Boundedness solutions of the complex Lorenz chaotic system

Applied Mathematics and Computation 243 (2014) 12–23

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Boundedness solutions of the complex Lorenz chaotic system Fuchen Zhang a,⇑, Guangyun Zhang b a b

College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China School of Foreign Languages, Southwest Petroleum University, Chengdu 610500, China

a r t i c l e

i n f o

Keywords: Lorenz system Global attractive set Generalized Lyapunov functions

a b s t r a c t This paper is concerned with the boundedness of solutions of the complex Lorenz system. We have obtained the global exponential attractive set Wk;m and the ultimate bound Xk;m for this system. Furthermore, we confirm that the rate of the trajectories of the system going from the exterior of the set Wk;m to the interior of the set Wk;m is an exponential rate. The rate of the trajectories is also obtained. Numerical simulations are presented to show the effectiveness of the proposed scheme. Ó 2014 Elsevier Inc. All rights reserved.

1. Introduction To estimate the boundedness of a chaotic system is a challenging but an interesting work in general [1–14]. A chaotic system is bounded, in the sense that its phase portraits are bounded in the phase space. And the ultimate boundedness of a chaotic system plays an important role in chaos control, chaos synchronization, and many other applications. If we can show that a chaotic or a hyperchaotic system has a global attractive set, then the system cannot possess hidden attractors outside the global attractive set. This is very important for engineering applications, since it is very difficult to predict the existence of hidden attractors and they can lead to crashes [15–17]. The boundedness of the Lorenz system were first studied by G.A. Leonov [18]. Then, the ultimate boundedness of other chaotic systems, including the synchronous motor system [19], a new chaotic system [20], the hyperchaotic Lorenz–Haken system [21], the Lü system [22] and the generalized Lorenz chaotic systems [23], was also studied and some important results were obtained. However, it is a very difficult task to estimate the boundedness of the chaotic systems [22,23]. The construction of new Lyapunov functions is always a piece of art, since there is no regular way to find one. Therefore, it is necessary to study the boundedness of the complex Lorenz chaotic system. Since Fowler et al. introduced the complex Lorenz equations [24], many complex chaotic systems have been proposed and studied in the last few decades. For example, Mahmoud et al. introduced the complex Chen and Lü systems [25]. It is well known that the complex chaotic systems have more widely applying space [25]. Such as secure communication, synchronization, control, etc. [25]. Another interesting application that discovered was the anti-synchronization, which has been investigated both experimentally and theoretically in many physical systems [26,27]. 2. Mathematical model State equations of the complex Lorenz chaotic system can be described as follows [28]:

⇑ Corresponding author. E-mail address: [email protected] (F. Zhang). http://dx.doi.org/10.1016/j.amc.2014.05.102 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.

F. Zhang, G. Zhang / Applied Mathematics and Computation 243 (2014) 12–23

13

8 > < y_ 1 ¼ aðy2  y1 Þ; y_ 2 ¼ by1  y2  y1 y3 ; > :_ 2 þ y 1 y2 Þ  cy3 ; y3 ¼ 12 ðy1 y

ð1Þ

8 u_ 1 > > > > > > < u_ 2 u_ 3 > > > > u_ 4 > > : u_ 5

ð2Þ

pffiffiffiffiffiffiffi 1 and y 2 are conjugate complex numbers of y1 and y2. Replacing comwhere y1 ¼ u1 þ ju2 ; y2 ¼ u3 þ ju4 ; y3 ¼ u5 ; j ¼ 1, y plex variables in system (1) with real number variables and imaginary number variables, Zhang et al. get an equivalent system as follows (see [28]):

¼ aðu3  u1 Þ; ¼ aðu4  u2 Þ; ¼ bu1  u3  u1 u5 ; ¼ bu2  u4  u2 u5 ; ¼ u1 u3 þ u2 u4  cu5 ;

where a, b, c are positive parameters of system (2). When a ¼ 35; b ¼ 55; c ¼ 83, the system (2) is chaotic [28]. Phase portraits of system (2) are shown in Figs. 1 and 2. Remark. While positive Lyapunov exponent is widely used as indication of chaos, rigorous consideration requires verification of additional properties of considered system (such as regularity, ergodicity), because of so-called Perron effects of Lyapunov exponents sign reversal (see excellent papers [29–31] for a detailed discussion of the attractor). Some basic dynamical properties of the complex Lorenz system (2) were studied in [28]. But many properties of the complex Lorenz remains unknown. In the following, we will discuss the boundedness of the complex Lorenz system (2). The following structure of this paper is organized as follows: In Section 3, we will study the ultimate boundedness of system (2). In Section 4, we will study the global exponential attractive set of system (2). Conclusion remarks will be given in Section 5. 3. The ultimate boundedness In this section, we will discuss the boundedness of the complex Lorenz system (2) for any a > 0, b > 0, c > 0. Before going into details, let us introduce the following lemma. Lemma 1. Define

(

) x22 ðz  cÞ2 y21 y22 ; þ þ þ þ ¼ 1; a – 0; b – 0; c – 0; d – 0; e – 0 2 a2 b2 c2 e2 d

 2  x1

P1 ¼ ðx1 ; x2 ; y1 ; y2 ; zÞ

ð3Þ

and

G1 ðx1 ; x2 ; y1 ; y2 ; zÞ ¼ x21 þ x22 þ y21 þ y22 þ z2 ; H1 ðx1 ; x2 ; y1 ; y2 ; zÞ ¼ x21 þ x22 þ y21 þ y22 þ ðz  2cÞ2 ;

ðx1 ; x2 ; y1 ; y2 ; zÞ 2 P1 :

Then, we have the conclusions that

max G1 ðx1 ;x2 ;y1 ;y2 ;zÞ2P1

¼

max H1 ðx1 ;x2 ;y1 ;y2 ;zÞ2P1

pffiffiffi 8 a4 ; a P b; a P d; a P e; a P 2c; > a2 c2 > > pffiffiffi > > b4 > ; b > a; b P d; b > e; b P 2c; > > < b2 c2 pffiffiffi d4 ¼ > d2 c2 ; d > a; d > b; d P e; d P 2c; > pffiffiffi > 4 > > > e2ec2 ; e > a; e P b; e > d; e P 2c; > > p ffiffiffi p ffiffiffi p ffiffiffi pffiffiffi : 2 4c ; a < 2c; b < 2c; d < 2c; e < 2c:

Proof. It can be easily proved by the Lagrange multiplier method. h Lemma 2. Define a set

(

)  2 2 x y2 ðz  ~cÞ w2 ~ – 0; ~c – 0; d ~–0 ; ~ – 0; b a þ þ þ ¼ 1; ~2 ~2 ~2 b ~c2 a d

C0 ¼ ðx; y; z; wÞ and

2 Gðx; y; z; wÞ ¼ x2 þ y2 þ z2 þ w2 ; Hðx; y; z; wÞ ¼ x2 þ y2 þ ðz  2~cÞ þ w2 ;

ðx; y; z; wÞ 2 C0 :

ð4Þ

F. Zhang, G. Zhang / Applied Mathematics and Computation 243 (2014) 12–23 40

u (3)

20 0 -20 -40 -40

30 20

-20

10

0

0 20

-10 40

u (2)

-20

u (1)

60 40

u (4)

20 0 -20 -40 40 20 0 -20 -40

u (3)

-20

0

-10

30

20

10

u (1)

60 40

u (4)

20 0 -20 -40 40 20

40 20

0 -20 -40

u (3)

-20 -40

0 u (2)

120 100 80 u (5)

14

60 40 60

20 0 -40

40 20 -20

0 0

20

-20 40

-40

u (4)

Fig. 1. Phase portraits of system (2) in 3D space.

u (3)

F. Zhang, G. Zhang / Applied Mathematics and Computation 243 (2014) 12–23 40 30 20

u (3)

10 0 -10 -20 -30 -20

-15

-10

-5

0

5

10

15

20

25

u (1)

50 40 30

u (4)

20 10 0 -10 -20 -30 -40 -30

-20

-10

0

10

20

10

20

30

40

u (2)

120

100

u (5)

80

60

40

20

0 -30

-20

-10

0

30

40

u (3) 120

100

u (5)

80

60

40

20

0 -40

-30

-20

-10

0

10

20

30

40

50

u (4)

Fig. 2. Projections of the phase portraits of (2) into planes.

15

16

F. Zhang, G. Zhang / Applied Mathematics and Computation 243 (2014) 12–23

Then, we have the conclusions that

pffiffiffi 8 a~4 ~ a ~ a ~ P b; ~ P d; ~ P 2~c; ; a > > > a~2 ~c2 p ffiffiffi > > ~>a ~ P d; ~ b ~ P 2~c; < b~4 ; b ~; b ~2 ~c2 b max G ¼ max H ¼ pffiffiffi ~4 > ðx;y;z;wÞ2C0 ðx;y;z;wÞ2C0 d ~ ~ ~ ~ ~ > 2~c; > ~ 2 ~c2 ; d > a; d > b; d P d > > pffiffiffi pffiffiffi pffiffiffi : 2 ~ ~ ~ < 2~c; b < 2~c; d < 2~c: 4~c ; a Proof. It can be easily proved by the Lagrange multiplier method. By Lemma 1 and 2, we can get the following theorem for the complex Lorenz system (2).

h

Theorem 3. For any m > 0; k > 0; a > 0; b > 0; c > 0; the following set

(

)  2  2 kb þ am 2 2 2 2  ðu1 ; u2 ; u3 ; u4 ; u5 Þ mu1 þ mu2 þ ku3 þ ku4 þ k u5  6R ; k

Xk;m ¼

ð5Þ

is the ultimate bound and positively invariant set of the complex Lorenz system (2), where,

R2 ¼

8 2 c ðbkþamÞ2 > ; a 6 1; c P 2a; > > < 4aðcaÞk 2

2

c ðbkþamÞ ; c P 2; a > 1; 4ðc1Þk > > > : ðbkþamÞ2 ; c < 2; c < 2a: k

Proof. Define the following generalized positively definite and radially unbounded Lyapunov function

 2 bk þ am V k;m ðUÞ ¼ V k;m ðu1 ; u2 ; u3 ; u4 ; u5 Þ ¼ mu21 þ mu22 þ ku23 þ ku24 þ k u5  ; k

ð6Þ

where k > 0; m > 0; U ¼ ðu1 ; u2 ; u3 ; u4 ; u5 Þ. Computing the derivative of V k;m ðUÞ along the trajectory of (2), we have,



dV k;m ðUÞ  dt

ð2Þ

  ¼ 2mu1 u_ 1 þ 2mu2 u_ 2 þ 2ku3 u_ 3 þ 2ku4 u_ 4 þ 2k u5  bkþam u_ 5 ; k ¼ 2amu1 ðu3  u1 Þ þ 2amu2 ðu4  u2 Þ þ 2ku3 ðbu1  u3  u1 u5 Þ þ 2ku4 ðbu2  u4  u2 u5 Þ   þ2k u5  kbþam ðu1 u3 þ u2 u4  cu5 Þ; k ¼ 2amu21  2amu22  2ku23  2ku24  2kcu25 þ 2cðbk þ amÞu5 ;  2 cðkbþamÞ2 þ 2k : ¼ 2amu21  2amu22  2ku23  2ku24  2kc u5  kbþam 2k

Obviously, V k;m ðUÞ is positively definite for a > 0; b > 0; c > 0; k > 0; m > 0. Let V_ k;m ðUÞ ¼ 0. Then, we can get the surface C:

(

)  2 2  kb þ am cðkb þ amÞ ; ðu1 ; u2 ; u3 ; u4 ; u5 Þamu21 þ amu22 þ ku23 þ ku24 þ ck u5  ¼ 2k 4k

ð7Þ

is an ellipsoid in R5 for a > 0; b > 0; c > 0; k > 0; m > 0. Outside C, V_ k;m ðUÞ < 0; while inside C, V_ k;m ðUÞ > 0. Since the V k;m ðUÞ is a continuous function and C is a bounded close set, then the function V k;m ðUÞ that defined in (6) can reach its max2 imum value max in (7).  2 CÞ on the surface C that defined  V k;m ðUÞ ¼ R ; ðU Obviously, ðu1 ; u2 ; u3 ; u4 ; u5 ÞV k;m ðUÞ 6 max V k;m ðUÞ; U 2 C contains the solutions of the system (2). By solving the following conditional extremum problem, one can get the maximum value of the function (6):

8 n   o 2 2 2 2 kbþam 2 > ; > < max V k;m ðUÞ ¼ max mu1 þ mu2 þ ku3 þ ku4 þ k u5  k > > : s:t:

mu21

cðkbþamÞ2 4ak

mu2

ku2

4ak

4k

2

ku2

kðu5 kbþam 2k Þ

4k

ðkbþamÞ2 4k

2 3 4 þ cðkbþamÞ 2 þ cðkbþamÞ2 þ cðkbþamÞ2 þ

ð8Þ

¼ 1:

pffiffiffi pffiffiffi pffiffiffi pffiffiffiffiffi pffiffiffiffiffi ~ 1 ; mu2 ¼ u ~ 2 ; ku3 ¼ u ~ 3 ; ku4 ¼ u ~ 4 ; ku5 ¼ u ~ 5 as the new variable, then (8) transforms into the Let us take mu1 ¼ u following form:

8

2 > 2 2 2 2 kbþam > p ffiffi ~ ~ ~ ~ ~ ; u u u u u max V ðUÞ ¼ max þ þ þ þ  > k;m 5 1 2 3 4 > k <

2 > ~ 5 kbþam pffi u > ~2 ~2 ~2 ~2 u u u u 2 k > 1 2 3 4 > ¼ 1: : s:t: cðbkþamÞ 2 þ cðbkþamÞ2 þ cðbkþamÞ2 þ cðbkþamÞ2 þ ðbkþamÞ2 4ak

4ak

4k

4k

4k

ð9Þ

F. Zhang, G. Zhang / Applied Mathematics and Computation 243 (2014) 12–23

17

By Lemma 1 and 2, we can easily get the above conditional extremum problem (9) as

max V k;m ðUÞ ¼ R2 ¼ U2C

8 2 c ðbkþamÞ2 > ; a 6 1; c P 2a; > > < 4aðcaÞk 2

2

c ðbkþamÞ ; c P 2; a > 1; 4ðc1Þk > > > : ðbkþamÞ2 ; c < 2; c < 2a: k

Finally, we should only show that (5) is the ultimate bound and positively invariant set of system (2). For Xk;m as shown in (5), the set (7) is contained in Xk;m . Next, we will show:

lim qðUðtÞ; Xk;m Þ ¼ 0:

ð10Þ

t!þ1

Using the reduction to absurdity, where U(t) = (u1(t), u2(t), u3(t), u4(t), u5(t)). If (10) does not hold, then we can conclude that the orbits of the system (2) are outside the set Xk;m permanently, thus V_ k;m < 0. Therefore, V k;m ðUðtÞÞ monotonously decreases outside the set Xk;m , which leads to the following result,

lim V k;m ðUðtÞÞ ¼ V~ k;m > R2 :

t!þ1

Let

 d ¼ inf V_ k;m ðUðtÞÞ ; U2D

n o ~ k;m 6 V k;m ðUðtÞÞ 6 V k;m ðUðt 0 ÞÞ , while t0 is the initial time. Consequently, we have the conclusion that where D ¼ UðtÞjV ~ k;m are both positive constants, and d; V

dV k;m ðUðtÞÞ 6 d: dt As t ? +1, we have

0 6 V k;m ðUðtÞÞ 6 V k;m ðUðt0 ÞÞ  dðt  t 0 Þ ! 1: That is a contradiction. Therefore, (10) holds, that is equivalent to say, the set Xk;m is the ultimate bound of system (2). Finally, we claim that Xk;m is also the positively invariant set of system (2), reasons as follows: suppose V k;m ðUðtÞÞ attains ~ 10 ; u ~ 20 ; u ~ 30 ; u ~ 40 ; u ~ 50 Þ. Since (7) is contained in Xk;m , for any point U(t) on its maximums value on surface (7) at the point P0 ðu (7) and U(t) – P0, we have V_ k;m < 0. Thus, any orbit U(t) (U(t) – P0) of system (2) will go into the set Xk;m . When U(t) = P0, by the Continuation Theorem [32], U(t) will also go into the set Xk;m . Summarizing the contents above, we conclude that Xk;m is also the positively invariant set of system (2). This completes the proof. h

Fig. 3. Phase portraits of system (2) in the (u2, u3, u4) space defined by X1,1.

18

F. Zhang, G. Zhang / Applied Mathematics and Computation 243 (2014) 12–23

Remark. (i) Let us take k ¼ 1; m ¼ 1; then we can get that

n

o



X1;1 ¼ ðu1 ; u2 ; u3 ; u4 ; u5 Þu21 þ u22 þ u23 þ u24 þ ðu5  b  aÞ2 6 l2 ; is the ultimate bound and positively invariant set of the complex Lorenz system (2), where,

2

l ¼

8 2 c ðbþaÞ2 > > > 4aðcaÞ ; a 6 1; c P 2a; < > > > :

c2 ðbþaÞ2 4ðc1Þ

; c P 2; a > 1; 2

ðb þ aÞ ; c < 2; c < 2a:

When a ¼ 35; b ¼ 55; c ¼ 83, we can obtain that

n

 

pffiffiffiffiffiffi 2 o ;

X1;1 ¼ ðu1 ; u2 ; u3 ; u4 ; u5 Þu21 þ u22 þ u23 þ u24 þ ðu5  90Þ2 6 ð24 15Þ

is the ultimate bound and positively invariant set of the complex Lorenz system (2). Fig. 3 shows phase portraits of system (2) in the (u2, u3, u4) space defined by X1,1. Fig. 4 shows the projections of X1,1 onto the planes (u2, u4), (u2, u3) and (u3, u4). Fig. 5 shows phase portraits of system (2) in the (u3, u4, u5) space defined by X1,1. Fig. 6 shows the projections of X1,1 onto the planes (u2, u4), (u2, u3) and (u3, u4). 100 80 60 40

u (2)

20 0 -20 -40 -60 -80 -100

-100

-80

-60

-40

-20

0 u (3)

20

40

60

80

100

-80

-60

-40

-20

0 u (2)

20

40

60

80

100

-80

-60

-40

-20

100 80 60 40

u (4)

20 0 -20 -40 -60 -80 -100 -100

100 80 60 40

u (4)

20 0 -20 -40 -60 -80 -100 -100

0 u (3)

20

40

60

80

Fig. 4. Projections of X1,1 onto the planes.

100

F. Zhang, G. Zhang / Applied Mathematics and Computation 243 (2014) 12–23

19

(ii) Let us take k ¼ 2; m ¼ 1; then we can get that

(

)  2  2 2b þ a 2 2 2 2  ðu1 ; u2 ; u3 ; u4 ; u5 Þ u1 þ u2 þ 2u3 þ 2u4 þ 2 u5  6L ; 2

X2;1 ¼

is the ultimate bound and positively invariant set of the complex Lorenz system (2), where,

L2 ¼

8 2 c ð2bþaÞ2 > ; a 6 1; c P 2a; > > < 8aðcaÞ 2

2

c ð2bþaÞ ; c P 2; a > 1; 8ðc1Þ > > > : ð2bþaÞ2 ; c < 2; c < 2a: 2

When a ¼ 35; b ¼ 55; c ¼ 83 ; we can obtain that

(

)  2  2 145 33640 2 2 2 2  ðu1 ; u2 ; u3 ; u4 ; u5 Þ u1 þ u2 þ 2u3 þ 2u4 þ 2 u5  6 ¼ 105:9 ; 2 3

X2;1 ¼

is the ultimate bound and positively invariant set of the complex Lorenz system (2). (iii) Let us take k ¼ 1; m ¼ 2, then we can get that

 

n

o

X1;2 ¼ ðu1 ; u2 ; u3 ; u4 ; u5 Þ2u21 þ 2u22 þ u23 þ u24 þ ðu5  b  2aÞ2 6 g2 ; is the ultimate bound and positively invariant set of the complex Lorenz system (2), where

g2 ¼

8 2 c ðbþ2aÞ2 > ; a 6 1; c P 2a; > > < 4aðcaÞ > > > :

c2 ðbþ2aÞ2 4ðc1Þ

; c P 2; a > 1; 2

ðb þ 2aÞ ; c < 2; c < 2a:

When a ¼ 35; b ¼ 55; c ¼ 83 ; we can obtain that



 

X1;2 ¼ ðu1 ; u2 ; u3 ; u4 ; u5 Þ2u21 þ 2u22 þ u23 þ u24 þ ðu5  125Þ2 6

50000 ¼ 129:12 ; 3

is the ultimate bound and positively invariant set of the complex Lorenz system (2). (iv) In fact, the set \k>0;m>0 Xk;m that is formed by the intersection of all the sets Xk;m ðk > 0; m > 0Þ is also the ultimate boundedness of system (2). (v) When a ¼ 35; b ¼ 55; c ¼ 83 ; we can get

 2 55k þ 35m 16ð55k þ 35mÞ2 mu21 þ mu22 þ ku23 þ ku24 þ k u5  6 ; k 15k

ðk > 0; m > 0Þ;

16ð55kþ35mÞ2 15km

from (5). Then, we can conclude that u21 6 holds for any k > 0; m > 0. In fact, in order to operate easily in chaos control and synchronization, we can find the minimum of the function

Fig. 5. Phase portraits of (2) in the (u3, u4, u5) space defined by X1,1.

20

F. Zhang, G. Zhang / Applied Mathematics and Computation 243 (2014) 12–23 200

150

u (5)

100

50

0

-50 -100

-80

-60

-40

-20

-80

-60

-40

-20

-80

-60

-40

-20

0

20

40

60

80

100

0

20

40

60

80

100

0 u (3)

20

40

60

80

u (4)

200

150

u (5)

100

50

0

-50 -100

u (3)

100 80 60 40

u (4)

20 0 -20 -40 -60 -80 -100 -100

100

Fig. 6. Projections of X1,1 onto the planes. 2

f ðmÞ ¼

16 ð55k þ 35mÞ ; 15k m

ðk > 0; m > 0Þ:

Let

16 ð55k þ 35mÞð35m  55kÞ 11k ¼ 0 ) m ¼ m0 ¼ > 0; 15k m2 7  3 19360k 19360 7 00 f 00 ðmÞ ¼ ) f ðm Þ ¼ > 0: 0 3m3 11 3k2 f 0 ðmÞ ¼

Then, the function f ðmÞ can reach its minimum when m ¼ m0 ¼ 11k for the given k > 0. Therefore, we can take 7 0 m ¼ m0 ¼ 11k for the given k0 > 0 in (6) in order to get a smaller bound of u21 . 7 (vi) Let us take k ¼ 1; m ¼ 0; then we can get that the following set

n

 

o

X2 ¼ ðu3 ; u4 ; u5 Þu23 þ u24 þ ðu5  bÞ2 6 c2 ;

ð11Þ

F. Zhang, G. Zhang / Applied Mathematics and Computation 243 (2014) 12–23

21

is the ultimate bound and positively invariant set of system (2), where,

8 c 2 b2 ; a 6 1; c P 2a; > > < 4aðcaÞ 2 2 2 c b c ¼ 4ðc1Þ ; c P 2; a > 1; > > : 2 b ; c < 2; c < 2a:

When a ¼ 35; b ¼ 55; c ¼ 83 ; we can obtain that



X2 ¼ ðu3 ; u4 ; u5 Þju23 þ u24 þ ðu5  55Þ2 6

9680 ¼ 56:82 ; 3

is the ultimate bound and positively invariant set of system (2). 4. Global attractive set Though Theorem 3 gives the ultimate bound and positively invariant set of system (2), it does not gives the rate of the trajectories going from the exterior of the trapping region into the interior trapping region. The rate of the trajectories going from the exterior of the trapping region into the interior trapping region of system (2) is given in the following Theorem 4. Theorem 4. For any k > 0; m > 0; a > 0; b > 0; c > 0 with

 2 bk þ am V k;m ðUÞ ¼ mu21 þ mu22 þ ku23 þ ku24 þ k u5  ; k 8 cðbkþamÞ2 > ; 1 6 c; 1 6 a; > > k 2 < cðbk þ amÞ 2 cðbkþamÞ Lk;m ¼ ¼ ; a 6 c; a 6 1; ka > kg > > cðbkþamÞ2 : ; c 6 a; c 6 1: kc

g ¼ minða; c; 1Þ > 0;

When V k;m ðUðtÞÞ > Lk;m ; V k;m ðUðt 0 ÞÞ > Lk;m , we can get an exponential estimate of system (2), given by

V k;m ðUðtÞÞ  Lk;m 6 ½V k;m ðUðt 0 ÞÞ  Lk;m egðtt0 Þ : Especially, the set

Wk;m ¼ fUjV k;m ðUÞ 6 Lk;m g

n   2 cðbkþamÞ2 o ¼ ðu1 ; u2 ; u3 ; u4 ; u5 Þmu21 þ mu22 þ ku23 þ ku24 þ k u5  bkþam 6 ; k kg

ð12Þ

is the global exponential attractive set of the complex Lorenz system (2). Proof. Define the following generalized positively definite and radially unbounded Lyapunov function

 2 bk þ am V k;m ðUÞ ¼ mu21 þ mu22 þ ku23 þ ku24 þ k u5  ; k where k > 0; m > 0; U ¼ ðu1 ; u2 ; u3 ; u4 ; u5 Þ. Computing the derivative of V k;m ðUðtÞÞ along the trajectory of the system (2), when V k;m ðUðtÞÞ > Lk;m ; V k;m ðUðt 0 ÞÞ > Lk;m , we have



dV k;m ðUÞ  dt

ð2Þ

  ¼ 2mu1 u_ 1 þ 2mu2 u_ 2 þ 2ku3 u_ 3 þ 2ku4 u_ 4 þ 2k u5  bkþam u_ 5 ; k ¼ 2amu1 ðu3  u1 Þ þ 2amu2 ðu4  u2 Þ þ 2ku3 ðbu1  u3  u1 u5 Þ þ 2ku4 ðbu2  u4  u2 u5 Þ   þ2k u5  kbþam Þðu1 u3 þ u2 u4  cu5 ; k ¼ 2amu21  2amu22  2ku23  2ku24  2kcu25 þ 2cðbk þ amÞu5 ; ¼ amu21  amu22  ku23  ku24  kcu25 þ 2cðbk þ amÞu5 amu21  amu22  ku23  ku24  kcu25 ;  2 cðbkþamÞ2 þ ¼ amu21  amu22  ku23  ku24  kc u5  bkþam k k amu21  amu22  ku23  ku24  kcu25 ;  2 cðbkþamÞ2 þ ; 6 amu21  amu22  ku23  ku24  kc u5  bkþam k k 2

; 6 gV k;m ðUÞ þ cðbkþamÞ k

 cðbkþamÞ2 < 0: ¼ g V k;m ðUÞ  kg

22

F. Zhang, G. Zhang / Applied Mathematics and Computation 243 (2014) 12–23

That is equivalent to say,

!  2 dV k;m  cðbk þ amÞ : 6  g V ðUÞ  k;m dt ð4Þ kg

ð13Þ

By comparison theorem and integrating both sides of formula (13) yields

Rt

2

egðtsÞ cðbkþamÞ ds kg   gðtt 0 Þ gðtt 0 Þ ¼ V k;m ðU 0 Þe þ Lk;m 1  e :

V k;m ðUðtÞÞ 6 V k;m ðU 0 Þegðtt0 Þ þ

t0

Thus, if V k;m ðUðtÞÞ > Lk;m ; V k;m ðUðt0 ÞÞ > Lk;m ; we have the following exponential estimate for system (2)

V k;m ðUðtÞÞ  Lk;m 6 ½V k;m ðU 0 Þ  Lk;m egðtt0 Þ : By the definition, taking limit on both sides of the above inequality as t ? +1 results in

lim V k;m ðUðtÞÞ 6 Lk;m :

t!þ1

Namely, the set Wk;m that defined in (12) is the global exponential attractive set of system (2). This completes the proof. h

Remark. (i) Let us take k ¼ 1; m ¼ 0; then we can get that the following set

n

 

o

W1;0 ¼ ðu3 ; u4 ; u5 Þu23 þ u24 þ ðu5  bÞ2 6 d2 ;

ð14Þ

where, 2

g ¼ minðc; 1Þ > 0; d2 ¼

cb

g

;

is the global exponential attractive set of system (2). The proved method is similar to Theorem 4.

5. Conclusions The ultimate boundedness of a chaotic systems is very important both in control theory and its applications. In this paper, the boundedness of solutions of the complex Lorenz system has been studied. Furthermore, the estimate of the trajectories going from the outside of the global exponential attractive set Wk;m to the interior of the global exponential attractive set Wk;m is also obtained. The corresponding boundedness is numerically verified by the computer. Numerical simulations are presented to show the effectiveness of the proposed scheme. The theoretical results obtained in this paper will find wide applications in chaos control and synchronization. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

[15]

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