Stability and Hopf bifurcation for Kaldor-Kalecki model of ... · Stability and Hopf bifurcation...

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.


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)


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.


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 .


.0)()( 22

abeaa

(2.10)

Let )0( 22 i is a root of (2.10). Then

.0)cossinsincos()( 22222222222

2

2 aaiaiia


,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



Page | 5


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


.0))(()(2 aebaa

(2.13)

Let )0( i is a root of (2.13). Then

.0)()(2

2 aebaiea


,0sin)(cos)(

,0cos)(sin

22

2

baa

aba

(2.14)


2222223 )(2))((2())(( baaabb

vab

av

,0))()(()()))(( 2222

abab

avbb

aa

(2.15)

where .2 v





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


Page | 6


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


,0sinsincos)(

,0coscossin)(

1

'

12

'

12

'

1

'

1

'

1

1

'

12

'

12

'

1

'

1

2'

1

baa

aba

(2.16)


2

'

1

2'

1

2'

12

'

1

2223'

12

'

1

4'

1 sin2))(cos2())(sin2()( aa

.0cos2)( 2

2

'

1

2222 baa

(2.17)





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




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)


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.


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



1)(

d

d.

)(

2 21

2

121

ea

beaee


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


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


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).


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


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


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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

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