stability and hopf bifurcation for kaldor-kalecki model of business cycles with two time delays
DESCRIPTION
Abstract— Papers investigate a Kaldor-Kalecki model of business cycle system with two different delays, which described the interaction of the gross product and the capital product . We derived the conditions for the local stability and the existence of Hopf bifurcation at the equilibrium of the system. By applying the normal form theory and center manifold theory, some explicit formulate for determining the stability and the direction of the Hopf bifurcation periodic solutions are obtained. Some numerical simulations by using Mathematica software supported the theoretical results. Finally, main conclusions are given.TRANSCRIPT
International Journal of Engineering Research & Science (IJOER) ISSN: [2395-6992] [Vol-2, Issue-5 May- 2016]
Page | 1
Stability and Hopf bifurcation for Kaldor-Kalecki model of
business cycles with two time delays Xiao-hong Wang
1, Yan-hui Zhai
2, Ka Long
3
School of Science, Tianjin Polytechnic University, Tianjin
Abstract— Papers investigate a Kaldor-Kalecki model of business cycle system with two different delays, which described
the interaction of the gross product Y and the capital product K . We derived the conditions for the local stability and the
existence of Hopf bifurcation at the equilibrium of the system. By applying the normal form theory and center manifold
theory, some explicit formulate for determining the stability and the direction of the Hopf bifurcation periodic solutions are
obtained. Some numerical simulations by using Mathematica software supported the theoretical results. Finally, main
conclusions are given.
Keywords— Two delays, Hopf bifurcation, Stability, Business cycle, Normal form.
I. INTRODUCTION
According to rational expectation hypothesis, the government will take into account the future capital stock in the process of
investment decision. Business cycle, named economic cycle, refers to the total alternate expansion and contraction in
economic activities, and these cyclical changes will appear in the form of fluctuation of the comprehensive economic activity
indicators such as gross national product, industrial production index, employment and income. In the early research of non-
linear business cycle theories, Kaldor [1] assumed that investment depended on gross product and capital stock and proved
that, through graph analysis, time-varying nonlinear investment and saving function could result in a business cycle. Chang
and Smyth [2] summarized Kaldor's theory on business cycle, established a nonlinear dynamic system which described the
gross national product and capital stock changed over time, gave necessary conditions of existence of limit cycles, and
provided a rigorous mathematical proof of Kaldor's model proposed in 1940. On this basis, Grasman and Wentzel [3] further
improved Kaldor's business cycle model by considering capital loss speed. Along another view, according to the IS-LM
model raised by J.R. Hicks and A.H. Hansen, Ackley [4] established a complete Keynes system which reflects the gross
domestic product and interest rate changes over time, which is also called standard IS-LM model. Kalecki [5] first considered
investment delay in the business cycle model in 1935, and he claimed capital equipment needed a conceived cycle or delay
from installation to production. As the theory of delay functional differential equations gradually become more accomplished
in 1990s, Krawiec and Szydlowski [6,7] first made a qualitative analysis of the impact of the investment delay on the
business cycle. In addition, some other scholars investigated the dynamic properties like stability, many types of bifurcation,
existence and stability of periodic solutions in Kaldor–Kalecki model with investment delay [8,9]. The record of business
cycle has been kept relatively well during the last 200 years, and business cycle theory, as the core issue of macroeconomics,
has been attracting the widespread interests of many economists. The modern business cycle theory can be traced back to the
masterpiece of Keynes's theory "The General Theory of Employment, Interest, and Money". Keynes discussed the formation
of the business cycle from the perspective of psychological factors based on national income theory. The following system
was formulated by Krawiec and Szydlowski [10] who combined two basic models of business cycle: the Kaldor model and
the Kalecki model. Kaldor [11] first treated the investment function as a nonlinear (s-shaped) function on Y so that the
system may create limit cycles, while Kalecki [12] assumed that the saving part is invested, and that there is a time delay due
to the past investment decision. Therefore, the gross product is available in the market after a time lag.
.)),(()(
)),,(),(()(
KKtYItK
KYSKYItY
(1.1)
Y is the gross product and K is the capital product of the business cycle; 0 measures the reaction of the system to the
difference between investment and saving; )1,0( is the depreciation rate of capital stock; RRRSI :, are
investment and saving functions of Y and K , respectively; is a time lag representing the delay for the investment due to
the past investment decision. In a business cycle system, delay occurred in the production of investment function not only,
also released on capital stock, so to introduce the following model.
International Journal of Engineering Research & Science (IJOER) ISSN: [2395-6992] [Vol-2, Issue-5 May- 2016]
Page | 2
.))(),(()(
)),,(),(()(
KtKtYItK
KYSKYItY
(1.2)
However, for the systems (1.2), the delay that occurs in investment function and happened in the capital stock is not
necessarily the same, so we set up two different time delay. By quoting the function of investment and capital saving
function in reference [1], KYIKYI )(),( , YKYS ),( , 0 , )1,0( . Following model be produced.
.))())(()(
),)(()(
21 KtKtYItK
YKYItY
(1.3)
The rest of this paper is organized as follows. In Section 2, we analyze the distribution of eigenvalues of the linearized
system of (1.3) and derived the conditions for the local stability and the existence of Hopf bifurcation at the equilibrium of
the system. In Section 3, by applying the normal form theory and center manifold theory, some explicit formulate for
determining the stability and the direction of the Hopf bifurcation periodic solutions are obtained. In Section 4, some
numerical simulations by using Mathematica software supported the theoretical results. Finally, main conclusions are given.
In Section 5, main conclusions are given.
II. STABILITY AND ANALYSIS OF LOCAL HOPF BIFURCATION
Let ),( KYE be an equilibrium point of Sys.(1.3), )( YII , and YYu1 ,
KKu2 ,
IYsIsi )()( . Then Sys.(1.3) can be transformed as
.))())((
),)((
222112
2111
ututuiu
uuuiu
(2.1)
Let the Taylor expansion [13] of i at 0 be
).|(|)(2
)()()( 32
''' uou
YIYIui
Then Sys.(2.1) can be transformed as
.))())((2
)()()(
),)(2
)()((
222
2
11
''
11
'
2
21
2
1
''
1
'
1
ututuYI
tuYIu
uuuYI
uYIu
(2.2)
Then the linear part of Sys.(2.2) at (0,0) becomes
.))()()(
),)((
22211
'
2
211
'
1
ututuYIu
uuuYIu
(2.3)
And the corresponding characteristic equation is
,0)()( 122
abeeaa
(2.4)
International Journal of Engineering Research & Science (IJOER) ISSN: [2395-6992] [Vol-2, Issue-5 May- 2016]
Page | 3
where ).(,))(( '' YIbYIa
Theorem 2.1 When 021 , the characteristic equation of system (2.4) is
.0)(2 aaba
(2.5)
All roots of the equation (2.5)has negative part if and only if
.0,0)( 111011 aabAaAH
Then the equilibrium point ),( KYE is locally asymptotically stable.
Theorem 2.2 When 0,0 21 , assume that )( 11H is satisfied. Then (2.2) has a pair of purely imaginary roots 10i
when 10 .
Proof.The characteristic equation of system (2.4) is
.0)()( 12
abea
(2.6)
Clearly, 1i is a root of Eq.(2.6) if and only if 1i satisfies
.0)()sin(cos)( 11111
2
1 aibia
Separating the real and imaginary parts, it follows
,sin)(
),(cos
111
112
ba
ab
(2.7)
According to conclusions of Beretta and Kuang [14], the stability of the system changes, when the real part of its
characteristic root passes through zero point. Therefore, considering the critical situation, let characteristic root real part
,0Re , adding up the squares of both Eq.(2.7), it yields
,0)( 22
1
22
1 bcavacv
(2.8)
where .)(, 22
11 cv
Hence Eq.(2.8) has solution 10v and 11v .Where
,2
)(4)()( 22222
10
bcacacav
.
2
)(4)()( 22222
11
bcacacav
Assume
,0)()( 22
1
22
11 bcavacvvh
(2.9)
Noting that 11v is negative. Eq.(2.9) has one positive root if and only if cabH 22
21)( . The characteristic Eq.(2.6) has a
pair of purely imaginary roots .1010 vii And, hence
).)()(
arccos(1 2
01
01
10b
a
The proof is completed.
International Journal of Engineering Research & Science (IJOER) ISSN: [2395-6992] [Vol-2, Issue-5 May- 2016]
Page | 4
Differentiating both sides of (2.6) with respect to 1 , it follows that
1
1
)(
d
d.
2 1
1
be
a
That is 01
101|)Re( 1
1
i
d
d
.)(
))((22
2224
b
a
Obviously, it is greater than zero. So, the transversality condition is meted. By applying Lemmas 2.1 and 2.2 and condition
)( 21H , we have the following results.
Theorem 2.3 For system (2.2), if the condition is satisfied, the equilibrium E of system (2.2) is asymptotically stable for
),0[ 101 . System (2.2) exhibits the Hopf bifurcation at the equilibrium E for 101 .
Theorem 2.4 When 0,0 21 , assume that )( 13H is satisfied. Then (2.2) has a pair of purely imaginary roots 02i
when 02 .
Proof.The characteristic equation of system (2.4) is
.0)()( 22
abeaa
(2.10)
Let )0( 22 i is a root of (2.10). Then
.0)cossinsincos()( 22222222222
2
2 aaiaiia
Separating the real and imaginary parts, it follows
,sincos)(
,0cossin
222222
22222
2
2
aa
baa
(2.11)
Adding up the squares of both Eq.(2.11), it yields
,0)()2(
))()(22()22(
2242
2
222222
2
2223
2
aabba
vbaaabvbav
(2.12)
where .2
2
2 v
From the analysis above, we can obtain that if all parameter values of the system (2.2) are given, the root of the equation
(2.12) can be easy to solved by use of the Mathematica software. Therefore, in order to give out the main conclusion in this
section, the following assumptions is given.
)( 31H The equation (2.12) at least has one negative root.
If the condition )( 31H is satisfied. Then, the characteristic Eq.(2.12) has an positive root 02v .So, the Eq.(2.10) has a pair of
purely imaginary roots .0202 vii And, hence
).)(
)(arccos(
12
20
2
2
20
2
02
20
a
aab
The proof is completed.
International Journal of Engineering Research & Science (IJOER) ISSN: [2395-6992] [Vol-2, Issue-5 May- 2016]
Page | 5
Differentiating both sides of (2.10) with respect to 2 , it follows that
1
2
)(
d
d.
)(
2 2
2
2
ea
ae
That is 02
202|)Re( 1
2
i
d
d
.))()((
)()3(222242
2
20
22224
20
2226
20
aa
aaa
Obviously, it is greater than zero. So the transversality condition is meted. By applying Lemmas 2.4 and condition )( 31H ,
we have the following results.
Theorem 2.5 For system (2.2), if the condition )( 31H is satisfied, the equilibrium E of system (2.2) is asymptotically
stable for ),0[ 202 . System (2.2) exhibits the Hopf bifurcation at the equilibrium E for .202
Theorem 2.6 When 21 , assume that )( 14H is satisfied. Then (2.2) has a pair of purely imaginary roots 0i
when .0
Proof.The characteristic equation of system (2.4) is
.0))(()(2 aebaa
(2.13)
Let )0( i is a root of (2.13). Then
.0)()(2
2 aebaiea
Separating the real and imaginary parts, it follows
,0sin)(cos)(
,0cos)(sin
22
2
baa
aba
(2.14)
Adding up the squares of both Eq.(2.14), it yields
2222223 )(2))((2())(( baaabb
vab
av
,0))()(()()))(( 2222
abab
avbb
aa
(2.15)
where .2 v
From the analysis above, we can obtain that if all parameter values of the system (2.2) are given, the root of the equation
(2.15) can be easy to solved by use of the Mathematica software. Therefore, in order to give out the main conclusion in this
section, the following assumptions is given.
)( 14H The equation (2.15) at least has one negative root.
If the condition )( 14H is satisfied. Then, the characteristic Eq.(2.15) has an positive root 0v . So, Eq.(2.13) has a pair of
purely imaginary roots .00 vii And, hence
).)(
)()(arccos(
122
0
2
2
0
0
0ba
abab
International Journal of Engineering Research & Science (IJOER) ISSN: [2395-6992] [Vol-2, Issue-5 May- 2016]
Page | 6
The proof is completed.
Differentiating both sides of (2.13) with respect to , it follows that
1)(
d
d.
))((
2
eba
ae
That is
0
0|)Re( 1
i
d
d
.
))((
)()())()(2(2222
0
2
2
0
2224
0
22226
0
2
ba
aabaab
Obviously, it is greater than zero. The transversality condition is meted. By applying Theorem 2.5 and condition )( 14H , we
havethe following results.
Theorem 2.7 For system (2.2), if the condition )( 32H is satisfied, the equilibrium E of system (2.2) is asymptotically
stable for ),0[ 0 . System (2.2) exhibits the Hopf bifurcation at the equilibrium E for .0
Theorem 2.8 When )(0,,0 2021 , assume that )( 51H is satisfied. Then (2.2) has a pair of purely imaginary roots
1i when .1
Proof. Let )0( '
1
'
1 i is a root of (2.4). Then
.0)sin(cos)sin)(cos()()( 1
'
11
'
12
'
12
'
1
'
1
'
1
2'
1 aibiaiia
Separating the real and imaginary parts, it follows
,0sinsincos)(
,0coscossin)(
1
'
12
'
12
'
1
'
1
'
1
1
'
12
'
12
'
1
'
1
2'
1
baa
aba
(2.16)
Adding up the squares of both Eq.(2.16), it yields
2
'
1
2'
1
2'
12
'
1
2223'
12
'
1
4'
1 sin2))(cos2())(sin2()( aa
.0cos2)( 2
2
'
1
2222 baa
(2.17)
From the analysis above, we can obtain that if all parameter values of the system (2.2) are given, the root of the equation
(2.17) can be easy to solved by use of the Mathematica software. Therefore, in order to give out the main conclusion in this
section, the following assumptions is given.
)( 51H The equation (2.17) at least has one negative root.
If the condition )( 51H is satisfied. So, the characteristic Eq.(2.17) has an positive root .1
i Eq.(2.4) has a pair of purely
imaginary roots .1
i And, hence
).)(
sincos))(2(arccos(
122
1
21121
22
1
1
1a
babaa
The proof is completed.
Differentiating both sides of (2.4) with respect to 1 , it follows that
International Journal of Engineering Research & Science (IJOER) ISSN: [2395-6992] [Vol-2, Issue-5 May- 2016]
Page | 7
1)(
d
d.
)(2 12
1
22
be
eaae
That is
1
11
|)Re( 1
1
i
d
d.
)(cos)(sin)1(sin)(cos2
1
21112211211111
b
aa
Obviously, 0|)Re( 1
11
1
1
i
d
d.The transversality condition is meted. By applying Lemmas 2.8 and condition )( 51H , we
have the following results.
Theorem 2.9 For system (2.2), if the condition )[0, 202 and )( 51H is satisfied, the equilibrium E of system (2.2) is
asymptotically stable for )[0, 11
. System (2.2) exhibits the Hopf bifurcation at the equilibrium E for .11
Theorem 2.10 When )[0,,0 1012 , assume that )( 61H is satisfied. Then (2.2) has a pair of purely imaginary roots
2i when .22
Proof.Let )0( 22 i is a root of (2.4). Then
.0)sin(cos)sin)(cos()()( 1
'
21
'
22
'
22
'
2
'
2
'
2
2'
2 aibiaiia Separating the real
and imaginary parts, it follows
,0sinsincos)(
,0coscossin)(
1
'
22
'
22
'
2
'
2
'
2
1
'
22
'
22
'
2
'
1
2'
2
baa
aba
(2.18)
Adding up the squares of both Eq.(2.18), it yields
0.)cos2(
sin)(2))(cos)(2)(2(
))(sin)(2())(cos22()(
1
'
2
2222
1
'
2
2'
2
2'
21
'
2
24222
3'
22
'
1
4'
21
'
2
22226'
2
abba
abaaabbaa
abba
(2.19)
In order to give out the main conclusion in this section, the following assumptions is given.
)( 61H The equation (2.19) at least has one negative root.
If the condition )( 61H is satisfied. Then, the characteristic Eq.(2.19) has an positive root '
2 .Eq.(2.4) has a pair of purely
imaginary roots .'
2i And, hence
).)(
sincos))(2(arccos(
122
2
12212
22
1
2
2a
babaa
The proof is completed.
Differentiating both sides of (2.4) with respect to 2 , it follows that
1)(
d
d.
)(
2 21
2
121
ea
beaee
International Journal of Engineering Research & Science (IJOER) ISSN: [2395-6992] [Vol-2, Issue-5 May- 2016]
Page | 8
That is
2
22
|)Re( 1
2
i
d
d,
))(()( 22
2
2
2
2 a
A
where
.)())(sin)(cos(
cos)()(sin))(2)((
2
2
2
122122212
22
2
222
2
22
a
aaaA
Obviously, 0|)Re( 2
22
1
2
i
d
d. So, the transversality condition is meted. By applying Theorem 2.10 and condition
)( 61H , we havethe following results.
Theorem 2.11 For system (2.2), if the condition )[0, 101 and )( 61H is satisfied, the equilibrium E of system (2.2) is
asymptotically stable for )[0, 22
. System (2.2) exhibits the Hopf bifurcation at the equilibrium E for .22
When
)[0,,0 1012 , assume that )( 61H is satisfied. Then (2.2) has a pair of purely imaginary roots 2i when
.22
III. DIRECTION AND STABILITY OF THE PERIODIC SOLUTIONS
In this section, by applying the normal form and center manifold theory [15,16], we discuss the direction and stability of the
bifurcating periodic solutions for )[0,,0 1012 . Throughout this section, we always assume that system (1.3) meets
the conditions of the Hopf bifurcation. In this section, we assume that )[0,, 10121
. For convenience, let
,22
clearly, 0 is the Hopf bifurcation value of system (1.3). Let /t and still denote t .Then the system
(1.3) can be rewritten as
),,()( tuFLtu
(3.1)
whereTtututu ))(),(()( 21 and
)),1()()0()(()( '
2
1''
2
CBAL
(3.2)
,)(),(
3
2
1
2
F
F
F
F
where
0
'a
A ,
0
00'
'
qB , and
0
00'C .
And .2
)(),(),0(),(
''''
2
12
1
''
2
2
1
''
1
'
YI
qqFqFYIq
By Riesz's representation theorem, there exists bounded variation functions
22]0,1[:),( R for )0,[ 1
,such that
.,)(),(0
1CdL
(3.3)
In fact, we take
International Journal of Engineering Research & Science (IJOER) ISSN: [2395-6992] [Vol-2, Issue-5 May- 2016]
Page | 9
.1,0
),,1(,)(
),0,[),)((
,0),)((
),(
2
1'
2
2
1''
2
'''
2
C
CB
CBA
(3.4)
For ]0,1[' C , we define that operator A and R is as follows form.
.0,))(),((
),0,1[,))((
)()(0
1
d
d
d
A
(3.5)
.0),,(
),0,1[,0)()(
FR (3.6)
Then the system (3.1) is equivalent to the following abstract differential equation
,)()( ttt uRuAu
(3.7)
where ))(),(),(( 321 tututuut for ].0,[ 0
2
Setting ],0[' C , we can define an adjoint operator )0(A corresponding to )0(A as the following form:
0
02
,0)),()0,((
],1,0(,)(
)(
sssd
sds
sd
sAT
For )],0,([ 20
2
' RC and ],0[ 0
2
' C , define the bilinear form:
,)()()()0()0()(),(0
1 0
dds
(3.8)
where ).0,()( After discussed above, we know that 22 is characteristic root of )0(A and
A . To determine
the normal form of operator A , we need to calculate the eigenvectors
22),1()( 2
iT e and
siT eD
22),1()( 2
of A and A corresponding to
22 and 22 , respectively. By the definition of the
)0(A and A . We calculate that
,222
ia.
22'
222
eq
ia
In addition, according to the equation (3.8), there is
)(),( s .)()()()0()0(0
1 0
dqd
]1
)(),1(1[2
0
1222
22
deDi
),1( 2222
222
'
122
ii
eeqD
International Journal of Engineering Research & Science (IJOER) ISSN: [2395-6992] [Vol-2, Issue-5 May- 2016]
Page | 10
Further, let 1)(),( qsq , there is .)1( 1
222
'
1222222
ii
eeqD
Next, according to the algorithm in paper [17] and the similar calculation process with the paper [18], following parameters
expression be getted.
),(2 212''
2
''
220
i
eqqDg
),2( 2121''
2
''
211
iieeqqDg
),(2 21''
2
''
202
i
eqqDg
)],)0()0(2())0(2
1)0(2([2 2
)1(
202
)1(
11
2
12
'')1(
20
)1(
11221
WWqWW
MDg
where
.)0()0()(
,)0(3
)0()(
2
22
11
22
1111
2
1
22
02
22
2020
2222
222222
Eegi
eig
W
eEei
gie
igW
ii
iii
(3.9)
While, 1E and 2E be determined by the following equation.
,2
22
222222 2''
''1
2
2
2'
2
1
iii
eq
q
eieq
aiE
.2
)( 1111''
''1
'2
ii
eeq
q
q
aE
In the end, we can calculate the following values of the coefficient.
.)})((Im{)}0(Im{
)},0(Re{2
,)}(Re{
)}0(Re{
,2
)3
12(
2)0(
22
2
'
212
12
2
'
12
212
02111120
22
1
CT
C
C
ggggg
iC
Theorem 3.1
(i) The direction of the Hopf bifurcation is decided by the symbol of 2 . If 02 , then it's supercritical. If 02 , it is
subcritical.
(ii) If 02 ( 02 ), then the bifurcating periodic solutions is stable (unstable).
(iii) If 02 T ( 02 T ), then the period of the bifurcating periodic solutions increase (decrease).
International Journal of Engineering Research & Science (IJOER) ISSN: [2395-6992] [Vol-2, Issue-5 May- 2016]
Page | 11
IV. NUMERICAL SIMULATION
In this section, we shall conduct some numerical simulations to verify the main conclusions of this paper, and discuss the
economic meanings of the different dynamic behaviors. In numerical simulations, according to literature [19], we choose the
parameter. Let 0.4,2.275,0.5, ' q ,088.0'' q ,1.0,8.0 then Eq.(2.2) can be written as
.1.0))((088.0)(8.0)(4.0
,)(2202.082.12275.0
2
2
1122112
2
1211
utututuu
uuuu
(4.1)
With calculation by Mathematica software, when 021 , we get 31992.110 A ,
61423.111 A . So the condition )( 10H is satisfied. When 0,0 21 , we get 10366.2,0.62444 1010 . If
),0[ 101 , the equilibrium E of system (2.2) is asymptotically stable. Once 1 more than critical value
2.1036610 , the system (2.2) produce a bunch of periodic solutions nearby the equilibrium point E . Numerical
simulation results are shown in Figs.1 and 2.Similarly, when 0,0 21 , we get 1.2149,1.47391 2020 (Figs.3
and 4).When 021 , we get 0.95603,1.1152 00 (see Figs.5 and 6).When ),0(2.1,0 2021 , we
get 0.2342,1.3327 11 (see Figs.7 and 8). Similarly, when ),0(8.1,0 0112 , we get
0.6286,0.775 22 (see Figs.9 and 10).
In addition, 06.13442 , .01632.1212 , 011.26632 T
By the theorem (3.1), when
),0(,0 0112 , the Hopf bifurcation produced by the system (2.2) is supercritical, and the period of the bifurcating
periodic solutions increase.
0.150.100.05 0.05 0.10 0.15u1
0.05
0.05
0.10
u2
20 40 60 80 100t
0.15
0.10
0.05
0.05
0.10
0.15
u1
20 40 60 80 100t
0.05
0.05
0.10
u2
FIGURE 1: THE EQUILIBRIUM E IS LOCALLY ASYMPTOTICALLY STABLE WITH 101 8.1
0.150.100.05 0.05 0.10 0.15u1
0.05
0.05
0.10
u2
20 40 60 80 100t
0.2
0.1
0.1
0.2
u1
20 40 60 80 100t
0.05
0.05
0.10
u2
FIGURE 2:THE EQUILIBRIUM E BECOMES UNSTABLE AND A HOPF BIFURCATION OCCURS WHEN 101 2.108
0.05 0.05 0.10u1
0.04
0.02
0.02
0.04
0.06
0.08
0.10
u2
20 40 60 80 100t
0.05
0.05
0.10
u1
20 40 60 80 100t
0.05
0.05
0.10
u2
FIGURE 3:THE EQUILIBRIUM E IS LOCALLY ASYMPTOTICALLY STABLE WITH 022 3.1
International Journal of Engineering Research & Science (IJOER) ISSN: [2395-6992] [Vol-2, Issue-5 May- 2016]
Page | 12
0.05 0.05 0.10u1
0.05
0.05
0.10
u2
20 40 60 80 100t
0.05
0.05
0.10
u1
20 40 60 80 100t
0.05
0.05
0.10
u2
FIGURE 4:THE EQUILIBRIUM E BECOMES UNSTABLE AND A HOPF BIFURCATION OCCURS WHEN 022 381.1
0.05 0.05 0.10u1
0.04
0.02
0.02
0.04
0.06
0.08
0.10
u2
20 40 60 80 100t
0.05
0.05
0.10
u1
20 40 60 80 100t
0.04
0.02
0.02
0.04
0.06
0.08
0.10
u2
FIGURE 5:THE EQUILIBRIUM E IS LOCALLY ASYMPTOTICALLY STABLE WITH 1021 0.9
0.05 0.05 0.10u1
0.04
0.02
0.02
0.04
0.06
0.08
0.10
u2
20 40 60 80 100t
0.05
0.05
0.10
u1
20 40 60 80 100t
0.05
0.05
0.10
u2
FIGURE 6: THE EQUILIBRIUM E BECOMES UNSTABLE AND A HOPF BIFURCATION OCCURS WHEN
1021 0.9565
0.05 0.05 0.10u1
0.04
0.02
0.02
0.04
0.06
0.08
0.10
u2
20 40 60 80 100t
0.05
0.05
0.10
u1
20 40 60 80 100t
0.05
0.05
0.10
u2
FIGURE 7:THE EQUILIBRIUM E IS LOCALLY ASYMPTOTICALLY STABLE WITH 2.1,0.1 211
0.05 0.05 0.10u1
0.05
0.05
0.10
u2
20 40 60 80 100t
0.05
0.05
0.10
u1
20 40 60 80 100t
0.05
0.05
0.10
u2
FIGURE 8:THE EQUILIBRIUM E BECOMES UNSTABLE AND A HOPF BIFURCATION OCCURS WHEN
2.1,0.235 211
0.15 0.10 0.05 0.05 0.10u1
0.05
0.05
0.10
u2
20 40 60 80 100t
0.15
0.10
0.05
0.05
0.10
0.15
u1
20 40 60 80 100t
0.05
0.05
0.10
u2
FIGURE 9:THE EQUILIBRIUM E IS LOCALLY ASYMPTOTICALLY STABLE WITH
221 2.0,1.8
International Journal of Engineering Research & Science (IJOER) ISSN: [2395-6992] [Vol-2, Issue-5 May- 2016]
Page | 13
0.10 0.05 0.05 0.10u1
0.05
0.05
0.10
u2
20 40 60 80 100t
0.10
0.05
0.05
0.10
u1
20 40 60 80 100t
0.05
0.05
0.10
u2
FIGURE 10:THE EQUILIBRIUM E BECOMES UNSTABLE AND A HOPF BIFURCATION OCCURS WHEN
221 629.0,1.8
V. CONCLUSION
Since the anticipation of capital stock and its future value are directly interrelated, the government should consider the
expectation of capital stock in investment decisions at present stage. At the same time, the implementation of past investment
decisions also need a pregnancy period which leads to production delay.
In this paper, the main contribution be lied in the following content: the first, we improved the traditional Kaldor-Kalecki
model with two delays in the gross product and the capital stock. We set up the Kaldor-Kalecki model of differential
equation with the two delays; The second, we study the stability and Hopf bifurcation. The results indicate that both capital
stock and investment lag are the certain factors leading to the occurrence of cyclical fluctuations in the macroeconomic
system. Moreover, the level of economic fluctuation can be dampened to some extent if investment decisions are made by the
reasonable short-term forecast on capital stock. And finally conduct numerical simulations to prove the conclusions.
The above arguments are well prepared to the further research work, but there are many undeveloped theory that need further
explore. The results of this paper can be used as qualitative analysis tool of mathematical economics and business
administration.
CONFLICT OF INTERESTS
The authors declare that there is no conflict of interests regarding the publication of this paper.
ACKNOWLEDGEMENTS
The authors are grateful to the referees for their helpful comments and constructive suggestions.
AUTHOR CONTRIBUTIONS
Conceived: Xiaohong Wang. Drawing graphics: Xiaohong Wang. Calculated: Xiaohong Wang. Modified: Yanhui Zhai.
Wrote the paper: Xiaohong Wang, Ka Long.
REFERENCES
[1] A. Krawiec, M. Szydlowski, The Kaldor-Kalecki business cycle model, Ann. Oper. Res. 89 (1999) 89-100.
[2] N. Kaldor, A model of the trade cycle, Econom. J. 40 (1940) 78-92.
[3] N. Kalecki, A macrodynamic theory of business cycles, Ecconometrica 3 (1935) 327-344.
[4] X. Xiuyan, Z. Jie, and W. Jiaqiu, “Hopf bifurcation analysis in Kaldor-Kalecki model of business cycle with two time delays,”
Journal of Harbin University of Commerce (Natural Science Edition), vol.28, no.4, pp.472–474, 2012.
[5] P. Xiaoqin, C Wua, W Liancheng, Multi-parameter bifurcations of the Kaldor-Kalecki model of business cycles with delay, Nonlinear
Analysis: Real World Applications 11 (2010) 869-887.
[6] T. Faria, L.T. Magalh?es, Normal forms for retarded functional differential equations and applications to Hopf bifurcation, J.
Differential Equations 122 (1995) 181-200.
[7] T. Faria, L.T. Magalh?es, Normal forms for retarded functional differential equations and applications to Bogdanov-Takens
singularity, J. Differential Equations 122 (1995) 201-224.
[8] J. Hale, L. Magalh?es, W. Oliva, Dynamics in Infinite Dimensions, Springer, 2002.
[9] Beretta E, Kuang Y. Geometric stability switch criteria in delay differential systems with delay dependent parameters. SIAM J Math
Anal 2002;33:1144–65.{2em}
[10] Kaldor N. A model of the trade cycle. Econ J 1940;50:78–92.
International Journal of Engineering Research & Science (IJOER) ISSN: [2395-6992] [Vol-2, Issue-5 May- 2016]
Page | 14
[11] Chang WW, Smyth DJ. The existence and persistence of cycles in a nonlinear model: Kaldor’s 1940 model re-examined. Rev Econ
Stud 1971;38:37–44.
[12] Grasman J, Wentzel JJ. Co-existence of a limit cycle and an equilibrium in Kaldor’s business cycle model and its consequences. J
Econ Behav Org 1994;24:369–77.
[13] Ackley G. Macroeconomic theory. New York: The MacMillan Company; 1961.
[14] Kalecki M. A macrodynamic theory of business cycles. Econometrica 1935;3:327–44.
[15] Krawiec A, Szydlowski M. The Kaldor–Kalecki business cycle model. Ann Oper Res 1999;89:89–100.
[16] Krawiec A, Szydlowski M. On nonlinear mechanics of business cycle model. Regul Chaotic Dyn 2001;6:101–18.
[17] Szydlowski M, Krawiec A. The stability problem in the Kaldor–Kalecki business cycle model. Chaos Solitons Fract 2005;25:299–305.
[18] Kaddar A, Alaoui HT. Local Hopf bifurcation and stability of limit cycle in a delayed Kaldor–Kalecki model. Nonlinear Anal: Modell
Control 2009;14:333–43.
[19] Wu XP. Codimension-2 bifurcations of the Kaldor model of business cycle. Chaos Solitons Fract 2011;44:28–42.