Hopf Bifurcation Analysis for the Comprehensive National Strength Model with Time Delay

International Journal of Engineering Research & Science (IJOER)

ISSN: [2395-6992]

[Vol-2, Issue-3 March- 2016]

Hopf Bifurcation Analysis for the Comprehensive National Strength Model with Time Delay Xiao-hong Wang1, Yan-hui Zhai2 School of Science, Tianjin Polytechnic University, Tianjin

Abstract— This paper mainly modifies and further develops the comprehensive national strength model. By modifying the basic comprehensive national strength model, it can more accurately illustrate the society phenomena with time delay. First, we research the dynamics of the modified with time delay. By employing the normal form theory and center manifold method, we obtain some testable results on these issues. The conclusion confirms that a Hopf bifurcation occurs due to the existence of stability switches when the delay varies. Finally, some numerical simulations are given to illustrate the effectiveness of our result.

Keywords— Hopf bifurcation, Stability, Comprehensive national strength model, Center manifold, Normal form.

I.

THE ESTABLISHMENT OF THE MODEL

Ordinary differential equation is used to study the comprehensive national strength of a country's overall situation, which is gradually developed in recent years. This article brings in time delay in the comprehensive national strength model. Besides, the comprehensive national strength model models are detailed analyzed. x  x (t ) is hard power strength in t , which is a composite indicator the level of the material civilization (resource, economy, military, science and technology, etc.) of a

x(t ) bigger, the material civilization more prosperous. y (t )  0 is soft power strength in t , which is the spiritual civilization of a country. y (t )  0 stand for social evils (decision-making errors, education failure, official corruption people steal, etc.). At this time, soft power has obstacle to the social development. y (t )  0 mean soft power is superior (wise decision-making, national quality, etc.), that has a promoting effect on social development.  ,  ,  , m , M country. As

is constant. So, in the literature [1] comprehensive national strength shown by the following differential equation:

M x   x  x( M )  y,    y  y   (m  x) x.

(1.1)

As is known to all, some change of the soft power can be reflected on the hard power after a certain period of time, the literature [2] has established the following delay model of the comprehensive national strength.

M x   x  x( M )   y (t   ),    y  y   (m  x) x,

(1.2)

where is positive, and the other parameters are the same as of (1.1). While model of (1.2) had considered the time delay, in the real world, hard power strength not only rely on hard power strength of soft power in the past, but also rely on soft power.

In order to make the model more accurate, we modify the system of (1.2) to the following form:

M x   x  x( M )   (ky(t   )  dy),    y  y   (m  x) x,

(1.3)

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International Journal of Engineering Research & Science (IJOER)

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where k  d  1 . The organization of this paper is as follows: regarding  as bifurcation parameter, we study the stability of the equilibrium point of the system (1.3) and Hopf bifurcation of the equilibrium depending on  . Then, based on the new normal form of the differential-algebraic system introduced by Chen et al. [3] and the normal form approach theory and center manifold theory introduced by Hassard et al. [4], we derive the formula for determining the properties of Hopf bifurcation of the system in the third section. Numerical simulations aimed at justifying the theoretical analysis will be reported in Section 4. Finally, this paper ends with a discussion.

II.

STABILITY AND LOCAL HOPF BIFURCATION ANALYSIS

The stability and Hopf conclusions for the system of (1.1) can be obtained directly from the literature [5,6]. From system (1.3), we can see that there exists equilibriums.

E0  ( X 0 , Y0 )  (

M (  m ) M (m  M )(  m ) , ), E1  (0,0).   M (  M ) 2

According to the practical significance of the model, here we only discuss the problems of the hopf bifurcation and stability for the sole positive equilibrium point E0 .

Set u1 (t )  x(t )  X 0 , u2 (t )  y (t )  Y0 . The system (1.3) becomes

 2  u1 (t )   (1  p)u1  M u1   (ku2 (t   )  du2 ),   2 u2 (t )  u2   (m  Mp)u1  u1 .

Where p 

(2.1)

2(  m ) . And, the linearization of system (2.1) at E0 is    M

 u1 (t )   (1  p)u1   (ku2 (t   )  du2 ),  u2 (t )  u2   (m  Mp)u1.

(2.2)

The characteristic equation of system (2.2) is as follows.

2  c  ge   f  0, where

(2.3)

c   (1  p )   (m  Mp), f   (1  p )(m  Mp)  d , g  k . To study the stability of the

equilibrium point E0 and bifurcation system, we only need to discuss the distribution of the roots for characteristic Eq. (2.3).

If the all roots of equation (2.3) have negative real part, the equilibrium point E0 is steady. If the equation has a root that contains positive real part, the equilibrium point E0 is not stable. In order to study the distribution of the roots for Eq. (2.3).

Considering

  0 in first, characteristic equation (2.3) is 2  c  q  0.

(2.4)

Where q   (1  p )(m  Mp )   .

( H1 )c  0, q  0.

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International Journal of Engineering Research & Science (IJOER)

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Obviously, all roots of equation of (2.3) has negative real part if

[Vol-2, Issue-3 March- 2016]

( H1 ) is satisfied. So the system in equilibrium E0 is

  0 . Now, we investigate the local stability around the positive equilibrium point for the system (1.3) and the existence of Hopf bifurcations occurring at the equilibrium point E0 when  0  0 . locally asymptotically stable for

Lemma 2.1 For the system (1.3), Eq.(2.2) has a part of purely imaginary roots for Proof. If

  i

0  0

when

( H1 ) is satisfied.

is a solution of the characteristic equation (2.2), when and only when  meet

  2  ci  g (cos  i sin  )  f  0.

The separation of the real and imaginary parts yields.

  2  f  g cos ,  c  g sin  ,

(2.5)

v 2  (c 2  2 f )v  f 2  g 2  0,

(2.6)

which lead to

where v   . We assume that the coefficients satisfies the following conditions.

2

( H 21 )c 2  2 f  0,4( f 2  g 2 )  (c 2  2 f ) 2 , ( H 22 )c 2  2 f  0,4( f 2  g 2 )  0 .

If the condition

( H 21 ) or ( H 22 ) is satisfied, then Eq. (2.6) has positive roots. Therefore, Eq. (2.2) have purely imaginary

roots. From (2.6) we obtain

2 f  c 2  (c 2  2 f ) 2  4( f 2  g 2 ) , 0  2

j  As a result, when

1

0

(arccos(

f  (0 ) 2 )  2 j ), j  0,1,2,   . g

   j , the characteristic equation have a pair of purely imaginary root. The proof is completed.

Lemma 2.2 The transversality conditions

Re(

d 1 )  0 or ( H 22 ) is satisfied. d 0

Proof. By differentiating both sides of Eq. (2.3) with regard to



and solving

' ( ) . We have

d  ge    , d 2  c  ge  then

Re(

2  c  ge    i0 d 1  i0 ) |  j ) |  j  Re(  ge  d Page | 76

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

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 20 cos0 j  c sin 0 j g0 c 2  2 f  2 2  . g2

We can get c  2 f  2  0 by condition 2

2

Lemma 2.3 For Eq. (2.3), if

  [0, 0 ) ,

( H 21 ) or ( H 22 ) , thus Re(

all of his roots have negative real part. Positive equilibrium is asymptotically

stable, and the positive equilibrium produces Hopf bifurcation in

III.

d 1 )  0 . The proof is completed. d 0

 0 .

THE DIRECTION OF THE HOPF BIFURCATION AND THE STABILITY OF PERIODIC SOLUTIONS

In §2, we obtain the conditions of Hopf bifurcation. In this section, we discuss the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions based on the normal form introduced by Chen et al. [3] and the center manifold theory introduced by Hassard et al. [4].

In the following part, we assume that the system (1.3) undergoes Hopf bifurcation at the equilibrium E0 for

   j , and we

let i 0 is the corresponding purely imaginary root of the characteristic equation at the equilibrium E0 . Set

   0   ,   R , clearly,   0 is the Hopf bifurcation value of system (3.2). Set t  s , u i (t )  ui (t ), i  1,2 , for ui (t ) said u i (t ) .Then the system (1.3) is equivalent to the following Functional

convenience, we continue to use

Differential Equation (FDE) system in C  C ([1,0], R ) , 2



u(t )  L  F ( , ut ). where u (t )  (u1 (t ), u 2 (t )) , L T

(3.1)

: C  R, F : R  C  R are given, respectively, by

  2    1 (0)  L ( )  ( 0   ) B1 (0)  ( 0   ) B2 (1), F (  ,  )  ( 0   ) M ,    2 (0)  2   where

L 为   (1 , 2 )T  C ([1,0], R 2 ) .

 d   0  k   (1  p) .  , B2   B1   0   (m  Mp)  0   By the Riesz representation theorem, there exists a matrix function whose components are bounded variation function

 ( ,  ) : [1,0]  R 22

such that 0

L   d ( ,  ) ( ),   C. 1

(3.2)

In fact, we take Page | 77

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  ,    B1 ( )  B2 (  1).   

is a Delta function. For

(3.3)

  C ' ([1,0], R 2 ) , we define operator A and R

 d ( ( )) ,  A(  ) ( )   d  0 d ( ( ,  ) ( )), 1

as follows.

  [1,0), (3.4)

  0.

 [1,0),   0.

0, R(  ) ( )   F ( ,  ),

(3.5)

Then the system (2.1) can be written as the following form: 

ut  A(  )ut  R(  )ut Setting

  C ' ([0,1], ( R 2 ) ) , the adjoint operator A

(3.6) of A is defined as

 d ( s) s  (0,1],  ds , A  ( s)    0 d ( T (t ,0) (t )), s  0, 1

(3.7)

and a bilinear inner product is given by

  ( s ), ( )   (0) (0)  

0

1

From the discussions in Section 2, we know that we calculate the eigenvector





0

 (   )d ( ) ( )d .

(3.8)

i j0 are eigenvalues of A . Thus, they are also eigenvalues of A . Next

q( ) of A belonging to i j0 and eigenvector q  (s ) of A belonging to the eigenvalue

 i j0 . By the definition of A , we have A(0)q(0) 



0

1

dq( ) i   and q( )  q(0)e j 0 . In addition, d

d ( )q ( )  B1q (0)  B2 q(1)  A(0)q (0)  i0 q (0),

(3.9)

Let q (0)  (1, q2 ) , Eq.(3.9) becomes T

  (1  p)   (d  kei0 j )  1  1     i0  , q     (m  Mp)  q2   2  where

q2 

 (1  p)  i0 . i   (d  ke ) 0 j

dq ( s ) i  s   By the definition of A , we have A (0) q ( s )   and q ( s )  q (0)e 0 0 . Then, ds 





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

0

1



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q  (t )d (t )  B1T q  (0)  B2T q  (1)  A q  (0)  i0 q  (0).

(3.10)

 T

Let q (0)  D(1, q2 ) , Eq.(3.10) becomes

 (1  p)  1   1       i0   , i0 j )  (m  Mp)  q2     (d  ke  q2  

where q2 

 (1  p)  i0 

.



 q  ( s ), q ( )   q (0)q(0)  

0

1







0



0



 D[1  q2 q2   (1, q 2 ) e

q (   )d ( )q( )d .

i j 0

1



 D[1  q2 q2   j e

i0 j



1 d ( ) ]  q2 

 1 (1, q 2 ) B2  ]  q2 

 D[1  q2 q 2  k j q2 e 

where  q ( s ), q ( )  1 , so D  [1  q2 q 2  k j q2 e 

 i0 j

],

 i0 j 1

] .

Next, we study the specific parameters of the direction and size for bifurcated periodic solutions. Using the same notations as in Hassard et al. [4], we first compute the coordinates to describe the center manifold C 0 at

On the center manifold

  0 . Define

z (t )  q, ut ,W (t , )  ut ( )  2 Re{z (t )q( )}.

(3.11)

W (t , )  W ( z(t ), z(t ), )

(3.12)

C0 ,we have

2

z2 z Where W ( z, z, )  W20 ( )  W11 ( ) z z  W02 ( )     .

2 2 In fact, z and

z are local coordinates for center manifold C0 in the direction of q and q . Note that W is real if u t is

real. We consider only real solutions. For the solution of Eq.(3.6) in center manifold

C0 , we have



z ' (t )   q ( s), t    q * ( ), ( A(0)  R(0))t  

  A q  ( s ), t   q ( s ) R(0) t  

0





1   0



q (   )d ( ) A(0) R(0) t ( )d 

 i j0 q  ( s), t   q (0) F (0, t ( ))

(3.13) Page | 79

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Let

z ' (t )  i0 z  g ( z, z ),

(3.14)

where 2

z2 z g ( z, z )  g 20  g11 z z  g02    , 2 2

(3.15)

   1  F  g ( z , z )  q (0) F0 ( z, z )   j V    1    j V ( F1  q 2 F2 ).  q 2  F2 

(3.16)

ut ( )  (u1 (t   ), u2 (t   ))  W (t , )  z (t )q( )  z (t )q( )

(3.17)

from (3.13) and (3.15), we have

By (3.11), we have

where (1)

u1 (t   )  W (1) (t , )  z (t )q (1) ( )  z (t )q ( ) , ( 2)

u2 (t   )  W ( 2 ) (t ,  )  z (t )q ( 2 ) ( )  z (t )q ( ) .

Then    2 2   2    q 2 q22 ) z 2  (  2 q 2 q2 q22 ) z z  (   q2 q2 )z  M M M   1 1 [ (2W11(1) (0)  W20(1) (0)  W20( 2) (0))   q 2 (2W11( 2) (0)q2  W20( 2) (0)q 2 )]z 2 z     M 2 2

g ( z, z )   j D[(

Comparing the coefficients with (3.16), it follows that

g 20  2 j D ( g11   j D (

  2  2 q 2 q2 q22 ) , M

g 02  2 j D (

g 21  2 j D[

    q 2 q22 ) , M

 2    q2 q2 ) , M

  1 1 (2W11(1) (0)  W20(1) (0)  W20( 2 ) (0))   q 2 (2W11( 2 ) (0)q2  W20( 2 ) (0)q 2 )] .

M 2 2

By Eq.(3.15), (3.16) and definition (3.11) of W (t ,  ) , we have

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    [1,0),  A(0)W  2 Re q (0) F0 ( z (t ), z (t ))q( ), W  ut  z q  z q      A(0)W  2 Re q (0) F0 ( z (t ), z (t ))q( )  F0 ,   0. 









 AW  H ( z, z, )

(3.18)



Let

W  A(0)W  H ( z, z, ) . Where 2

z2 z H ( z, z, )  H 20 ( )  H11 ( ) z z  H 02 ( )     2 2

(3.19)

From the definition (3.11) of W (t ,  ) , we have 





W  Wz z  Wz z

(3.20)

By (3.18), (3.19) and (3.20), we have

H 20 ( )  2i 0W20 ( )  A(0)W20 ( ),

(3.21)

H11 ( )   A( )W11 ( ).

For

  [1,0] , by (3.14), (3.18) and (3.19), we get 

H ( z, z,0)  2 Re[q (0) F0 q( )] 2

z2 z zz 2  ( g 20  g11 z z  g02  g 21 )q( ) 2 2 2 2

 ( g 20

z2 z zz 2  g 11 z z  g 02  g 21 )q( ) 2 2 2

(3.22)

Comparing the coefficients with (3.19), we gives that

H 20 ( )   g 20 q( )  g 02 q( ), H11 ( )   g11q( )  g 11 q( ).

(3.23)

It follows from (3.21) that 

W 20 ( )  2i 0W20 ( )  A(0) g 20 q( )  g 20 q( ), 

(3.24)

W 11 ( )  g11q( )  g 11 q( ).

Solving for W20 ( ) and

W11 ( ) , we obtain

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International Journal of Engineering Research & Science (IJOER)

W20 ( ) 

ig 20

 0

q(0)ei 0  

ig11

W11 ( )  

ISSN: [2395-6992]

 0

q(0)e

i 0

i g 02 q(0)e i 0   E1e 2i 0  , 3 0



i g 11

 0

q(0)e

i 0

[Vol-2, Issue-3 March- 2016]

(3.25)

 E2 .

By (3.4) and (3.21), we get



d ( )W20 ( )  2i j0W20 (0)  H 20 (0) ,

0 1



d ( )W11 ( )   H11 (0)

0 1

(3.26)

From (3.17) and (3.18), we get

H 20 ( )   g 20 q(0)  g 02 q (0)  2 j ( E11 , E12 , E13 )T , H11 ( )   g11q (0)  g 11 q (0)   j ( E21 , E22 , E23 )T .

(3.27)

According to Eq.(3.25), (3.26) and (3.27), we get

(2i j0 I   01 e

2i j0

d ( ))E1  2 j ( E11 , E12 , E13 )T ,

(3.28)

Then

 2i0   (1  p)  (d  ke 2i0 j )   E1  2   2 i    ( m  Mp ) 0  

   (1  p)   E2    (m  Mp)   

1

1

      M 2 ,   q2 

  2    2  .  2  q q 2 2 

According to the above Proposition and [7,11], we can compute the following parameters:

C1 (0) 

2  

i 2 j0

( g 20 g11  2 g11 

1 g 2 g 02 )  21 , 3 2

Re{C1 (0)} , Re{' ( j )}

 2  2 Re{C1 (0)}, T2  

Im{C1 (0)}   2 (Im{' ( j )})

IV.

0 j

.

NUMERICAL SIMULATION

Numerical simulation shows from stable to unstable complex transformation process the system (1.3). We give a concrete example to show the dynamic behavior of comprehensive national strength model.

We take   0.1, M  10,   0.3, k  0.8, d  0.2,   1   2, m  9

in the system (1.3). So we can obtain Page | 82

International Journal of Engineering Research & Science (IJOER)

ISSN: [2395-6992]

[Vol-2, Issue-3 March- 2016]

equilibrium E0  (11.8,12.93) . Through calculating of the Mathematical software, we obtain lemma 2.2 and above conclusion, we can obtain that equilibrium E0 is stable if the equilibrium E0 is unstable if

  1.1   0 ,

0  2.62, 0  0.88 . By

  0.7   0 , (see Figure 1). By contrast,

(see Figure 2). Besides, when  0  0.886 , the periodic solutions occur

from the equilibrium E0 (see Figure 3-4). However our analysis indicates that the dynamics of the comprehensive national





strength model with time delay can be much more complicated than we may have expected. It is still interesting and inspiring to research. yt 30

yt 20

25

18

20

16

15

14

10

12



5

5

10

15

20

10

xt

25

8



FIGURE 1 THE EQUILIBRIUM E0 OF SYSTEM (1.3) IS STABLE WITH

10

12

14

16

18



xt

FIGURE 2 THE EQUILIBRIUM E0 OF SYSTEM (1.3) IS STABLE WITH 

  0.7   0

 1.1   0

yt 25 20 15 10

5

10

15

20



xt

FIGURE 3 SYSTEM (1.3) PRODUCE THE PERIODIC SOLUTIONS WITH  0  0.886

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

International Journal of Engineering Research & Science (IJOER)



xt

ISSN: [2395-6992]

[Vol-2, Issue-3 March- 2016]

yt

20

25

15

20

10

15 10

5 5

20

40

60

80

100

120

140

t 20

40

60

80

100

120

140

t

FIGURE 4 TIME SEQUENCE DIAGRAM OF FIGURE 3 V.

CONCLUSION

These papers apply delay in the comprehensive national strength model, which show rich dynamics behavior. Different from previous studies, we added the influence of time delay feedback in the system (1.3). Dynamic behavior of comprehensive national strength model with time delay is analyzed by using the method of quantitative. When delay $\tau$ across a series of critical value, nonlinear dynamic system generate the Hopf bifurcation. In addition, by employing the normal form theory and center manifold method, we obtain some testable results on these issues. We use normative theory and center of popular theorem obtained the calculating method of the direction of Hopf bifurcation and stability of periodic solutions. Finally, the above theoretical analysis is verified by numerical simulation. The dynamic behavior of the comprehensive national strength model is rich. Many aspects is not mining, yet to be. References

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.

REFERENCES [1] S.H. Wang, “Differential equation model and chaos,” Journal of China Science and Technology University, pp. 312–324, 1999.

[2] X.M Xin. Hopf Bifurcation Analysis in Comprehensive National Power Model with Two Delays[J].Journal of Yili Normal University(Natural Science Editio), vol.21, no.3, pp.1–4, 2007.

[3] B.S. Chen, X.X. Liao, Y.Q. Liu, Normal forms and bifurcations for the difference-algebraic systems, Acta Math. Appl. Sin. 23 (2000) 429–443 (in Chinese).

[4] B. Hassard, D. Kazarinoff, Y. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981.

[5] T.H. L and Z.W. Liu, “Hopf bifurcation of price Reyleigh equation with time delay,” Journal of Jilin University, vol.47, no.3, 2009.

[6] G.D. Zhang, L.L. Zhu, B.S. Chen, Hopf bifurcation and stability for a differential–algebraic biological economic system, Appl. Math.

Comput 217 (1) (2010) 330–338.

[7] Y. Wang and Y.H. Z, “Stability and Hopf bifurcation of differential equation model of price with time delay,” Highlights of Sciencepaper Online, vol.4, no.1, 2011.

[8] Y.H. Z, H.Y. B, Y. Xiong, and X.N. M, “Hopf bifurcation analysis for the modified Rayleigh price model with time delay,” Abstract and Applied Analysis, vol.2013,Article ID 290497, 6 pages, 2013.

[9] D. D. Hassard, N. D. Kazarinoff, and Y.H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, UK, 1981.

[10] J.H. Ma, Y.Q. Chen, L.X Liu.Hopf bifurcation and chaos of financial system on condition of specific combination of parameters.

Journal of Systems Science and Complexity 2008, Vol.21, no.2, pp.250-259, 2004.

[11] J. Hale, Theory of Functional Differential Equations, Springer, New York, NY, USA, 2nd edition, 1977.

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