- Email: [email protected]
Hopf bifurcation analysis of the Lü system
Chaos, Solitons and Fractals 21 (2004) 1215–1220 www.elsevier.com/locate/chaos
Hopf bifurcation analysis of the L€ u system q Yongguang Yu *, Suochun Zhang Institute of Applied Mathematics, Academy of Mathematics and System Science, Chinese Academy of Sciences, Beijing 100080, PR China Accepted 10 December 2003
Abstract In this paper, we introduce a new practical method for distinguishing the chaotic, periodic and quasi-periodic orbits, and analysis the Hopf bifurcation using an analytic technique for the L€ u system. As a result, we have further explored the dynamical behaviors. Ó 2004 Elsevier Ltd. All rights reserved.
1. Introduction Chaos, as a very interesting nonlinear phenomenon, has been intensively studied in the last decades [1,2]. It is found to be either useful or has great potential in many fields, such as in engineer, power network and secret communication technology. In 1963, Lorenz found the first canonical chaotic attractor [3], which is a third-order autonomous system with only two multiplication terms but displays very complex dynamical behaviors. Chen et al. found another similar but nonequivalent chaotic attractor in 1999 [4,5]. It has recently been proved that it is a dual system to the Lorenz system in a mathematical sense defined by Vanecek and Celikovsk y [6]: When these systems are separated into linear and nonlinear parts Ax þ f ðxÞ with A ¼ ½aij 33 , the linear part of the Lorenz system satisfies a12 a21 > 0 while the Chen system satisfies a12 a21 < 0. Recently L€ u et al. found another chaotic system [7,8], which satisfies the condition a12 a21 ¼ 0. So it represents the transition between the Lorenz and the Chen attractors [7,8]. For statement convenience, we call it the L€ u system. There has been some detailed investigations of the L€ u system [7–12]. Especially, it is found that the L€ u chaotic attractor has a compound structure by merging together two simple attractors after performing a mirror operation [9]. These results are obtained essentially from numerical simulations, and more theoretical analysis are still in order. In Section 2 we provide a new practical method for distinguishing the chaotic, periodic and quasi-periodic orbits. In this method, we first select a subsystem of the original system, then analyze the dynamical behaviors of the subsystem in a lower-dimension space, and finally analyze the dynamical behaviors of the whole system based on the subsystem. In Section 3 we use an analytical technique to analyze the Hopf bifurcation and in Section 4 we present a summary and conclusion.
2. Lu¨ system and its dynamical analysis The L€ u system can be described by the following differential equation:
q
Supported by the National Nature Science Foundation of China (Grant No. 10171099). Corresponding author. E-mail address: [email protected] (Y. Yu).
*
0960-0779/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2003.12.063
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Y. Yu, S. Zhang / Chaos, Solitons and Fractals 21 (2004) 1215–1220
8 < x_ ¼ aðy xÞ; y_ ¼ xz þ cy; : z_ ¼ xy bz;
ð1Þ
where a > 0, b > 0, which has a chaotic attractor as shown in Fig. 1 when a ¼ 36, b ¼ 3 and c ¼ 20. Next, we distinguish the chaotic, periodic and quasi-periodic orbits of the L€ u system (1) using the so-called complementary-cluster energy-barrier criterion [13]. Consider the first two equations of the L€ u system (1) x_ ¼ aðy xÞ; ð2Þ y_ ¼ xz þ cy; where z is regards as a known function of the time t. When t ¼ t0 , z is a constant number, then the system (2) is a twodimensional linear system with constant coefficients. Therefore its dynamical behavior is very simple and global. If z 6¼ c, then the system (2) only has an equilibrium Oð0; 0Þ and the corresponding characteristic polynomial is k2 þ ða cÞk þ aðz cÞ ¼ 0:
ð3Þ
We have the following results: (I) if z < c, then Eq. (3) has a pair of real roots with opposite sign, i.e. k1 k2 < 0. Therefore the equilibrium Oð0; 0Þ is a saddle in (x; y)-plane. 2 (II) if c < z < c þ ðacÞ and a > c, Eq. (3) has two negative real roots, i.e. k1 < 0, k2 < 0. Therefore the equilibrium 4a Oð0; 0Þ is a stable node in (x; y)-plane. 2 (III) if z > c þ ðacÞ and a > c, Eq. (3) has a pair of complex conjugate eigenvalues with negative real part, i.e. the equi4a librium Oð0; 0Þ is 2a stable focus. (IV) if c < z < c þ ðacÞ and a < c, Eq. (3) has a pair of positive real roots, then the equilibrium Oð0; 0Þ is a unstable 4a node. 2 (V) if z > c þ ðacÞ and a < c, Eq. (3) has a pair of complex conjugate eigenvalues with positive real part, i.e. the equi4a librium Oð0; 0Þ is a unstable focus. Next, under the assumption that a > c, we take a closer look to the dynamical behaviors of the L€ u system (1). In Fig. 2 2 we can see that the orbit zðtÞ goes through the straight lines z ¼ c and z ¼ cþ ðacÞ alternatively, 4a and repeat all the time. The z-axis is partitioned into three disjoint domains: ð1; cÞ, c; c þ ðacÞ 4a
2
2
and c þ ðacÞ ; þ1 by the two lines. 4a
According to the results above-obtained, we know that the system (2) has different dynamical behaviors in these three
30
20
y
10
0
–10
–20
–30 40 30
30 20 20
10 0 10
–10 –20
z
0
–30
x
Fig. 1. The L€ u chaotic attractor with a ¼ 36, b ¼ 3, c ¼ 20.
Y. Yu, S. Zhang / Chaos, Solitons and Fractals 21 (2004) 1215–1220
1217
2
Fig. 2. The chaotic time series of trajectory zðtÞ (a ¼ 36, b ¼ 3, c ¼ 20), z ¼ c ¼ 20 and z ¼ c þ ðacÞ ¼ 22. 4a
intervals. Therefore the L€ u system (1) changes the dynamical behaviors and zðtÞ goes through the three hyperplane repeatedly, i.e. it has complex dynamics such as the appearance of bifurcations and chaos. Note that the system (2) is a time-dependent system when zðtÞ is varying with time t.2 From Fig. 2, one can know that the system (1) is chaotic and zðtÞ goes through the hyperplane z ¼ c and z ¼ c þ ðacÞ alternatively. Furthermore, the 4a time-delay are not same and distinct relatively when zðtÞ goes through these two planes, so it is different from a periodic 2 orbit. However the time-delay is always same when zðtÞ goes through the hyperplane z ¼ c itself (or z ¼ c þ ðacÞ ). For 4a quasi-period orbits the time-delay is not same but distinct with one another, i.e. there is multiple correlation with one another.
3. Hopf bifurcation In this section we will deal with Hopf bifurcation of the L€ u system (1) using an analytical method. We first introduce several definitions. T Suppose that C n is a linear space defined on the complex number Pn field C. For any vectors x ¼ ðx1 ; x2 ; . . . ; xn Þ , T y ¼ ðy1 ; y2 ; . . . ; yn Þ where xp yi 2 ffiC (i ¼ 1; 2; . . . ; n), we call hx; yi ¼ i¼1 xi yi inner product of the vectors x; y. If we i ; ffiffiffiffiffiffiffiffiffiffi introduce the norm kxk ¼ hx; xi, then C n is a Hibert space. Next, we consider the following nonlinear system x_ ¼ Ax þ F ðxÞ x 2 Rn ;
ð4Þ
2
where F ðxÞ ¼ Oðkxk Þ is a smooth function, and it can be expanded into 1 1 F ðxÞ ¼ Bðx; xÞ þ Cðx; x; xÞ þ Oðkxk4 Þ 2 6
ð5Þ
in which Bðx; xÞ and Cðx; x; xÞ are bilinear and trilinear functions, respectively. In Eq. (4), if the matrix A only has a pair of pure imaginary eigenvalues k1;2 ¼ ai, a > 0, let q 2 C n be a complex eigenvector corresponding to the eigenvalue k1 , then we have Aq ¼ iaq, A q ¼ ia q. At the same time, we introduce the adjoint eigenvector p 2 C n which satisfies the following conditions AT p ¼ iap;
AT p ¼ iap
ð6Þ
and hp; qi ¼ 1: According to Ref. [14], the first Lyapunov coefficient of the system (4) at the origin point O can be written as
ð7Þ
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Y. Yu, S. Zhang / Chaos, Solitons and Fractals 21 (2004) 1215–1220
l1 ð0Þ ¼
1 q; ð2iaE AÞ1 Bðq; qÞÞiÞ: Reðhp; Cðq; q; qÞi 2hp; Bðq; A1 Bðq; qÞÞi þ hp; Bð 2a
ð8Þ
Eq. (8) is very convenient for analytical treatment of the Hopf bifurcation in a n-dimensional nonlinear system (n P 2) [14]. Regarding the L€ u system (1), we can obtain the following results easily. (I) If c < 0, the L€ u system (1) has only one equilibrium Oð0; 0; 0Þ, and it is an asymptotically stable node. pffiffiffiffiffi (II) If c > 0, the L€ u system has three equilibria Oð0; 0; 0Þ, S ðx0 ; y0 ; z0 Þ and Sþ ðx0 ; y0 ; z0 Þ where x0 ¼ y0 ¼ bc and z0 ¼ c, and the origin point Oð0; 0; 0Þ is a saddle. So the L€ u system displays bifurcation at c ¼ 0. The Jacobian matrix of the L€ u system (1) at the point (x; y; z) is 0 1 a a 0 @ z c x A: y x b
ð9Þ
The above-matrix at Oð0; 0; 0Þ has no eigenvalues such as k1;2 ¼ ixðx > 0Þ, so Oð0; 0; 0Þ is not the Hopf bifurcation point of the L€ u system (1). Through simple calculation, we know that the system (1) has the same characteristic polynomial at the points S and Sþ , which is f ðkÞ ¼ k3 þ ða þ b cÞk2 þ abk þ 2abc ¼ 0:
ð10Þ
Suppose that Eq. (10) has pure imaginary roots k1;2 ¼ ixðx > 0Þ, then we have x3 i ða þ b cÞx2 þ abxi þ 2abc ¼ 0: From Eq. (11), 2abc ða þ b cÞx2 ¼ 0; abx x3 ¼ 0:
ð11Þ
ð12Þ
Solve the Eq. (12), we have c ¼ ch ¼
aþb : 3
ð13Þ
The Equation of Hopf bifurcation plane corresponding to the system (1) is given by Eq. (10), that is Eq. (13). To keep the presentation short, we will only consider the case b ¼ a, we can similarly study the bifurcation of the L€ u system (1) at the other points on the Hopf bifurcation plane (13). When b ¼ a, the Eq. (13) can be written as c ¼ 2a3 . Next, we study the Hopf bifurcation of the L€ u system (1) on the line b ¼ a, c ¼ 2a3 except the origin point. From Eq. (13), we know that Eq. (10) has the purely conjugate imaginary roots k1;2 ¼ ix (x ¼ a), and they satisfy k0 ðcÞ ¼
k2 2ab ; 3k þ 2ða þ b cÞk þ ab
ð14Þ
2
thus a0 ð0Þ Re k0 ðch Þ ¼
27ab 18ab þ 8ða þ bÞ2
6¼ 0:
ð15Þ
6¼ 0. Besides, the third root k3 ¼ 2ðaþbÞ 3 Due to the symmetry of S and Sþ , it is sufficient to analyze the Hopf bifurcation at the point Sþ . When c ¼ ch , the Jacobian matrix of the L€ u system (1) at Sþ is 0 1 a a 0pffiffi 2a 2a 36a A: ð16Þ A ¼ @ pffiffi 3 p3ffiffi 6a 6a a 3 3
Y. Yu, S. Zhang / Chaos, Solitons and Fractals 21 (2004) 1215–1220
1219
Next, we will calculate the corresponding vectors p, q of the matrix A as in Eq. (8). By tedious calculations, we have 0 1 1 ð17Þ q ¼ @ pffiffi1 þpiffiffi A 6 6 i 2 6 and 0 pffiffi pffiffi 1 6 pffiffiffi þ 6i 6 B 2pffiffi 6pffiffi C 6 6 A p¼ @ 6 8i 2 þ 6 i 1 which satisfy Aq ¼ iaq, At p ¼ iap and hp; qi ¼ 1. For the L€ u system (1), the bilinear and trilinear functions are 0 1 0 BðX ; X 0 Þ ¼ @ xz0 A xy 0
ð18Þ
ð19Þ
and 0 1 0 CðX ; X ; X Þ ¼ @ 0 A; 0 0
00
respectively, where X ¼ ðx; y; zÞT , X 0 ¼ ðx0 ; y 0 ; z0 ÞT and X 00 ¼ ðx00 ; y 00 ; z00 ÞT . By some tedious manipulations, from (16)–(20), we have 0 1 pffiffi 0 pffiffi Bðq; qÞ ¼ @ 26 þ 66 i A; 1þi 0 1 pffiffi 0 pffiffi Bðq; qÞ ¼ @ 26 66 i A 1i
ð20Þ
ð21Þ
ð22Þ
and 0 B A1 ¼ @
4a3 p4a3ffiffi a6
0 1 paffiffi 6 3a
pffiffi 1 6 4affiffi p 6 C: A 4a
ð23Þ
0
Let s ¼ A1 Bðq; qÞ, then from (22) and (23), we have 0 1 0 3 1 A Bðq; sÞ ¼ @ p2a ffiffi p2affiffi i ; 5 6 8a6 i 8a
ð24Þ
therefore 27 þ 21i : 8að3 þ 4iÞ
hp; Bðq; sÞi ¼
Again, let s0 ¼ ð2ixE AÞ1 Bðq; qÞ, we can obtain 1 0 0 1 C B Bðq; s0 Þ ¼ @ pffiffi 3a A;
ð25Þ
ð26Þ
6ð1þ3iÞ 4að23iÞ
hp; Bðq; s0 Þi ¼
219 57i : 25að2 þ 3iÞ
ð27Þ
1220
Y. Yu, S. Zhang / Chaos, Solitons and Fractals 21 (2004) 1215–1220
Consequently, from (24) and (26), we have 1 q; ð2ixE AÞ1 Bðq; qÞÞiÞ Reðhp; Cðq; q; qÞi 2hp; Bðq; A1 Bðq; qÞÞiÞ þ hp; Bð 2a 1 1 27 þ 21i 219 57i 1077 þ ¼ < 0: ¼ Reð2hp; Bðq; si þ hp; Bðq; s0 ÞiÞ ¼ Re 2a 2a 4að3 þ 4iÞ 25að2 þ 3iÞ 2600a2
l1 ð0Þ ¼
ð28Þ
According to the method in Ref. [14], we know that the system (1) displays a Hopf bifurcation when l1 ð0Þ < 0, which is non-degenerate and supercritical. When a þ b c 6¼ 0, on the Hopf bifurcation plane (13) we can similarly study the bifurcation of the L€ u system (1) at the other points. If a þ b c ¼ 0, the system (1) will display Fold-Hopf bifurcation.
4. Conclusion This paper investigates the Hopf bifurcation of the L€ u system by an analytical method, and introduces a new practical technique for distinguishing the chaotic, periodic and quasi-periodic orbits. Yet, there are still many unknown dynamics of the chaotic system studied in this paper which deserve further studies in the near future.
Acknowledgement The authors are grateful to Dr. L€ u Jinhu for helpful papers and suggestions.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
Chen G, Dong X. From chaos to order: Methodologies, perspectives and applications. Singapore: World Scientific; 1998. L€ u J, Lu J, Chen S. Chaotic time series analysis and its application. China: Wuhan University Press; 2002. Lorenz EN. Deterministic non-periodic flows. J Atmos Sci 1963;20:130–41. Chen G, Ueta T. Yet another chaotic attractor. Int J Bifurcat Chaos 1999;9:1465–6. Ueta T, Chen G. Bifurcation and chaos in Chen’s equation. In: Proc IEEE Int Symposium on Circuits and Systems, Geneva, Switzerland, 29–31 May 2000, vol. V. p. 505–8. Vanecek A, Celikovsk y S. Control systems: From linear analysis to synthesis of chaos. London: Prentice-Hall; 1996. L€ u J, Chen G. A new chaotic attractor coined. Int J Bifurcat Chaos 2002;12(3):659–61. L€ u J, Chen G, Zhang S. Dynamical analysis of a new chaotic attractor. Int J Bifurcat Chaos 2002;12(5):1001–15. L€ u J, Chen G, Zhang S. The compound structure of a new chaotic attractor. Chaos, Solitons & Fractals 2002;14(5):1–4. Yu Y, Zhang S. Adaptive backstepping control of the uncertain L€ u system. Chinese Phys 2002;11(12):1249–53. L€ u J, Chen G, Cheng D, Celikovsky S. Bridge the gap between the Lorenz system and the Chen system. Int J Bifurcat Chaos 2002;12(12). Yu Y, Zhang S. Controlling uncertain L€ u system using backstepping design. Chaos, Solitons & Fractals 2003;15(5):897–902. Xue Y. Quantitative study of general motion stability and an example on power system stability. China: Jiangsu Science and Technology Press; 1999. Kuznetsov YA. Elements of applied bifurcation theory. 2nd ed. New York: Springer-Verlag; 1998.
Hopf bifurcation analysis of the L€ u system q Yongguang Yu *, Suochun Zhang Institute of Applied Mathematics, Academy of Mathematics and System Science, Chinese Academy of Sciences, Beijing 100080, PR China Accepted 10 December 2003
Abstract In this paper, we introduce a new practical method for distinguishing the chaotic, periodic and quasi-periodic orbits, and analysis the Hopf bifurcation using an analytic technique for the L€ u system. As a result, we have further explored the dynamical behaviors. Ó 2004 Elsevier Ltd. All rights reserved.
1. Introduction Chaos, as a very interesting nonlinear phenomenon, has been intensively studied in the last decades [1,2]. It is found to be either useful or has great potential in many fields, such as in engineer, power network and secret communication technology. In 1963, Lorenz found the first canonical chaotic attractor [3], which is a third-order autonomous system with only two multiplication terms but displays very complex dynamical behaviors. Chen et al. found another similar but nonequivalent chaotic attractor in 1999 [4,5]. It has recently been proved that it is a dual system to the Lorenz system in a mathematical sense defined by Vanecek and Celikovsk y [6]: When these systems are separated into linear and nonlinear parts Ax þ f ðxÞ with A ¼ ½aij 33 , the linear part of the Lorenz system satisfies a12 a21 > 0 while the Chen system satisfies a12 a21 < 0. Recently L€ u et al. found another chaotic system [7,8], which satisfies the condition a12 a21 ¼ 0. So it represents the transition between the Lorenz and the Chen attractors [7,8]. For statement convenience, we call it the L€ u system. There has been some detailed investigations of the L€ u system [7–12]. Especially, it is found that the L€ u chaotic attractor has a compound structure by merging together two simple attractors after performing a mirror operation [9]. These results are obtained essentially from numerical simulations, and more theoretical analysis are still in order. In Section 2 we provide a new practical method for distinguishing the chaotic, periodic and quasi-periodic orbits. In this method, we first select a subsystem of the original system, then analyze the dynamical behaviors of the subsystem in a lower-dimension space, and finally analyze the dynamical behaviors of the whole system based on the subsystem. In Section 3 we use an analytical technique to analyze the Hopf bifurcation and in Section 4 we present a summary and conclusion.
2. Lu¨ system and its dynamical analysis The L€ u system can be described by the following differential equation:
q
Supported by the National Nature Science Foundation of China (Grant No. 10171099). Corresponding author. E-mail address: [email protected] (Y. Yu).
*
0960-0779/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2003.12.063
1216
Y. Yu, S. Zhang / Chaos, Solitons and Fractals 21 (2004) 1215–1220
8 < x_ ¼ aðy xÞ; y_ ¼ xz þ cy; : z_ ¼ xy bz;
ð1Þ
where a > 0, b > 0, which has a chaotic attractor as shown in Fig. 1 when a ¼ 36, b ¼ 3 and c ¼ 20. Next, we distinguish the chaotic, periodic and quasi-periodic orbits of the L€ u system (1) using the so-called complementary-cluster energy-barrier criterion [13]. Consider the first two equations of the L€ u system (1) x_ ¼ aðy xÞ; ð2Þ y_ ¼ xz þ cy; where z is regards as a known function of the time t. When t ¼ t0 , z is a constant number, then the system (2) is a twodimensional linear system with constant coefficients. Therefore its dynamical behavior is very simple and global. If z 6¼ c, then the system (2) only has an equilibrium Oð0; 0Þ and the corresponding characteristic polynomial is k2 þ ða cÞk þ aðz cÞ ¼ 0:
ð3Þ
We have the following results: (I) if z < c, then Eq. (3) has a pair of real roots with opposite sign, i.e. k1 k2 < 0. Therefore the equilibrium Oð0; 0Þ is a saddle in (x; y)-plane. 2 (II) if c < z < c þ ðacÞ and a > c, Eq. (3) has two negative real roots, i.e. k1 < 0, k2 < 0. Therefore the equilibrium 4a Oð0; 0Þ is a stable node in (x; y)-plane. 2 (III) if z > c þ ðacÞ and a > c, Eq. (3) has a pair of complex conjugate eigenvalues with negative real part, i.e. the equi4a librium Oð0; 0Þ is 2a stable focus. (IV) if c < z < c þ ðacÞ and a < c, Eq. (3) has a pair of positive real roots, then the equilibrium Oð0; 0Þ is a unstable 4a node. 2 (V) if z > c þ ðacÞ and a < c, Eq. (3) has a pair of complex conjugate eigenvalues with positive real part, i.e. the equi4a librium Oð0; 0Þ is a unstable focus. Next, under the assumption that a > c, we take a closer look to the dynamical behaviors of the L€ u system (1). In Fig. 2 2 we can see that the orbit zðtÞ goes through the straight lines z ¼ c and z ¼ cþ ðacÞ alternatively, 4a and repeat all the time. The z-axis is partitioned into three disjoint domains: ð1; cÞ, c; c þ ðacÞ 4a
2
2
and c þ ðacÞ ; þ1 by the two lines. 4a
According to the results above-obtained, we know that the system (2) has different dynamical behaviors in these three
30
20
y
10
0
–10
–20
–30 40 30
30 20 20
10 0 10
–10 –20
z
0
–30
x
Fig. 1. The L€ u chaotic attractor with a ¼ 36, b ¼ 3, c ¼ 20.
Y. Yu, S. Zhang / Chaos, Solitons and Fractals 21 (2004) 1215–1220
1217
2
Fig. 2. The chaotic time series of trajectory zðtÞ (a ¼ 36, b ¼ 3, c ¼ 20), z ¼ c ¼ 20 and z ¼ c þ ðacÞ ¼ 22. 4a
intervals. Therefore the L€ u system (1) changes the dynamical behaviors and zðtÞ goes through the three hyperplane repeatedly, i.e. it has complex dynamics such as the appearance of bifurcations and chaos. Note that the system (2) is a time-dependent system when zðtÞ is varying with time t.2 From Fig. 2, one can know that the system (1) is chaotic and zðtÞ goes through the hyperplane z ¼ c and z ¼ c þ ðacÞ alternatively. Furthermore, the 4a time-delay are not same and distinct relatively when zðtÞ goes through these two planes, so it is different from a periodic 2 orbit. However the time-delay is always same when zðtÞ goes through the hyperplane z ¼ c itself (or z ¼ c þ ðacÞ ). For 4a quasi-period orbits the time-delay is not same but distinct with one another, i.e. there is multiple correlation with one another.
3. Hopf bifurcation In this section we will deal with Hopf bifurcation of the L€ u system (1) using an analytical method. We first introduce several definitions. T Suppose that C n is a linear space defined on the complex number Pn field C. For any vectors x ¼ ðx1 ; x2 ; . . . ; xn Þ , T y ¼ ðy1 ; y2 ; . . . ; yn Þ where xp yi 2 ffiC (i ¼ 1; 2; . . . ; n), we call hx; yi ¼ i¼1 xi yi inner product of the vectors x; y. If we i ; ffiffiffiffiffiffiffiffiffiffi introduce the norm kxk ¼ hx; xi, then C n is a Hibert space. Next, we consider the following nonlinear system x_ ¼ Ax þ F ðxÞ x 2 Rn ;
ð4Þ
2
where F ðxÞ ¼ Oðkxk Þ is a smooth function, and it can be expanded into 1 1 F ðxÞ ¼ Bðx; xÞ þ Cðx; x; xÞ þ Oðkxk4 Þ 2 6
ð5Þ
in which Bðx; xÞ and Cðx; x; xÞ are bilinear and trilinear functions, respectively. In Eq. (4), if the matrix A only has a pair of pure imaginary eigenvalues k1;2 ¼ ai, a > 0, let q 2 C n be a complex eigenvector corresponding to the eigenvalue k1 , then we have Aq ¼ iaq, A q ¼ ia q. At the same time, we introduce the adjoint eigenvector p 2 C n which satisfies the following conditions AT p ¼ iap;
AT p ¼ iap
ð6Þ
and hp; qi ¼ 1: According to Ref. [14], the first Lyapunov coefficient of the system (4) at the origin point O can be written as
ð7Þ
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Y. Yu, S. Zhang / Chaos, Solitons and Fractals 21 (2004) 1215–1220
l1 ð0Þ ¼
1 q; ð2iaE AÞ1 Bðq; qÞÞiÞ: Reðhp; Cðq; q; qÞi 2hp; Bðq; A1 Bðq; qÞÞi þ hp; Bð 2a
ð8Þ
Eq. (8) is very convenient for analytical treatment of the Hopf bifurcation in a n-dimensional nonlinear system (n P 2) [14]. Regarding the L€ u system (1), we can obtain the following results easily. (I) If c < 0, the L€ u system (1) has only one equilibrium Oð0; 0; 0Þ, and it is an asymptotically stable node. pffiffiffiffiffi (II) If c > 0, the L€ u system has three equilibria Oð0; 0; 0Þ, S ðx0 ; y0 ; z0 Þ and Sþ ðx0 ; y0 ; z0 Þ where x0 ¼ y0 ¼ bc and z0 ¼ c, and the origin point Oð0; 0; 0Þ is a saddle. So the L€ u system displays bifurcation at c ¼ 0. The Jacobian matrix of the L€ u system (1) at the point (x; y; z) is 0 1 a a 0 @ z c x A: y x b
ð9Þ
The above-matrix at Oð0; 0; 0Þ has no eigenvalues such as k1;2 ¼ ixðx > 0Þ, so Oð0; 0; 0Þ is not the Hopf bifurcation point of the L€ u system (1). Through simple calculation, we know that the system (1) has the same characteristic polynomial at the points S and Sþ , which is f ðkÞ ¼ k3 þ ða þ b cÞk2 þ abk þ 2abc ¼ 0:
ð10Þ
Suppose that Eq. (10) has pure imaginary roots k1;2 ¼ ixðx > 0Þ, then we have x3 i ða þ b cÞx2 þ abxi þ 2abc ¼ 0: From Eq. (11), 2abc ða þ b cÞx2 ¼ 0; abx x3 ¼ 0:
ð11Þ
ð12Þ
Solve the Eq. (12), we have c ¼ ch ¼
aþb : 3
ð13Þ
The Equation of Hopf bifurcation plane corresponding to the system (1) is given by Eq. (10), that is Eq. (13). To keep the presentation short, we will only consider the case b ¼ a, we can similarly study the bifurcation of the L€ u system (1) at the other points on the Hopf bifurcation plane (13). When b ¼ a, the Eq. (13) can be written as c ¼ 2a3 . Next, we study the Hopf bifurcation of the L€ u system (1) on the line b ¼ a, c ¼ 2a3 except the origin point. From Eq. (13), we know that Eq. (10) has the purely conjugate imaginary roots k1;2 ¼ ix (x ¼ a), and they satisfy k0 ðcÞ ¼
k2 2ab ; 3k þ 2ða þ b cÞk þ ab
ð14Þ
2
thus a0 ð0Þ Re k0 ðch Þ ¼
27ab 18ab þ 8ða þ bÞ2
6¼ 0:
ð15Þ
6¼ 0. Besides, the third root k3 ¼ 2ðaþbÞ 3 Due to the symmetry of S and Sþ , it is sufficient to analyze the Hopf bifurcation at the point Sþ . When c ¼ ch , the Jacobian matrix of the L€ u system (1) at Sþ is 0 1 a a 0pffiffi 2a 2a 36a A: ð16Þ A ¼ @ pffiffi 3 p3ffiffi 6a 6a a 3 3
Y. Yu, S. Zhang / Chaos, Solitons and Fractals 21 (2004) 1215–1220
1219
Next, we will calculate the corresponding vectors p, q of the matrix A as in Eq. (8). By tedious calculations, we have 0 1 1 ð17Þ q ¼ @ pffiffi1 þpiffiffi A 6 6 i 2 6 and 0 pffiffi pffiffi 1 6 pffiffiffi þ 6i 6 B 2pffiffi 6pffiffi C 6 6 A p¼ @ 6 8i 2 þ 6 i 1 which satisfy Aq ¼ iaq, At p ¼ iap and hp; qi ¼ 1. For the L€ u system (1), the bilinear and trilinear functions are 0 1 0 BðX ; X 0 Þ ¼ @ xz0 A xy 0
ð18Þ
ð19Þ
and 0 1 0 CðX ; X ; X Þ ¼ @ 0 A; 0 0
00
respectively, where X ¼ ðx; y; zÞT , X 0 ¼ ðx0 ; y 0 ; z0 ÞT and X 00 ¼ ðx00 ; y 00 ; z00 ÞT . By some tedious manipulations, from (16)–(20), we have 0 1 pffiffi 0 pffiffi Bðq; qÞ ¼ @ 26 þ 66 i A; 1þi 0 1 pffiffi 0 pffiffi Bðq; qÞ ¼ @ 26 66 i A 1i
ð20Þ
ð21Þ
ð22Þ
and 0 B A1 ¼ @
4a3 p4a3ffiffi a6
0 1 paffiffi 6 3a
pffiffi 1 6 4affiffi p 6 C: A 4a
ð23Þ
0
Let s ¼ A1 Bðq; qÞ, then from (22) and (23), we have 0 1 0 3 1 A Bðq; sÞ ¼ @ p2a ffiffi p2affiffi i ; 5 6 8a6 i 8a
ð24Þ
therefore 27 þ 21i : 8að3 þ 4iÞ
hp; Bðq; sÞi ¼
Again, let s0 ¼ ð2ixE AÞ1 Bðq; qÞ, we can obtain 1 0 0 1 C B Bðq; s0 Þ ¼ @ pffiffi 3a A;
ð25Þ
ð26Þ
6ð1þ3iÞ 4að23iÞ
hp; Bðq; s0 Þi ¼
219 57i : 25að2 þ 3iÞ
ð27Þ
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Y. Yu, S. Zhang / Chaos, Solitons and Fractals 21 (2004) 1215–1220
Consequently, from (24) and (26), we have 1 q; ð2ixE AÞ1 Bðq; qÞÞiÞ Reðhp; Cðq; q; qÞi 2hp; Bðq; A1 Bðq; qÞÞiÞ þ hp; Bð 2a 1 1 27 þ 21i 219 57i 1077 þ ¼ < 0: ¼ Reð2hp; Bðq; si þ hp; Bðq; s0 ÞiÞ ¼ Re 2a 2a 4að3 þ 4iÞ 25að2 þ 3iÞ 2600a2
l1 ð0Þ ¼
ð28Þ
According to the method in Ref. [14], we know that the system (1) displays a Hopf bifurcation when l1 ð0Þ < 0, which is non-degenerate and supercritical. When a þ b c 6¼ 0, on the Hopf bifurcation plane (13) we can similarly study the bifurcation of the L€ u system (1) at the other points. If a þ b c ¼ 0, the system (1) will display Fold-Hopf bifurcation.
4. Conclusion This paper investigates the Hopf bifurcation of the L€ u system by an analytical method, and introduces a new practical technique for distinguishing the chaotic, periodic and quasi-periodic orbits. Yet, there are still many unknown dynamics of the chaotic system studied in this paper which deserve further studies in the near future.
Acknowledgement The authors are grateful to Dr. L€ u Jinhu for helpful papers and suggestions.
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