the linear part of Sys.1.4 may have a simple-zero root, a double-zero root, or a simple zero root and a pair of purely imaginary roots.In this paper, simple-zero fold and double-zero Bogdanov-Takens singularities for Sys.1.4 and their corresponding dynamical behaviors are investigated by using k and τ as bifurcation parameters where k is defined in Section 2 .The discussion of zero-Hopf singularity will be addressed in a coming paper.
The rest of this manuscript is organized as follows.In Section 2, a detailed presentation is given for the distribution of eigenvalues of the linear part of Sys.1.4 at an equilibrium point in the k, τ -parameter space.In Section 3, the theory of center manifold reduction for general delayed differential equations DDEs is briefly introduced.In Sections 4 and 5, center manifold reduction is performed for Sys.1.4 ; and hence, the normal forms for simple-zero and double-zero singularities are obtained on the center manifold, respectively.In Section 6, the normal forms for the double-zero singularity are used to predict the bifurcation diagrams such as Hopf, homoclinic, and double limit cycle bifurcations for the original Sys. of 1.4 .Finally in Section 7, some numerical simulations are presented to confirm the theoretical results.

Distribution of Eigenvalues
Throughout the rest of this paper, we assume that α, β > 0, q,γ ∈ 0,

2.2
Let the Taylor expansion of i at 0 be where The linear part of Sys.2.2 at 0, 0 is and the corresponding characteristic equation is where For τ 0, 2.6 becomes
Lemma 2.2.Suppose that k k * .Then the following are considered.
ii Let τ * > 0. Then the following are given.a Equation 2.6 has a simple root 0 if and only if τ / τ * , b Equation 2.6 has a double root 0 if and only if τ τ * and g q 2 / 0. Let ωi ω > 0 be a purely imaginary root of 2.6 .After plugging it into 2.6 and separating the real and imaginary parts, we have that ω 2 αβγ αβγ cos ωτ βω sin ωτ , q 2 − αβγ q ω αβγ sin ωτ − βω cos ωτ .

2.14
Then 2.14 has a nonzero solution if and only if g q 2 < 0 and does not have a nonzero solution if and only if g q 2 ≥ 0. If g q 2 < 0, from 2.14 , we solve ω as follows: and from 2.13 , we solve cos ω 0 τ , sin ω 0 τ as:
Based on Lemma 2.2, we have the following result.Lemma 2.3.Suppose that k k * and 0 < q < 1.Then the following are obtained.
i Under one of the conditions (H1), (H2), and (H4), 2.6 has a simple zero root and does not have other roots in the imaginary axis.
ii Under the condition (H5), 2.6 has a simple zero root and a pair of purely imaginary roots ±ω 0 i in the imaginary axis if τ τ j , j 0, 1, 2, . . . .
iii Under one of the conditions (H3) and (H6), then 2.6 has a double root 0 and does not have other roots in the imaginary axis.
iv Under the condition (H7), 2.6 has a double zero root and a pair of purely imaginary roots ±ω 0 i in the imaginary axis if τ * τ j for some j.
Now we use the roots of f x 0, g x 0 to give a more detailed discussion for the roots of 2.6 .Define

2.19
Clearly q 0 is the positive root of f x 0 and q 1 , q 2 are two positive roots of g x 0 if β > 2αγ.Note that f x ≤ 0 if 0 < x ≤ q 0 , andf x > 0 if x > q 0 , g x ≥ 0 if 0 < x ≤ q 1 , or x ≥ q 2 , then g x < 0 if q 1 < x < q 2 .Also note that as well as if β > 2αγ, q 2 0 < q 1 .In fact it is based on the following calculation: > 0.
i Suppose that q 0 ≥ 1.Then for 0 < q < 1, then 2.6 has a simple zero root and does not have roots in the imaginary axis.
ii Suppose that q 0 < 1 ≤ √ q 1 < √ q 2 .If 0 < q ≤ q 0 , then 2.6 has a simple zero root and does not have roots in the imaginary axis.And if q 0 < q < 1, 2.6 has a double zero root and does not have roots in the imaginary axis.
iii Suppose that q 0 < √ q 1 < 1 < √ q 2 .If 0 < q ≤ q 0 , then 2.6 has a simple zero root and does not have roots in the imaginary axis.If q 0 < q ≤ √ q 1 , then 2.6 has a double zero root and does not have roots in the imaginary axis.And if √ q 1 < q < 1, then 2.6 has a double zero root and has a pair of purely imaginary roots.
iv Suppose that √ q 2 ≥ 1.Then if 0 < q ≤ q 0 , then 2.6 has a simple zero root and does not have roots in the imaginary axis.If q 0 < q ≤ √ q 1 , then 2.6 has a double zero root and does not have roots in the imaginary axis.If √ q 1 < q < √ q 2 , then 2.6 has a double zero root and has a pair of purely imaginary roots when τ * τ j for some j.And if √ q 2 ≤ q < 1,

has a double zero root and does not have a pair of purely imaginary roots.
Define λ τ σ τ iω τ to be the root of 2.6 such that σ τ j 0 and ω τ j ω 0 .Then we have the following result.Lemma 2.5.Suppose that k k * and g q 2 < 0. Then σ τ j > 0.
Proof.Differentiating 2.6 with respect to τ yields dλ dτ and a simple calculation gives Re dλ dτ thus completing the proof.
Next we discuss the distribution of other roots of 2.6 .We need the following lemma due to Ruan and Wei 19 .
Lemma 2.6.Consider the exponential polynomial P λ, e −λτ p λ q λ e −λτ , 2.24 where p, q are real polynomials such that deg q < deg p and τ ≥ 0. As τ varies, the sum of the order of zeros of P λ, e −λτ on the open right half-plane can change only if a zero appears on or crosses the imaginary axis.
i If q > q 0 , then all roots of 2.6 except 0 and purely imaginary roots have negative real parts, ii If 0 < q ≤ q 0 , then 2.6 has at least one positive root.

2.25
Also noting that B C 0 when k k * , we have that This proves the second part of the lemma and completes the proof of the lemma.

Center Manifold Reduction
In this section, we briefly summarize the theory of center manifold reduction for general DDEs.The material is mainly taken from 20, 21 .Consider the following DDE: which can be written as Define the bilinear form between C and C * C 0, τ , R n p * where R n p * is the space of all row n p -vectors by The adjoint of A 0 is defined by A * 0 as In our setting, 3.3 has p trivial components.Assume that the characteristic equation of 3.3 has eigenvalue zero with multiplicity 2p and all other eigenvalues have negative real parts.
Then L has a generalized eigenspace P which is invariant under the flow 3.4 .Let P * be the space adjoint with P in C * .Then C can be decomposed as Choose the bases Φ and Ψ for P and P * , respectively, such that where J is Jordan matrix associated with the eigenvalue 0. To consider Sys.3.3 , we need to enlarge the space C to the following BC:

3.10
Discrete Dynamics in Nature and Society The elements of BC can be expressed as ψ ϕ X 0 α with ϕ ∈ C, α ∈ R n p , and where I is the n × n identity matrix.Define the projection π : BC → P by Then the enlarged phase space BC can be decomposed as

3.13
where A is an extension of the infinitesimal generator A 0 from C 1 to BC, defined by for ϕ ∈ C 1 and its adjoint by A * is defined by

3.17
On the center manifold, 3.16 can be approximated as

Simple-Zero Singularity
In this section, we assume that the condition H2 holds.From the definition of τ * , we know that τ * > 0 if and only if q > q 0 .Therefore H2 is equivalent to From ii of Lemma 2.4 and ii of Lemma 2.7, we know that, at 0, 0 , the characteristic equation of the linear part of Sys.2.5 has a simple zero root and the rest of roots have negative parts.We treat k as a bifurcation parameter near Then Sys.2.5 can be rewritten as

4.3
Let η θ Aδ θ Bδ θ τ where Then Sys.4.2 becomes Ẋ t LX t F X t .4.7 From 3.7 , the bilinear form can be expressed as It is not hard to see that the infinitesimal generator A : for ϕ ∈ C 1 and its adjoint A * by Next we obtain the bases for the center space P and its adjoint space P * , respectively.Let Aϕ 0 for ϕ ∈ C 1 , that is, φ θ 0 for − τ ≤ θ < 0, Aϕ 0 Bϕ −τ 0 for θ 0. 4.11 then we know that ϕ is a constant vector a 1 , a 2 , a 3 Then we have two linearly independent solutions ϕ 1 q, γ, 0 T , ϕ 2 0, 0, 1 T which are bases for the center space P .Let Φ ϕ 1 , ϕ 2 .Similarly, let A * ψ 0 for ψ ∈ C 1 * , that is, From this we have two linearly independent solutions ψ 1 − q β , αβ, 0 and ψ 2 0, 0, 1 which are bases for the center space P * .Let Ψ rψ 1 , ψ 2 T with r being determined such that Clearly r is well defined since τ − τ * / 0. It is not hard to check that Φ ΦJ, Ψ −JΨ and Ψ, Φ I, where J 0 0 0 0 .Let u Φx y.Then Sys.4.2 can be decomposed as ẋ ΨF Φx y , i Suppose that μ 0. Then if i 2 / 0, the equilibrium Y * , K * is unstable, and if i 2 0 and i 3 / 0, then the equilibrium Y * , K * is asymptotically stable for τ − iii At Y * , K * , k * , Sys. 1.4 undergoes a transcritical bifurcation if i 2 / 0 and a pitchfork bifurcation if i 2 0 and i 3 / 0.

Double-Zero Singularity
In this section, we assume that one of the conditions H3 and H6 holds and g q 2 > 0, or equivalently, as k k * , τ τ * , q > q 0 , g q 2 > 0.

5.1
From Section 2, we can see that, at 0, 0 , the characteristic equation of Sys.2.5 has a double root 0 and all other roots have negative real parts if k k * and τ τ * .We treat k, τ as a bifurcation parameter near k * , τ * .By scaling t → t/τ, Sys.2.2 can be written as

5.4
Let where

5.8
Then Sys.5.3 can be transformed into Ẋ t LX t F X t .5.9 Let C * C 0, 1 , R 4 * .From 3.7 , the bilinear inner product between C and C * can be expressed by for ϕ ∈ C 1 and its adjoint by for ψ ∈ C 1 * .From Section 2, we know that 0 is an eigenvalue of A and A * with multiplicity 4. Now we compute eigenvectors of A and A * associated with 0, respectively.

5.13
From this we obtain that ϕ θ ϕ 0 is a constant vector in R 4 satisfying A B ϕ 0 0.

5.18
It is easy to see that 5.18 has no solution if ϕ 0 1 is either a 3 or a 4 .For ϕ 0 1 a 1 , setting ϕ 0 2 0, l, 0, 0 T in 5.18 , we obtain and hence a 2 θ θq, l γθ, 0, 0 T .Thus we obtain bases a 1 , a 2 , a 3 , a 4 for the center space P .

5.24
Since we have

5.32
Next we use techniques of nonlinear transformations in 22 to transform Sys.5.31 into normal forms.If i 2 / 0, then up to the second order, Sys.5.31 can be written as μ1 0, μ2 0.

5.33
This system can be transformed into the following normal form:

5.37
This system can be transformed into the following normal form:

Bifurcation Diagrams
In this section, we will use the truncated systems 5.34 and 5.38 to obtain bifurcation diagrams of Sys.5.3 .First, we consider the truncated system of 5.34 : where a 2 and b 2 are in Section 5. Note that a We may assume that i 2 > 0. After the change of coordinates we have still using x 1 , x 2 for simplicity that where Simple calculation shows that s sign δ where 6.4 Now take s −1, namely δ < 0. The complete bifurcation diagrams of Sys.6.3 can be found in 22 .Here, we just briefly list some results.For ν 1 , ν 2 small enough, consider the following.
i Sys.6.3 undergoes a fold bifurcation when ν 1 , ν 2 is on the curves ii Sys.6.3 undergoes a Hopf bifurcation when ν 1 , ν 2 is on the half-line and the Hopf bifurcation gives rise to a stable limit cycle.
iii Sys.6.3 undergoes a homoclinic loop bifurcation when ν 1 , ν 2 is on the curve Moreover, when ν 1 , ν 2 is in the region between the curves H and P , Sys. 6.1 has a unique stable periodic orbit.
For s 1, under the transformation t → −t, x 1 → −x 1 , we can get Sys.6.12 whose parametric portrait remains as it was but the cycle becomes unstable.Applying the above results and using the expressions of ν 1 , ν 2 , we obtain the following result regarding Sys.5.3 .Theorem 6.1.Suppose that i 2 > 0 and δ < 0. For sufficiently small μ 1 , μ 2 , consider the following i Sys.5.3 undergoes a fold bifurcation in the half-lines ii Sys.5.3 undergoes a Hopf bifurcation on the curve 6.9 iii Sys.5.3 undergoes a saddle of homoclinic bifurcation on the curve , μ 2 > 0 .

6.10
Moreover, if μ 1 , μ 2 is in the region between the curves H and P , Sys. 5.3 has a unique stable periodic orbit.
Next, we consider the truncated system of 5.38 : where a 2 , b 2 are in Section 5.The bifurcation diagrams of this system are more complicated and interesting.We must consider two cases.
i When ε 1 , ε 2 is in the line Sys. 6.12 undergoes a pitchfork bifurcation.
ii Sys.6.12 undergoes a stable Hopf bifurcation for the trivial equilibrium point on the half-line iii On the curve Sys. 6.12 undergoes a heteroclinic bifurcation.Moreover, if ε 1 , ε 2 is in the region between the curves H 1 and C then Sys.6.12 has a unique stable periodic orbit.
i Sys.5.3 undergoes a pitchfork bifurcation in the line 6.17 ii Sys.5.3 undergoes a stable Hopf bifurcation in the half-line iii Sys.5.3 undergoes a branch of homoclinic bifurcation on the curve Moreover, if μ 1 , μ 2 is in the region between the curves H 1 and C, Sys.5.3 has a unique stable periodic orbit.
i When ε 1 , ε 2 is in the line Sys. 6.20 undergoes a pitchfork bifurcation.
ii When ε 1 , ε 2 is in the half-line Sys. 6.20 undergoes a stable Hopf bifurcation at E 1,2 and the bifurcation is subcritical.
iii When ε 1 , ε 2 is on the curve Sys. 6.20 has a unique homoclinic orbit connecting E 1 and E 2 and two homoclinic orbits simultaneously at E 0 .Moreover, if ε 1 , ε 2 is in the region between the curves H 2 and C , Sys. 6.20 has three limit periodic orbits: a "large" one and two "small" ones.
iv When ε 1 , ε 2 is on the curve where c ≈ 0.752, Sys.6.20 undergoes a double limit cycle bifurcation.Moreover, if ε 1 , ε 2 is in the region between the curves C and C d , then Sys.6.20 has two large limit cycles: the outer one which is stable and the inner one which is unstable, and these two cycles collide on C d .
i Sys.5.3 undergoes a pitchfork bifurcation in the line ii Sys.5.3 undergoes a branch of stable Hopf bifurcation on the curve iii Sys.5.3 has two small homoclinic orbits simultaneously at Y * , K * and a large homoclinic orbit on the curve Moreover, if μ 1 , μ 2 is in the region between the curves H 2 and C , then Sys.5.3 has three limit periodic orbits: a "large" one and two "small" ones.
iv Sys.5.3 undergoes a branch of a double limit cycle bifurcation on the curve where the constant c ≈ 0.752.Moreover, if μ 1 , μ 2 is in the region between the curves C and C d , Sys. 5.3 has two large different limit cycles.The outer one is stable, the inner one is unstable, and these two cycles collide on C d .

Numerical Simulations
In this section, we give some examples to verify the theoretical results obtained in Section 6.
For simplicity, we assume that 0,0 is one of the equilibrium points.

7.4
Take μ 1 −0.000035, μ 2 0.0001, and hence μ 1 , μ 2 is in the region between H 1 and C. Figure 2 shows that there is a limit cycle which is stable according to Theorem 6.2.If we take μ 1 0.0000861, μ 2 −0.0001, then μ 1 , μ 2 is in the region between H 2 and C , and hence there are two small limit cycles and a large limit cycle Figure 3 .If we take μ 1 0.0001, μ 2 −0.0001, then μ 1 , μ 2 is in the region between C and C d , and hence, there are two large limit cycles Figure 4 .

5 KFigure 1 :
Figure 1:A stable limit cycle is generated when μ 1 , μ 2 is in the region between H and P .

KFigure 2 :Figure 3 :Example 7 . 3 .
Figure 2:A stable periodic orbit is generated when μ 1 , μ 2 is located in the region between H and P .

KFigure 4 :
Figure 4: Two large limit cycles are generated when μ 1 , μ 2 is in the region between C and C d .
and Szydłowski 3-5 incorporated the idea of Kalecki into the model of Kaldor by proposing the following Kaldor-Kalecki model of business cycles: .1 leads to the following Kaldor-Kalecki model of business cycles: where β > 0 and γ ∈ 0, 1 are constants, we obtain the following system:dY t dt α I Y t − βK t − γY t , dK t dt I Y t − τ − βK t − τ − qK t .