Department of Mathematics, Computer & Information Sciences, Mississippi Valley State University, Itta Bena, MS 38941, USA
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.
1. 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 and Szydłowski [3–5] incorporated the idea of Kalecki into the model of Kaldor by proposing the following Kaldor-Kalecki model of business cycles:
where is the gross product, is the capital stock, is the adjustment coefficient in the goods market, is the depreciation rate of capital stock, and are investment and saving functions, and 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 in the investment function of the second equation of Sys. (1.1) leads to the following Kaldor-Kalecki model of business cycles:
As in [3]; also see [10, 16, 17], using the following saving and investment functions and , respectively,
where and are constants, we obtain the following system:
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 as a bifurcation parameter to show that the Hopf bifurcation may occur under some conditions as 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 .
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 and as bifurcation parameters (where 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 -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.
2. Distribution of Eigenvalues
Throughout the rest of this paper, we assume that
and that is an equilibrium point of Sys. (1.4). Let , , , and . Then Sys. (1.4) can be transformed as
Let the Taylor expansion of at 0 be
where
The linear part of Sys. (2.2) at is
and the corresponding characteristic equation is
where
For , (2.6) becomes
Define
Theorem 2.1. Let . If , then all roots of (2.8) have negative real parts, and hence is asymptotically stable. If , then (2.8) has a positive root and a negative root, and hence, is unstable.
Now assume . Clearly if and only if . Next we always assume that . It is easy to attain
Define . Then we have that,
Define
Hence if , , and hence , and if , , and hence if and only if . Also if and only if . Thus we obtain the following result.
Lemma 2.2. Suppose that . Then the following are considered. (i)If , then (2.6) has a simple root 0 for all . (ii)Let . Then the following are given. (a)Equation (2.6) has a simple root 0 if and only if , (b)Equation (2.6) has a double root 0 if and only if and .
Let () be a purely imaginary root of (2.6). After plugging it into (2.6) and separating the real and imaginary parts, we have that
Adding squares of two equations yields
Then (2.14) has a nonzero solution if and only if and does not have a nonzero solution if and only if . If , from (2.14), we solve as follows:
and from (2.13), we solve , as:
Define
From (2.16), we obtain
Clearly if , then has two positive roots, and if , then . Now, under , we impose the following conditions:
(H1), , (H2), , , (H3), , , (H4), , ,(H5), , ,(H6), , ,(H7), , .
Based on Lemma 2.2, we have the following result.
Lemma 2.3. Suppose that and . 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 in the imaginary axis if , (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 in the imaginary axis if for some .
Now we use the roots of , to give a more detailed discussion for the roots of (2.6). Define
Clearly is the positive root of and , are two positive roots of if . Note that if if , if , or then if . Also note that as well as if , . In fact it is based on the following calculation:
Thus for , we always have . Noting that , we have the following result.
Lemma 2.4. Let . Then the following are given. (i)Suppose that . Then for , then (2.6) has a simple zero root and does not have roots in the imaginary axis. (ii)Suppose that . If , then (2.6) has a simple zero root and does not have roots in the imaginary axis. And if , (2.6) has a double zero root and does not have roots in the imaginary axis. (iii)Suppose that . If , then (2.6) has a simple zero root and does not have roots in the imaginary axis. If , then (2.6) has a double zero root and does not have roots in the imaginary axis. And if , then (2.6) has a double zero root and has a pair of purely imaginary roots. (iv)Suppose that . Then if , then (2.6) has a simple zero root and does not have roots in the imaginary axis. If , then (2.6) has a double zero root and does not have roots in the imaginary axis. If , then (2.6) has a double zero root and has a pair of purely imaginary roots when for some . And if , (2.6) has a double zero root and does not have a pair of purely imaginary roots.
Define to be the root of (2.6) such that and . Then we have the following result.
Lemma 2.5. Suppose that and . Then .
Proof. Differentiating (2.6) with respect to yields
and a simple calculation gives
which gives
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
where , are real polynomials such that and . As varies, the sum of the order of zeros of on the open right half-plane can change only if a zero appears on or crosses the imaginary axis.
Lemma 2.7. Let and . Then, the following are obtained. (i)If , then all roots of (2.6) except 0 and purely imaginary roots have negative real parts,(ii)If , then (2.6) has at least one positive root.
Proof. Note that, for , if or , has a zero root and a negative root. Using Lemmas 2.2 and 2.6, we obtain claim (i). For , has a zero root and a positive root if or . For , let
Also noting that when , we have that
and . This proves the second part of the lemma and completes the proof of the lemma.
3. 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:
where , . This equation is equivalent to
which can be written as
where , , and . Define with supreme norm and is defined by , ; is a bounded linear operator; and is a function with , . Consider the following linear system:
Since is a bounded linear operator, then can be represented by a Riemann-Stieltjes integral
by the Riesz representation theorem, where () is an matrix function of bounded variation. Let be the infinitesimal generator for the solution semigroup defined by Sys. (3.4) such that
Define the bilinear form between and (where is the space of all row -vectors) by
The adjoint of is defined by as
In our setting, (3.3) has trivial components. Assume that the characteristic equation of (3.3) has eigenvalue zero with multiplicity and all other eigenvalues have negative real parts. Then has a generalized eigenspace which is invariant under the flow (3.4). Let be the space adjoint with in . Then can be decomposed as where , . Choose the bases and for and , respectively, such that
where is Jordan matrix associated with the eigenvalue 0.
To consider Sys. (3.3), we need to enlarge the space to the following :
The elements of can be expressed as with and
where is the identity matrix. Define the projection by
Then the enlarged phase space can be decomposed as Let with and . Then (3.3) can be decomposed as
where is an extension of the infinitesimal generator from to , defined by
for and its adjoint by is defined by
for . Let . Then Sys. (3.13) becomes
where
On the center manifold, (3.16) can be approximated as
4. Simple-Zero Singularity
In this section, we assume that the condition (H2) holds. From the definition of , we know that if and only if . Therefore (H2) is equivalent to
From (ii) of Lemma 2.4 and (ii) of Lemma 2.7, we know that, at , 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 as a bifurcation parameter near .
Set , . Let . Then Sys. (2.5) can be rewritten as
The linearization of Sys. (4.2) at is
Let where
Let and
Define
Then Sys. (4.2) becomes
From (3.7), the bilinear form can be expressed as
It is not hard to see that the infinitesimal generator is given by
for and its adjoint by
for .
Next we obtain the bases for the center space and its adjoint space , respectively. Let for , that is,
then we know that is a constant vector such that
Then we have two linearly independent solutions , which are bases for the center space . Let .
Similarly, let for , that is,
then we know that is a constant vector such that
From this we have two linearly independent solutions and which are bases for the center space . Let with being determined such that . In fact
Clearly is well defined since . It is not hard to check that , and , where .
Let . Then Sys. (4.2) can be decomposed as
Write . Note that
Here represents higher-order terms. Thus, for sufficiently small , on the center manifold, if , then Sys. (4.2) becomes
If and , then Sys. (4.2) can be transformed into the following form:
Thus we have the following results.
Theorem 4.1. Let be small. Then consider what follows. (i)Suppose that . Then if , the equilibrium is unstable, and if and , then the equilibrium is asymptotically stable for and unstable if .(ii)The equilibrium is asymptotically stable if and unstable if .(iii)At , Sys. (1.4) undergoes a transcritical bifurcation if and a pitchfork bifurcation if and .
5. Double-Zero Singularity
In this section, we assume that one of the conditions (H3) and (H6) holds and , or equivalently, as
From Section 2, we can see that, at , the characteristic equation of Sys. (2.5) has a double root 0 and all other roots have negative real parts if and . We treat as a bifurcation parameter near .
By scaling , Sys. (2.2) can be written as
Let . Let , . Then on we have
The linearization of Sys. (5.3) at is
Let
where
Define
Let . Let and where
Then Sys. (5.3) can be transformed into
Let . From (3.7), the bilinear inner product between and can be expressed by
for and . As in Section 4, the infinitesimal generator associated with is given by
for and its adjoint by
for . From Section 2, we know that is an eigenvalue of and with multiplicity 4. Now we compute eigenvectors of and associated with , respectively.
Next we obtain the bases for the center space and its adjoint space , respectively. Let for . This means that
From this we obtain that is a constant vector in satisfying
This equation has three linearly independent solutions: , , . Let be one of those. Suppose that for , namely,
This implies that there is a constant vector in such that and
Since
we have that
It is easy to see that (5.18) has no solution if is either or . For , setting in (5.18), we obtain
and hence . Thus we obtain bases , , , for the center space . Let . Then we have that where .
Similarly, let for , that is,
which means that is a constant vector satisfying
This equation has three linearly independent solutions: , , . Asserting that gives
Let be one of , , . Suppose , that is,
which implies that there is such that satisfying
Since
we have
It is not hard to check that (5.26) has no solution if or . Letting , setting in (5.26) and using , we can get and :
Hence
Then , , , are bases of the center space . Let . Then , and .
Let , namely,
Then Sys. (5.9) can be decomposed as
Write . Then, on the center manifold, Sys. (5.30) becomes
where
Next we use techniques of nonlinear transformations in [22] to transform Sys. (5.31) into normal forms. If , then up to the second order, Sys. (5.31) can be written as
This system can be transformed into the following normal form:
where
Since
we have that is regular and hence the transversality condition holds.
If and , then up to the third order, Sys. (5.31) becomes
This system can be transformed into the following normal form:
where , .
6. 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 and are in Section 5. Note that under the transformation . We may assume that . After the change of coordinates
we have (still using for simplicity) that
where , and . Simple calculation shows that where
Now take , namely . The complete bifurcation diagrams of Sys. (6.3) can be found in [22]. Here, we just briefly list some results. For small enough, consider the following.
(i)Sys. (6.3) undergoes a fold bifurcation when is on the curves
(ii)Sys. (6.3) undergoes a Hopf bifurcation when 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 is on the curve
Moreover, when is in the region between the curves and , Sys. (6.1) has a unique stable periodic orbit.
For , under the transformation , , 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 , we obtain the following result regarding Sys. (5.3).
Theorem 6.1. Suppose that and . For sufficiently small , 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
(iii)Sys. (5.3) undergoes a saddle of homoclinic bifurcation on the curve
Moreover, if is in the region between the curves and , Sys. (5.3) has a unique stable periodic orbit.
Next, we consider the truncated system of (5.38):
where are in Section 5. The bifurcation diagrams of this system are more complicated and interesting. We must consider two cases.
Case 1 ( so that ). We can assume which is equivalent to in (6.4). Then Sys. (6.11) can be transformed as
where
The complete bifurcation diagrams of Sys. (6.12) can be found, for example, in [22–24]. For , under the transformation and , we can get Sys. (6.12). Here, we briefly list some results: for small , as follows. (i)When 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 is in the region between the curves and then Sys. (6.12) has a unique stable periodic orbit.
Applying the above results and using the expressions of , , , , we obtain the following theorem regarding Sys. (5.3).
Theorem 6.2. Suppose that , and . For sufficiently small , , the following are given. (i)Sys. (5.3) undergoes a pitchfork bifurcation in the line
(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 is in the region between the curves and , Sys. (5.3) has a unique stable periodic orbit.
Case 2 ( so that ). We assume , namely . After rescaling, then Sys. (6.11) can be transformed as
where
For , under the transformation , we can get Sys. (6.20). The complete bifurcation diagrams of this system can be found, for example, in [22–24]. Here, we briefly list some results: for small , , as follows.(i)When is in the line
Sys. (6.20) undergoes a pitchfork bifurcation.(ii)When is in the half-line
Sys. (6.20) undergoes a stable Hopf bifurcation at and the bifurcation is subcritical.(iii)When is on the curve
Sys. (6.20) has a unique homoclinic orbit connecting and and two homoclinic orbits simultaneously at . Moreover, if is in the region between the curves and , Sys. (6.20) has three limit periodic orbits: a “large” one and two “small” ones.(iv)When is on the curve
where , Sys. (6.20) undergoes a double limit cycle bifurcation. Moreover, if is in the region between the curves and , 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 .
Applying the above results and using the expressions of , , , , we obtain the following theorem regarding Sys. (5.3).
Theorem 6.3. Suppose that , and . For sufficiently small , , considered the following. (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 and a large homoclinic orbit on the curve
Moreover, if is in the region between the curves and , 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 . Moreover, if is in the region between the curves and , 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 .
7. 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.
Example 7.1. This example demonstrates the result of Theorem 6.1. Let , , , . Then , . Take
Then (0,0) is the trivial equilibrium point and and . Simple calculation shows that
Take , . Then and is in the region between and . According to Theorem 6.1, Sys. (5.3) has a unique stable periodic orbit (see Figure 1).
Figure 1: A stable limit cycle is generated when
is in the region between
and
.
Example 7.2. This example supports the result of Theorem 6.2. Let , , , . Then , and . Take
Then , and hence, the condition of Theorem 6.2 is satisfied. After using the algorithm in Section 6, we have
Take , and hence is in the region between and . Figure 2 shows that there is a limit cycle which is stable according to Theorem 6.2.
Figure 2: A stable periodic orbit is generated when
is located in the region between
and
.
Example 7.3. This example verifies the result of Theorem 6.3. Let , , , . Then , and . Take
and hence, , . Thus the condition of Theorem 6.3 holds. Simple calculation shows
If we take , , then is in the region between and , and hence there are two small limit cycles and a large limit cycle (Figure 3). If we take , , then is in the region between and , and hence, there are two large limit cycles (Figure 4).
Figure 3: Three limit (two small and one large) cycles are generated when
is in the region between
and
.
Figure 4: Two large limit cycles are generated when
is in the region between
and
.
Acknowledgment
The author would like to thank the referees for their valuable suggestions.