ISSN 1749-3889(print),1749-3897(online)International Journal of Nonlinear Science

Vol. 1 (2006) No. 1, pp. 49-53

Hopf Bifurcation Analysis ofthe Energy Resource Chaotic System

Mei Sun,Lixin Tian 1 , Jian YinNonlinear Scientific Research Center, Faculty of Science, Jiangsu University

Zhenjiang, Jiangsu, 212013, P.R. China(Received 22 September 2005, accepted 30 November 2005)

Abstract: In this paper, the Hopf bifurcation of the energy resource chaotic system is studiedby using an analytical method. It is shown that the energy resource chaotic system will displaya subcritical Hopf bifurcation under certain conditions.Keywords: energy resource chaotic system; Lyapunov coefficient; subcritical Hopf bifurca-tion

1 Introduction

Since Lorenz found the first classical chaotic attractor in 1963 [1], chaos, as a very interesting nonlinearphenomenon, has been intensively studied in the last decades [2-6]. It is found to be either useful or hasgreat potential in many fields, such as in engineering, biology and economics.

In paper [7], we set up a new system was named the energy resource system which is a third orderautonomous system. This new system displays very complex 2-scroll attractor. The value of Lyapunovexponents of this system is obtained as (0.068, 0.016, -0.016). This new attractor is different from the well-known Lorenz attractor, Rossler attractor, Chua’s attractor, Chen attractor, Lu attractor and so on. In paper[8], we further explored its dynamical behaviors based on two subsystems of the energy resource systemand achieved chaotic synchronization for the energy resource system by applied the modified adaptive syn-chronization. In paper [9], chaos in energy resource chaotic system is controlled to equilibrium points orperiodic orbits by using feedback control and adaptive control.

In this paper, Hopf bifurcation of the energy resource system is analyzed by using an analytical method.Analysis results show that this system will display Hopf bifurcation at two equilibria O(0, 0, 0), S2(x2, y2, z2)and Hopf bifurcation at equilibrium O(0, 0, 0) is nondegenerate and subcritical. It is shown that the energyresource chaotic system has complex dynamics with some interesting characteristics.

2 Energy resource chaotic system

Energy resource chaotic system [7] is described by the following system of differential equations:

dxdt = a1x

(1− x

M

)− a2 (y + z)dydt = −b1y − b2z + b3x[N − (x− z)]dzdt = c1z (c2x− c3)

(2.1)

where x(t) the energy resource shortage in A region, y(t) the energy resource supply increment in B region,z(t) the energy resource import in A region; ai, bi, ci, M, N are positive real constants. This system has

1Corresponding author. E-mail address: tianlx@ujs.ed.cn

Copyright c©World Academic Press, World Academic UnionIJNS.2006.01.15/07

50 International Journal of Nonlinear Science,Vol,1(2006),No.1,pp. 49-53

a chaotic attractor shown in Fig.1, when a1 = 0.09, a2 = 0.15, b1 = 0.06, b2 = 0.082, b3 = 0.07, c1 =0.2, c2 = 0.5, c3 = 0.4,M = 1.8, N = 1 and initial condition [0.82, 0.29, 0.48]. The value of Lyapunovexponents of this system is obtained as (0.068, 0.016, -0.016). This system has three equilibria:

O (0, 0, 0) , S1 = (x1, y1, z1) , S2 = (x2, y2, z2) , where x1 = a2b3MN−a1b1Ma2b3M−a1b1

y1 = a1b3(M−N)(a2b3MN−a1b1M)

(a2b3M−a1b1)2, z1 = 0, x2 = c3

c2

y2 =

ha1a2

(1−x2M )(b3x2−b2)−b3x2+b3N

ix2

b1+b3x2−b2, z2 =

a1b1a2

x2(1−x2M )−b3Nx2+b3x2

2

b1+b3x2−b2.

Figure 1: Energy resource attractor.

3 Hopf bifurcation

In this section we will deal with Hopf bifurcation of the energy resource chaotic system (2.1) using ananalytical method. We first introduce several definitions.

Let Cn is a linear space defined on the complex number field C. For any vectors x = (x1, x2, · · · , xn)T ,

y = (y1, y2, · · · , yn)T where xi, yi ∈ C (i = 1, 2, · · ·n), we call < x, y >=n∑

i=1xiyi inner product of the

vectors x, y. If we introduce the norm ‖x‖ =√

< x, x > , then Cn is a Hilbert space.Next, we consider the following nonlinear system

x = Ax + F (x) x ∈ Rn (3.2)

where F (x) = O(‖x‖2

)is a smooth function, which can be expanded as

F (x) =12B (x, x) +

16C (x, x, x) + O

(‖x‖4

)(3.3)

where B (x, x) and C (x, x, x) are bilinear and trilinear functions, respectively.In Eq. (3.2), if the matrix A has a pair of pure imaginary eigenvalues: λ1,2 = ±iω, ω > 0, let q ∈ Cn

be a complex eigenvector corresponding to the eigenvalue λ1, then we have

Aq = iωq, Aq = −iωq (3.4)

At the same time, we introduce the adjoint eigenvector p ∈ Cn which satisfies the following conditions

AT p = −iωp, AT p = iωp (3.5)

where < p, q >= 1.According to [10], the first Lyapunov coefficient of the system (2.1) at the origin point O can be written

as

l1(0) = 12ωRe[< p, C(q, q, q) > −2 < p, B(q, A−1B(q, q)) >

+ < p, B(q, (2iωE −A)−1B(q, q)) >](3.6)

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M. Sun,L. Tian, J. Yin: Hopf Bifurcation Analysis of the Energy Resource Chaotic System 51This formula is very convenient for analytical treatment of the Hopf bifurcation in n-dimensional sys-

tems with n ≥ 2. When l1 (0) < 0, the Hopf bifurcation is nondegenerate and supercritical; when l1 (0) > 0,the Hopf bifurcation is nondegenerate and subcritical.

The Jacobian of system (2.1) at the point O (0, 0, 0) is given by

a1 −a2 −a2

b3N −b1 −b2

0 0 −c1c3

For simplicity, we fix parameters:a2 = 0.15, b2 = 0.082, b3 = 0.07, c1 = 0.2, c2 = 0.5, c3 = 0.4, M = 1.8, N = 1.By calculations, we can obtain the characteristic polynomical of Jacobian matrix of the system (2.1) at

O (0, 0, 0) is

λ3 + (b1 − a1 + 0.08) λ2 + (0.0105− a1b1 + 0.08b1 − 0.08a1) λ− 0.08a1b1 + 0.00084 = 0 (3.7)

When b1 = bh = a1, Eq. (3.7) has pure imaginary roots λ1,2 = ±iω = i√

0.0105− a21 (if a1 <√

0.0105 = 0.1025), and they satisfyλ′b1 = λ2+(0.08−a1)λ−0.08a1

3λ2+2(b1−a1+0.08)λ+0.0105−a1b1+0.08b1−0.08a1,

thusα′ (0) = Reλ′b1 (bh) = −(a1−0.0715)(a1+0.1915)

4(0.0505−a21)

,

When 0.0715 < a1 < 0.1025, α′ (0) 6= 0.Besides, the third root λ3 = −0.08 < 0. Thus, a Hopf bifurcation takes place.In order to analyze the bifurcation (i.e., to determine the direction of the limit cycle bifurcation), we will

computer the first Lyapunov coefficient l1 (0) of the restricted system on the center manifold at the criticalparameter values.

When b1 = bh, the Jacobian of system (2.1) at the point O (0, 0, 0) is

A =

a1 −0.15 −0.150.07 −a1 −0.0820 0 −0.08

(3.8)

Next, we will calculate the corresponding vectors p, q of the matrix A. By tedious calculations, we have

q =

(1,

a1 − i√

0.0105− a21

0.15, 0

)T

(3.9)

and

p =0.00525

0.0105− a21 − a1i

√0.0105− a2

1

(1,−a1 + i

√0.0105− a2

1

0.07, 0

)T

(3.10)

which satisfy Aq = iωq, AT p = −iωp and < p, q >= 1.For the energy resource chaotic system (2.1), the bilinear and trilinear functions are

B(X, X ′) =

−2a1

M xx′

−2b3xx′ + b3 (xz′ + zx′)c1c2 (xz′ + zx′)

=

− a1

0.9xx′

−0.14xx′ + 0.07 (xz′ + zx′)0.1 (xz′ + zx′)

(3.11)

and

C(X, X ′, X ′′) =

000

(3.12)

respectively, where X = (x, y, z)T , X ′ = (x′, y′, z′)T and X ′′ = (x′′, y′′, z′′)T .

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52 International Journal of Nonlinear Science,Vol,1(2006),No.1,pp. 49-53

By some tedious manipulations, from (3.8) - (3.12), we have

B(q, q) = B(q, q) =

− a1

0.9−0.140

(3.13)

and

A−1 =1

0.08(a21 − 0.0105)

0.08a1 −0.012 0.0123− 0.15a1

0.0056 −0.08a1 0.082a1 − 0.01050 0 0.0105− a2

1

(3.14)

Let s = A−1B (q, q), then from (3.13) and (3.14), we have

B (q, s) =− a2

10.9 + 0.021(

a21 − 0.0105

)− a1

0.9−0.140

(3.15)

therefore

< p, B (q, s) >=0.00525

(− a2

10.9 + 0.021

)

a21 − 0.0105

·− a1

0.9 + 2(a1 − i

√0.0105− a2

1

)

0.0105− a21 + a1i

√0.0105− a2

1

(3.16)

Again, let s′ = (2iωE −A)−1 B (q, q), we can obtains′ = 1

3�2i√

0.0105−a21+0.08

�(a2

1−0.0105)B

where

B =

− a10.9

[4a2

1 + 0.08a1 − 0.042 + (0.16 + 2a1)√

0.0105− a21i

]

+0.14(0.012 + 0.3

√0.0105− a2

1i)

− a10.9

(0.056 + 0.14

√0.0105− a2

1i)

+0.14[0.042− 4a2

1 + 0.08a1 − (0.16− 2a1)√

0.0105− a21i

]

0

Let∆ = − a1

0.9

[4a2

1 + 0.08a1 − 0.042 + (0.16 + 2a1)√

0.0105− a21i

]

+0.14(0.012 + 0.3

√0.0105− a2

1i)

then

B(q, s′

)=

− a1

0.9∆−0.14∆0

(3.17)

< p, B(q, s′

)>=

0.00525∆

3(2i

√0.0105− a2

1 + 0.08) (

a21 − 0.0105

) ·− a1

0.9 + 2(a1 − i

√0.0105− a2

1

)

0.0105− a21 + a1i

√0.0105− a2

1

(3.18)Consequently, from (3.16) and (3.18), we have

l1 (0) =12ω

Re[−2 < p, B (q, s) > + < p, B

(q, s′

)>

]

=1.0288a1

(a2

1 + 0.2711) (

0.1235− a21

)(0.0105− a2

1

) 32(0.0121− a2

1

) . (3.19)

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M. Sun,L. Tian, J. Yin: Hopf Bifurcation Analysis of the Energy Resource Chaotic System 53When 0.0715 < a1 < 0.1025(=

√0.0105), the Lyapunov coefficient is clearly positive, so the Hopf

bifurcation is nondegenerate and subcritical, i.e. a unique and unstable cycle bifurcates from the equilibriumvia the Hopf bifurcation for b1 > bh = a1.

In paper [7], we consider the Hopf bifurcation of system (2.1) at the point S2 (x2, y2, z2) and get thefollowing conclusion: if b2 = 0.1013, then the equilibrium point S2 (x2, y2, z2) is a Holf bifurcation point.Similarly to the case of the equilibrium point O (0, 0, 0), Hopf bifurcation at equilibrium point S2 (x2, y2, z2)can be further explored.

4 Conclusion

This paper investigates the Hopf bifurcation of the energy resource system by an analytical method. Yet,there are still many unknown dynamics of the chaotic system studied in this paper which deserve furtherstudies in the near future.

5 Acknowledgements

Research were supported by the National Nature Science Foundation of China ( No. 90210004 ) andNational Social Science Foundation of China ( No. 02BJY051 ) and Social Science Foundation of JiangsuProvince of China ( No. 04SJD790010 ) and Nature Science Foundation of Jiangsu Province of China(No.05KJB110018).

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