Triple-zero singularity of a Kaldor – Kalecki model of business cycles with delay

Abstract. In this manuscript, we study triple zero singularity of a Kaldor–Kalecki model of business cycles with delay in both the gross product and the capital stock. By using the frameworks of Campbell–Yuan [1] and Faria–Magalhães [2, 3], the normal form on the center manifold is derived for this singularity and hence the corresponding bifurcation diagrams such as Hopf, BT, zero-Hopf, and homoclinic bifurcations are obtained. An example is given to verify some theoretical results.


Introduction
In 1940, Kaldor [4] first proposed a system of differential equations to model business cycles with nonlinear investment and saving functions so that the system may oscillate cyclically.Krawiec and Szydlowski [5][6][7] combined Kaldor's model and Kalecki's idea [8] that there is a time delay for investment after a business decision has been made by proposing the following Kaldor-Kalecki model of business cycles: ( Here Y is the gross product, K the capital stock, α > 0 the adjustment coefficient in the goods market, q ∈ (0, 1) the depreciation rate of capital stock, I(Y, K) and S(Y, K) investment and saving functions, and τ 0 a time lag representing delay for the investment on the capital stock due to the past investment decision.
Kaddar and Talibi Alaoui [9] noted that the past decision on the investment also has influence on the gross product by proposing the following Kaldor-Kalecki model c Vilnius University, 2013 X.P. Wu of business cycles: with delay in both the gross product and the capital stock.In this research, as in [5][6][7], we assume that the investment and saving functions have the following forms: where β > 0 and γ ∈ (0, 1), respectively.Thus system (2) becomes The dynamical behaviors and bifurcations of system (3) has been studied extensively [9][10][11][12].In [9], Kaddar and Talibi Alaoui found a critical of τ such that system (3) undergoes a Hopf bifurcation.In [10], Wang and Wu refined Kaddar and Talibi Alaoui's result and obtained the normal form of Hopf bifurcation which can be used to determine the stability and the direction of Hopf bifurcation.In [11], Wu studied simple zero, and double zero singularities of system (3) and obtained bifurcation diagrams, from which double limit and heteroclinic bifurcations were obtained.In [12], Wu studied zero-Hopf singularity of system (3) and obtained its corresponding bifurcation diagrams.
Note that all the results mentioned above depend on the distribution of roots of the characteristic equation of the linear part of system (3) at the equilibrium point.If the characteristic equation has a pair of purely imaginary roots, Hopf singularity occurs and hence a limit cycle may bifurcate from the equilibrium point; if the characteristic equation has a simple zero or double zero root, simple zero or double zero singularity occurs; so does zero-Hopf singularity if the characteristic equation has a simple zero root and a pair of purely imaginary roots.However, under certain conditions, the characteristic equation may have a triple zero root and this has not been studied in the literature.For a double zero root or a zero root with algebraic multiplicity 2 and geometric multiplicity 1, the corresponding Jordan matrix is 0 1 0 0 .For a triple zero root or a zero root with algebraic multiplicity 3, there are three cases for geometric multiplicities: (i) geometric multiplicity 1, (ii) geometric multiplicity 2, (iii) geometric multiplicity 3.
www.mii.lt/NAMore specifically, we use (k, β, τ ) as bifurcation parameter (where k is the increasing rate of the investment function at the equilibrium point (see the detail in Section 2)) to obtain the critical value (k * , β * , τ * ) such that the characteristic equation has a triple zero root with geometric multiplicity 1 and then investigate its corresponding dynamical behaviors.Note that we can find the conditions such that the equilibrium point is asymptotically stable.But this is not practical since business cycles in real world seem to change cyclically.This leads us to study Hopf singularity.But the condition for Hopf singularity is not always satisfied.We show that, for triple zero singularity, we still can obtain limit cycles under small perturbations of (k * , β * , τ * ) and certain conditions despise the fact that the condition for Hopf singularity is violated.The rest of this manuscript is organized as follows.In Section 2, the detailed conditions are given for the linear part of system (3) at an equilibrium point in the (k, β, τ )parameter space to have a triple zero eigenvalue and other eigenvalues with negative real parts.In Section 3, the normal form of triple zero singularity for system (3) is obtained on the center manifold by using the frameworks from [1] and [2,3].In Section 4, the normal form in Section 3 is used to obtain bifurcation diagrams of the original system (3) such as Hopf and homoclinic bifurcations.Finally in Section 5, an example is presented to confirm some theoretical results.

Distribution of eigenvalues
Throughout the rest of this paper, we assume that α, β > 0, q, γ ∈ (0, 1), and I(s) is a nonlinear C 4 function, and that (Y * , K * ) is an equilibrium point of system (3).Let I * = I(Y * ), u 1 = Y − Y * , u 2 = K − K * , and i(s) = I(s + Y * ) − I * .Then system (3) can be transformed as Let the Taylor expansion of i at 0 be where The linear part of system (4) at (0, 0) is where We only consider the case of τ > 0. It is easy to attain It is not hard to check that, if k = k * , β = β * , τ = τ * and q > αγ, then where Let q = αγ sec(σ) where σ ∈ (0, π/2) such that q ∈ (0, 1).Then k * , β * and τ * can be simply expressed as Thus we obtain the following result.
Let ωi (ω > 0) be a purely imaginary root of Eq. ( 6).Then we have After separating the real and imaginary parts, we obtain It is not hard to check that for ω > 0. Thus Eq. ( 6) does not have purely imaginary roots.For the other roots of Eq. ( 6), we need the following lemma from [11].Let Lemma 2. Let k = k * and τ > 0.
From this lemma, we know that if k = k * , β = β * , τ = τ * and q > max{αγ, q 0 (β * )}, Eq. ( 6) has a triple zero root and all other roots are in the left half plane of the imaginary axis.
Remark.The condition q > max{αγ, q 0 (β * )} states that if system (3) exhibits triplezero singularity, the depreciation q can not be very small.

The computation of the normal form
In the rest of this manuscript, we always assume We treat (k, β, τ ) as a bifurcation parameter near (k * , β * , τ * ).By scaling t → t/τ , system (4) can be written as Nonlinear Anal.Model.Control, 2013, Vol. 18, No. 3, 359-376 and the linear operator From Section 2, we see that L has a triple zero eigenvalue and all the other eigenvalues have negative real parts.It is easy to see that , where Then system (7) can be written as whose corresponding linear part at 0 is From [2,3], the bilinear inner product between C and C * can be expressed by Then L has a generalized eigenspace P which is invariant under the flow (8).Let P * be the space adjoint with P in C * .Then C can be decomposed as C = P ⊕ Q, where Q = {ϕ ∈ C: ψ, ϕ = 0 ∀ψ ∈ P * }.Furthermore, we can choose the bases Φ and Ψ for P and P * , respectively, such that where I is the identity matrix and J = 0 1 0 0 0 1 0 0 0 the Jordan matrix associated with the triple zero eigenvalue with geometric multiplicity 1.This guarantees that cases (ii) and (iii) will not happen for system (3) and hence triple-zero bifurcation occurs.
Next, we obtain the explicit expressions of Φ and Ψ .According to Campbell and Yuan [1], the basis Φ for P can be chosen as and the basis Ψ for P * as Note that ( 9) is equivalent to from which we obtain Similarly, (10) is equivalent to, respectively, From w 1 (A + B) = 0 we obtain w 1 = (a 1 , a 2 ), a 1 and a 2 will be determined later.Since and then using Ψ, Φ = I, we obtain the expressions of a 1 , a 2 , a 3 , a 4 , a 5 and a 6 after long and tedious calculations and Thus we obtain the bases Φ and Ψ of P and P * such that Φ = ΦJ and Ψ = −JΨ .
Next we compute the corresponding normal form.Let u = Φx + y (here x = (x 1 , x 2 , x 3 ) T ∈ R 3 and y = (y 1 , y 2 ) T ∈ C); namely Then, on the center manifold y = g(x(t), θ), system (7) becomes where α j , β j , γ j are linear functions of µ (j = 1, 2, 3) and will be given later.Denote the coefficient of x j 1 x k 2 x l 3 by (a jkl , b jkl , c jkl ) T in the right side of system (11).Then If i (2) = 0, after projection on the center manifold and truncation up to the second order, system (11) can be written as the following: If i (2) = 0, i (3) = 0, after projection on the center manifold and truncation up to the third order, then system (11) can be written as the following: From [1], system (12) can be transformed as the following normal form, in which χ j and A jkl are given by After long computation, we obtain the explicit expressions of α 1 , β 1 , γ 1 , β 2 , γ 3 , and A jkl in ( 14) , and also the explicit expressions of A ljk ) is regular and hence the transversality condition holds.
If i (2) = 0 and i (3) = 0, then A ijk = 0 for i + j + k = 2. Using a result from [13], we transform system (13) into the following normal form: where Using the expressions of a ijk , b ijk and c ijk , we have Note that

Bifurcation diagrams
In this section, we truncate higher order terms from system ( 14) and ( 15) to obtain bifurcation diagrams of system (3).
= 0, we consider the truncated system of ( 14) where A 200 , A 110 , A 101 and A 020 are in Section 3. Note that ).We may assume that i (2) > 0.
The complete bifurcation diagrams of system ( 16) can be found in [1].Here, we just briefly list some results.
(ii) The origin undergoes a Hopf bifurcation on the curve (iii) For χ 1 = 0, there is a nontrivial equilibrium point at x * = (− χ 1 /A 200 , 0, 0).Moreover, the nontrivial equilibrium point x * undergoes a Hopf bifurcation on the curve (iv) The origin undergoes a BT bifurcation on the curve The origin and the nontrivial equilibrium point x * undergo zero-Hopf bifurcation on the curve Note that for case (iv), when χ 1 = χ 2 = 0 and χ 3 < 0, system (16) undergoes BT bifurcation.Using the technique in [14] (we omit the detail), it is not hard to obtain that ( 16) is equivalent to the following normal form: www.mii.lt/NAwhere Since χ 3 is small, we can see that r 1 s 1 < 0. For the bifurcation diagram of this system, we need the following result from [15].
Moreover if (ρ 1 , ρ 2 ) is between l h and l ∞ , there is a unique stable limit cycle.
Case (v) is very interesting and we can further find the normal form for this bifurcation: Since x 1 = x2 , through the change of variables x 1 = w 1 −iw 2 , x 2 = w 1 +iw 2 , x 3 = w 3 , and then a change to cylindrical coordinates according to w 1 = r cos ξ, w 2 = r sin ξ, w 3 = ζ, system (18) becomes Nonlinear Anal.Model.Control, 2013, Vol. 18, No. 3, 359-376 (iv) The origin undergoes a BT bifurcation on the curve The origin and the nontrivial equilibrium points x * ± undergo zero-Hopf bifurcation on the curve We can use the same technique as in the previous subsection to study BT and zero-Hopf bifurcations and here we omit the details.Note that for case (iv) , when χ 1 = χ 2 = 0 and χ 3 < 0, the system undergoes BT bifurcation.Using the technique in [14], it is not hard to obtain that (21) is equivalent to the following normal form: where Since χ 3 is small, we can see that r 2 s 2 < 0. Without loss of generality, we assume i (3) < 0 so that r 2 > 0 and s 2 < 0. System (22) can be transformed as where The complete bifurcation diagrams of system (23) can be found, for example, in [14,17].
Here, we just list two results.Lemma 6.For small ε 1 , ε 2 : (i) System (23) undergoes a Hopf bifurcation for the trivial equilibrium point on the line 23) undergoes a heteroclinic bifurcation.Moreover, if (ε 1 , ε 2 ) is in the region between the curves H 1 and C system (23) has a unique stable periodic orbit.
For (v)', we can use the same technique in [14] to find the following normal form for zero-Hopf bifurcation at the origin (for simplicity): where This system is a so-called predator-prey system which has been well studied and here we omit the detail.

Numerical simulations
In this section, we give some example to verify some theoretical results obtained Section 4. For simplicity, we assume that (0,0) is one of the equilibrium points.
Then (0, 0) is the trivial equilibrium point and i is on the curve C 2 so that there is a unique limit cycle bifurcation from the nontrivial equilibrium point.It is not hard to use the technique in [14] to show that the first Lyapunov coefficient 1 (0) = −2.26094× 10 10 < 0 in this setting.Thus the limit cycle is stable (see Fig. 1).It is easy to check that (ρ 1 , ρ 2 ) is between two curves l h and l ∞ and thus there is a unique stable limit cycle bifurcating from the trivial equilibrium point (see Fig. 2).

Conclusion
Since Krawiec and Szydlowski [5][6][7] introduced Kalecki's idea that there is a time delay for investment before a business decision to Kaldor's model, the Kaldor-Kalecki model has been studied extensively.In this presentation, we studied the triple-zero singularity for the Kaldor-Kalecki model of business cycle with delay in both the gross product and the capital stock.First we analyzed the characteristic equation at the equilibrium point and gave the condition such that it has a triple-zero root.The normal form for this singularity was presented.By using this normal form, the bifurcation diagrams were given.Some examples were given to confirm the theoretic result.