new estimations for globally attractive and positive ... - Semantic Scholar
International Journal of Bifurcation and Chaos, Vol. 16, No. 11 (2006) 3383–3390 c World Scientific Publishing Company
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NEW ESTIMATIONS FOR GLOBALLY ATTRACTIVE AND POSITIVE INVARIANT SET OF THE FAMILY OF THE LORENZ SYSTEMS PEI YU∗ Department of Applied Mathematics, The University of Western Ontario, London, Ontario N6A 5B7, Canada [email protected] XIAOXIN LIAO† Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan, Hubei 430074, P. R. China Received April 22, 2005; Revised October 3, 2005 In this paper, we employ generalized Lyapunov functions to derive new estimations of the ultimate boundary for the trajectories of two types of Lorenz systems, one with parameters in finite intervals and the other in infinite intervals. The new estimations improve the results reported so far in the literature. In particular, for the singular cases: b → 1+ and a → 0+ , we have obtained the estimations independent of a. Moreover, our method using elementary algebra greatly simplifies the proofs in the literature. This is an interesting attempt in obtaining information of the attractors which is difficult when merely based on differential equations. It indicates that Lyapunov function is still a powerful tool in the study of qualitative behavior of chaotic systems. Keywords: The Lorenz system; generalized Lyapunov function; globally attractive set; ultimate boundary.
1. Introduction The discovery of the first chaotic attractor, now called Lorenz attractor (also known as butterfly attractor), by Lorenz in 1963, has created a new era of nonlinear dynamical systems (e.g. see [Lorenz, 1963, 1993; Ruelle, 1976; Sparrow, 1982; Pecora & Carroll, 1990; Cuomo & Oppenhein, 1993; Chen & Dong, 1998; Liao & Chen, 2003; Liao, 2004; Liao & Yu, 2005]). The well-known physicist, Ford, considered the discovery as “the third revolution of physics in 20th century”. Since then although
the fundamental and complex behavior of chaotic systems has been extensively studied and many results have been published, only until 1999 was the existence of the Lorenz attractor rigorously proved [Tucker, 1999, 2002; Stewart, 2002]. It was pointed out by Tucker [1999, 2002] that it was very difficult to obtain information of the attractor directly from the differential equation itself. To study the qualitative behavior of a chaotic system, the property of ultimate boundary plays a very important role in the study. This is because
∗
Author for correspondence. Currently, the author is also with School of Automatics, Wuhan University of Technology, Wuhan, Hubei 430070, P. R. China.
†
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the existence of such an ultimate boundary rules out the possibility of equilibrium points, periodic or quasi-periodic solutions, or other chaotic attractors existing outside the attractive set. The Russian scholar, G. Leonov was the first one to investigate the estimation of the global attractive set of the Lorenz system and obtained two important results [Leonov, 1988, 2001; Leonov et al., 1987a, 1987b]. Recently, Li et al. [2005] applied the method of Lagrange multiplier to obtain an estimation of the global attractive set for the Lorenz system which allows b taking the values 1 ≤ b ≤ 2. This improved Leonov’s estimation [Leonov, 1987a] which becomes singular when b → 1+ . We also obtained [Liao, 2004; Yu & Liao, 2006] some new results on the global attractive set of the Lorenz system, which improved Leonov’s results. Our methods simplified Leonov’s proofs. As a continuation of our previous work [Liao, 2004; Yu & Liao, 2006], in this paper, we want to further consider the estimation of the ultimate boundary (or the estimation of the global attractive set) of the Lorenz system for two types of parameter intervals. Our results improve and generalize the results obtained in [Leonov, 1988, 2001; Leonov et al., 1987a, 1987b]. We will give elementary but rigorous proofs for the results presented in this paper. Our method may provide a useful technique in further studying the ultimate boundary of chaotic systems, showing that the generalized Lyapunov function is still a powerful tool in the study of the qualitative behavior of chaotic systems.
a = 10, b = 8/3 and c = 28 [Lorenz, 1963]. The Lorenz system (1) is chaotic when the three parameters take certain values, and regular for other values. Regardless whether the Lorenz system is chaotic or not, we shall uniformly consider the ultimate boundary of the Lorenz system (1) (or the Lagrange asymptotic stability), and derive the estimation of the ultimate boundary. Theorem 1. For the first type of Lorenz system (1), we have the following estimations for the ultimate boundary:
(1) when b ≥ 2, ∀ a ∈ (0, +∞) and ∀ c ∈ [1, +∞), b2 c2 2 (t) + (z(t) − c)2 ≤ := R2 , lim y t→+∞ 4(b − 1) b2 c2 := R2 ; lim x2 (t) ≤ t→+∞ 4(b − 1) (2) (2) when 0 < b ≤ 2, ∀ a ∈ (0, +∞) and ∀ c ∈ [1, +∞), ˜ 2, lim y 2 (t) + (z(t) − c)2 ≤ c2 := R t→∞ (3) lim x2 (t) ≤ c2 := R ˜2. t→∞
The first estimation in Eq. (2) was obtained in [Leonov, 2001], and the second estima˜ 2 = σc2 (σ > 1) was given in tion in Eq. (3) with R [Liao, 2004]. Remark.
For system (1), with respect to the partial variables y and z, we construct the following generalized positive definite and radially unbounded Lyapunov function: V1 = y 2 +(z−c)2 . Differentiating V1 with respect to time along the trajectory of system (1) yields dV1 = −y 2 − bz 2 + bcz ≤ 0 dt (1) Proof.
2. The First Type of Lorenz System Consider the first type of Lorenz system, given by [Lorenz, 1963] x˙ = a(y − x), y˙ = cx − xz − y, z˙ = xy − bz,
(1)
where the dot denotes differentiation with respect to time t, the parameters a, b and c are assumed positive real numbers, satisfying a, b ∈ (0, +∞) and c ∈ [1, +∞), because when c ∈ (0, 1) it is easy to show that the origin (0, 0, 0) is the unique equilibrium point of the system and is globally, exponentially stable. Therefore, the parameters a, b and c are varied in the specified intervals, and we may call this Lorenz system as an interval Lorenz system. The typical set of parameter values for this Lorenz system to exhibit a chaotic attractor is:
when y 2 + bz 2 − bcz ≥ 0. Define an ellipse Q : y 2 + bz 2 − bcz = 0, then dV1 /dt = 0 on the ellipse Q, while dV1 /dt < 0 outside the ellipse Q. Thus, on any circle: V1 = l2 which encloses the ellipse Q, we have dV1 /dt ≤ 0. That is, the direction of any trajectory of system (1) starting from the circle V1 = l2 must move to inside of the circle. Now, we want to select the smallest circle which encloses the ellipse Q, i.e. to find the smallest radius,
Globally Attractive Set of the Lorenz Systems
l0 , of the circle V1 = l2 . To achieve this, first note that l0 ≥ c and z0 ≥ 0, where z0 is the coordinate of the intersection point. Thus, consider the following system of equations: 2 y + (z − c)2 = l2 , (4) y 2 + bz 2 − bcz = 0,
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from which it is easy to obtain l2 = (1 − b)z 2 + (b − 2)cz + c2 (b − 2)c 2 b2 c2 = −(b − 1) z − + 2(b − 1) 4(b − 1) 2 b2 c2 (2 − b)c + = −(b − 1) z + 2(b − 1) 4(b − 1) 2 b2 c2 (2 − b)c . − = (1 − b) z − 2(1 − b) 4(1 − b)
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3. The Second Type of Lorenz System Chen and L¨ u [2003] proposed the following family of Lorenz systems (or called unified system): x˙ = (25α + 10)(y − x), y˙ = (28 − 35α)x − xz − (1 − 29α)y,
(6)
8+α z, z˙ = xy − 3 where 0 ≤ α ≤ 1. Then,
(5)
Then, based on Eq. (5), we have the following results. (i) When b ≥ 2, the second line of Eq. (5) clearly shows that the unique real solution for z0 is obtained when l2 = b2 c2 /(4(b − 1)) ≥ c2 at which z = z0 = (b − 2)c/(2(b − 1)). Thus, l0 = b2 c2 /(4(b − 1)) for b ≥ 2. (ii) When 1 < b ≤ 2, it is easy to see from the third line of Eq. (5) that l2 ≤ c2 for any values of c and z ≥ 0, and l2 = c2 at z = z0 = 0. Thus, l0 = c for 1 < b ≤ 2. (iii) When 0 < b < 1, the fourth line of Eq. (5) indicates that the minimum value of l2 (≥ 0) is obtained when z = z0 = 0 at which l2 = c2 . Hence, l0 = c for 0 < b < 1. (iv) When b = 1, it is straightforward to obtain l0 = c, since l2 = c2 − cz ≤ c2 , and l2 = c2 at z = z0 = 0. The above results prove the first statement in Eqs. (2) and (3). Next, for system (1), construct the positive definite and radially unbounded Lyapunov function for the variable x : V2 = x2 /2. Differentiating V2 along the trajectory of system (1) results in dV2 = −a(x2 − xy) ≤ −a|x|(|x| − |y|) ≤ 0 dt (1) when |x| ≥ |y|, which shows that the second statement in Eqs. (2) and (3) are true. This completes the proof of Theorem 1.
(1) when α = 0, system (6) is the classical Lorenz system, and when 0 ≤ α < 0.8, it is called generalized Lorenz system [Chen & L¨ u, 2003]; (2) when α = 1, system (6) is the Chen system, and when 0.8 < α ≤ 1, it is called generalized Chen system [Chen & L¨ u, 2003]; (3) when α = 0.8, system (6) is the L¨ u system, which is also called generalized L¨ u system. For Chen and L¨ u systems, no correct results have been reported on the estimations of their global attractive and positive invariant sets. In fact, the ultimate boundedness of these two systems has not been proved. For the family of generalized Lorenz systems, no proof has been given for its global attractive and positive invariant set. However, for a subset of the family of generalized Lorenz systems (i.e. when 0 < α < 1/29), Li et al. [2003] have investigated the estimation on the ultimate boundary of the trajectories. In order to distinct this subset from the Lorenz system and the generalized Lorenz system, we call this subset of the family of generalized Lorenz system as the second type of Lorenz system with finite parameter intervals. Although all the parameters in the second type of Lorenz system are bounded, it has one more coefficient for the y variable in the second equation, compared to the classical Lorenz system. Therefore, the second type and the first type (which only contains three parameters) of Lorenz systems do not overlap. They need to be studied separately. For convenience, introduce the following notations: a(α) := 25α + 10, c(α) := 28 − 35α
b(α) :=
α+8 , 3
and d(α) := 1 − 29α.
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lim y 2 (t) + [z(t) − c(α)]2 t→∞ b2 (α)c2 (α) := R 2 , ≤ 4[b(α) − d(α)]d(α) b2 (α)c2 (α) lim x2 (t) ≤ := R 2 . t→∞ 4[b(α) − d(α)]d(α)
Then, for 0 < α < 1/29, we have 10 ≤ a(α) ≤
315 , 29
777 ≤ c(α) ≤ 28 29
and
8 ≤ b(α) ≤ 3, 3 0 < d(α) ≤ 1,
(8)
and thus system (6) can be rewritten as
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x˙ = a(α)(y − x), y˙ = c(α)x − xz − d(α)y, z˙ = xy − b(α)z.
For system (7), with respect to the partial variables y and z, construct the following generalized positive definite and radially unbounded Lyapunov function: 1 (9) V1 = {y 2 + [z − c(α)]2 }. 2 Differentiating V1 with respect to time, with the aid of the second and third equations of (7), yields Proof.
(7)
Theorem 2. For the second type of Lorenz system (7), let X ∗ = (0, 0, c(α)). We have the following estimations for the ultimate boundary:
dV1 = y y˙ + [z − c(α)]z˙ = c(α)xy − xyz − d(α)y 2 + [z − c(α)][xy − b(α)]z dt (7)
= −d(α)y 2 − d(α)z 2 + 2d(α)c(α)z − [b(α) − d(α)]z 2 + [b(α)c(α) − 2d(α)c(α)]z = −d(α)y 2 − d(α)[z − c(α)]2 + d(α)c2 (α) − [b(α) − d(α)] [b(α) − 2d(α)]c(α) 2 {[b(α) − 2d(α)]c(α)}2 + × z− 2[b(α) − d(α)] 4[b(α) − d(α)] ≤ −d(α)y 2 − d(α)[z − c(α)]2 + d(α)c2 (α) + = −d(α)y 2 − d(α)[z − c(α)]2 +
×
{[b(α) − 2d(α)]c(α)}2 4[b(α) − d(α)]
b2 (α)c2 (α) 4[b(α) − d(α)]
< 0
for y 2 + [z − c(α)]2 >
= 0
b2 (α)c2 (α) . for y 2 + [z − c(α)]2 = 4[b(α) − d(α)]d(α)
Therefore, we obtain the first estimation of (8), which implies that |y(t)| ≤ R. Finally, for the first equation of (7), similarly construct the positive definite and radially unbounded Lyapunov function x : V2 = x2 /2. Then, along the trajectory of system (7), by noting that y(t)| ≤ R, we have dV2 = xx ˙ = a(α)(yx − x2 ) ≤ a(α)(|y||x| − |x|2 ) dt (11) ≤ a(α)(R − |x|)|x| ≤ 0 for |x| ≥ R. Hence, the second estimation in (8) holds. The proof of Theorem 2 is complete. In the following, we give a more general result.
b2 (α)c2 (α) , 4[b(α) − d(α)]d(α)
(10)
Theorem 3. For the second type of Lorenz system (7), and for an arbitrary λ ≥ 0, let X ∗ = (0, 0, λa(α) + c(α)). Then, we have the following estimation for the ultimate boundary:
lim λx2 (t) + y 2 (t) + [z(t) − λa(α) − c(α)]2
t→∞
≤ Proof.
b2 (α)[λa(α) + c(α)]2 . 4[b(α) − d(α)]d(α)
(11)
Let
f (z) = −[b(α) − d(α)]2 z 2 + [b(α) − 2d(α)] × [λa(α) + c(α)]z + d(α)[λa(α) + c(α)]2 . Then, df /dz = −2[b(α) − d(α)]z + [b(α) − 2d(α)] × [λa(α) + c(α)] = 0 leads to z0 = ([b(α) −
Globally Attractive Set of the Lorenz Systems
2d(α)][λa(α) + c(α)])/(2[b(α) − d(α)]) at which (df /dz)(z0 ) = 0. Since b(α) > 1 > d(α), we have z0 > 0. Further, noticing that d2 f/dz 2 = −2[b(α) − d(α)] > 0, we obtain sup f (z) = max f (z) = z∈R
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z∈R
b2 (α)[λa(α) + c(α)]2 . 4[b(α) − d(α)]
Next, for system (7), construct the following generalized positive definite and radially unbounded Lyapunov function: 1 Vλ = {λx2 + y 2 + [z − λa(α) − c(α)]2 }, 2 where λ ≥ 0. At λ = 0, Vλ becomes the generalized positive definite and radially unbounded Lyapunov function V1 for the variables y and z. Computing dVλ /dt along the trajectory of system (7) and noticing that a(α) > 1, 0 < d(α) ≤ 1 results in dVλ dt
× [λa(α) + c(α)]z − [b(α) − d(α)]z 2 + [b(α) − 2d(α)][λa(α) + c(α)]z ≤ −λd(α)x2 − d(α)y 2 − d(α)[z − λa(α) − c(α)]2 + d(α)[λa(α) + c(α)]2 + f (z) ≤ −λd(α)x2 − d(α)y 2 − d(α)[z − λa(α) − c(α)]2 + d(α)[λa(α) + c(α)]2 + f (z0 ) = −λd(α)x2 − d(α)y 2 − d(α)[z − λa(α) − c(α)]2 +
b2 (α)[λa(α) + c(α)]2 4[b(α) − d(α)]
≤ 0 when λx2 + y 2 + [z − λa(α) − c(α)]2 ≥
2
2
(12)
lim λx2 (t) + y 2 (t) + [z(t) − λa(α) − c(α)]2
t→+∞
b2 (α)[λa(α) + c(α)]2 , 4[b(α) − d(α)]d(α)
≤
which is Eq. (11). This finishes the proof of Theorem 3.
2
= −λd(α)x − d(α)y − d(α)z + 2d(α)
z
z 50 40 30 20 10 0
-20-15
b2 (α)[λa(α) + c(α)]2 . 4[b(α) − d(α)]d(α)
Hence, we obtain
(7)
= λxx˙ + y y˙ + [z − λa(α) − c(α)]z˙
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60 40 20 0
-10 -5 x
-5 -15 0 5 10 15 -25 20
5
15 y
25
40 -25 -15
-5 x
(a)
5
-20 15
25
20 0 y
-40
(b)
z
z
250 200 150 100 50 0
300 200 100 0
-40 -20
0 x
-50 20 40 60 -100 (c)
150 100 50 y 0
200 -60 -40
-20 0 -100 20 40 60 80 -150 x
0
100 y
(d)
Fig. 1. Simulated phase portraits of the first type of Lorenz system (1) with the initial conditions x(0) = y(0) = z(0) = 10 for parameter values: (a) a = 10, b = 8/3, c = 28; (b) a = 10, b = 8/3, c = 40; (c) a = 10, b = 8/3, c = 130; (d) a = 10, b = 8/3, c = 180; (e) a = 20, b = 12, c = 130; (f) a = 0.5, b = 1.1, c = 1.0.
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z 250 200 150 100 50 0
5 y 0 -5
150 100
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-80 -60
0 -50 0 20 -100 40 60 x 80 -150
-40 -20
50 y
-10
-2
0
2
(e)
4
x
6
8
10
12
(f) Fig. 1.
(Continued )
Theorem 3 is a general result, and Eq. (11) covers many special cases considered so far, for example, as listed below.
Remarks.
(1) Take α = 0, λ ≥ 0 in Theorem 3, then Eq. (11) is the result given in the main theorem of [Liao, 2004].
(2) Take α = 0, λ = 1 in Theorem 3, then Eq. (11) gives the estimation obtained by Leonov [1987a]. (3) Take α = 0, λ = 0 in Theorem 3, then Eq. (11) becomes the estimation obtained by Leonov [1988]. z
z 90 70 50 30 10
40 30 20 10 20 -15 -10
-5 0 -10 5 10 15 20 -20 x
10 0 y
40 20 y
-15-10 -5 0 0 5 10 -20 15 20 25 x 30 -30
(a)
(b)
10
z 250 200 150 100 50 0
5 y 0
-25 -15 -5
5 x
15 25 (c)
-50 -100
150 100 50 y 0
-5 -10
-2
0
2
4
x
6
8
10
12
(d)
Fig. 2. Simulated phase portraits of the first type of Lorenz system (1) with the initial conditions x(0) = y(0) = z(0) = 10 for parameter values: (a) a = 10, b = 1.0, c = 28; (b) a = 10, b = 0.5, c = 50; (c) a = 2, b = 0.1, c = 130; (d) a = 0.1, b = 1.0, c = 1.0.
Globally Attractive Set of the Lorenz Systems
(4) Take 0 ≤ α ≤ 1, λ = 1 in Theorem 3, then Eq. (11) leads to the result given in Theorem 1 of [Li et al., 2005].
4. Numerical Simulation Results
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In this section, we present some numerical simulation results under the assumptions that a > 0, b > 0 and c ≥ 1. For the first type of Lorenz systems, a total of ten cases for choosing different values of
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the parameters, a, b and c are given. The 3-D phase portraits of the simulated trajectories are depicted in Fig. 1 (for the first six cases) and Fig. 2 (for the last four cases). For the first four cases, the typical values for a and b are chosen as a = 10 and b = 8/3, and the values of c are varied from 28 to 180. For other cases all the three parameters are varied. Note that the sixth and the last cases take the critical value c = 1. This type of Lorenz system exhibits chaotic motions in eight cases, except the sixth and
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 3. Simulated phase portraits of the second type of Lorenz system (6) with the initial conditions x(0) = y(0) = z(0) = 10 for α to equal: (a) 0.03; (b) 0.034; (c) 0.3; (d) 0.5; (e) 0.9; and (f) 0.95.
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the last cases which show convergence to stable points. However, it should be pointed out that these stable points, as shown in Figs 1(f) and 2(d), are not equilibrium points. For the second type of Lorenz system, a total of six cases are presented for different values of α. The simulated phase portraits are shown in Fig. 3. For the first two cases: α = 0.03 and α = 0.034, both of them satisfy α ∈ (0, 1/29). Formula (8) given in Theorem 2 can provide estimations for the boundaries of the attractive sets. The numerical results indicate that except for the case α = 0.9, all other cases exhibit either generalized Lorenz attractor or generalized Chen attractor. When α = 0.9, the trajectory [see Fig. 3(e)] shows a quasi-periodic motion. When α is further increased from 1, it is found that for the values of α > 1, but very close to 1, the motion is chaotic; while when α is slightly greater than 1, the motion is quasi-periodic, and then quickly becomes periodic.
5. Conclusion In this paper, we have given a survey on the results obtained so far on the estimations of the global attractive and positive invariant set of the Lorenz attractor. We have considered two types of Lorenz systems and obtained new estimations. We apply generalized Lyapunov functions to give simple proofs based on elementary algebra. The simple approach shows that Lyapunov function is still a useful tool in the study of complex nonlinear systems. Numerical simulations are presented to support our theoretical results.
Acknowledgments This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC, No. R2686A02)), and the National Natural Science Foundation of China (NNSFC, No. 60274007, 60474011).
References Chen, G. & Dong, X. [1998] From Chaos to Order (World Scientific, Singapore).
Chen, G. R. & L¨ u, J. H. [2003] Dynamical Analysis, Control and Synchronization of Lorenz Families (Chinese Science Press, Beijing). Cuomo, K. M. & Oppenhein, A. V. [1993] “Circuit implementation of synchronization chaos with application to communication,” Phys. Rev. Lett. 71, 65–68. Leonov, G., Bunin, A. & Koksch, N. [1987a] “A tractor localization of the Lorenz system,” ZAMM 67, 649– 656. Leonov, G. & Reitmann, V. [1987b] Attraktoreingrenzung fur Nichtlineare System (Tenbner-Verlag, Leipzing). Leonov, G. [1988] “On estimates of attractors of Lorenz system,” Vestnik Leningradskogo Universiten Matematika 21, 32–37. Leonov, G. [2001] “Bound for attractors and the existence of homoclinic orbits in the Lorenz system,” J. Appl. Math. 65, 19–32. Liao, X. X. [1993] Absolute Stability of Nonlinear Control Systems (Kluwer Academic Pub., China Science Press, Beijing). Liao, X. X. & Chen, G. R. [2003] “Some new results on chaos synchronization,” Contr. Th. Appl. 20, 254– 258. Liao, X. X. [2004] “On the new results of global attractive set and positive invariant set of the Lorenz chaotic system and the application to chaos control and synchronization,” Sci. China 34, 1–16. Liao, X. X. & Yu, P. [2005] “Analysis on the global exponent synchronization of Chua’s circuit using absolute stability theory,” Int. J. Bifurcation and Chaos 15, 3687–3881. Lorenz, E. N. [1963] “Deterministic nonperiodic flow,” J. Atmos. Sci. 20, 130–141. Lorenz, E. N. [1993] The Essence of Chaos (University of Washington Press, USA). Pecora, L. M. & Carroll, T. L. [1990] “Synchronization in chaotic circuits,” Phys. Rev. Lett. 64, 821–824. Sparrow, C. [1982] The Lorenz Equations: Bifurcation Chaos and Strange Attractors (Springer Press, NY). Stewart, I. [2002] “The Lorenz attractor exists,” Nature 406, 948–949. Tucker, W. [1999] “The Lorenz attractor exists,” C. R. Acad. Sci. Paris 328, 1197–1202. Tucker, W. [2002] “A rigorous ODE solver and Smale’s 14th problem,” Found. Comput. Meth. 2, 53–117. Ruelle, D. [1976] Lorenz Attractor and Problem of Turbulence, Lecture Notes of Mathematics, Vol. 565 (Springer-Verlag, NY). Yu, P. & Liao, X. X. [2006] “Globally attractive and positive invariant set of the Lorenz system,” Int. J. Bifurcation and Chaos 16, 757–764.
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NEW ESTIMATIONS FOR GLOBALLY ATTRACTIVE AND POSITIVE INVARIANT SET OF THE FAMILY OF THE LORENZ SYSTEMS PEI YU∗ Department of Applied Mathematics, The University of Western Ontario, London, Ontario N6A 5B7, Canada [email protected] XIAOXIN LIAO† Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan, Hubei 430074, P. R. China Received April 22, 2005; Revised October 3, 2005 In this paper, we employ generalized Lyapunov functions to derive new estimations of the ultimate boundary for the trajectories of two types of Lorenz systems, one with parameters in finite intervals and the other in infinite intervals. The new estimations improve the results reported so far in the literature. In particular, for the singular cases: b → 1+ and a → 0+ , we have obtained the estimations independent of a. Moreover, our method using elementary algebra greatly simplifies the proofs in the literature. This is an interesting attempt in obtaining information of the attractors which is difficult when merely based on differential equations. It indicates that Lyapunov function is still a powerful tool in the study of qualitative behavior of chaotic systems. Keywords: The Lorenz system; generalized Lyapunov function; globally attractive set; ultimate boundary.
1. Introduction The discovery of the first chaotic attractor, now called Lorenz attractor (also known as butterfly attractor), by Lorenz in 1963, has created a new era of nonlinear dynamical systems (e.g. see [Lorenz, 1963, 1993; Ruelle, 1976; Sparrow, 1982; Pecora & Carroll, 1990; Cuomo & Oppenhein, 1993; Chen & Dong, 1998; Liao & Chen, 2003; Liao, 2004; Liao & Yu, 2005]). The well-known physicist, Ford, considered the discovery as “the third revolution of physics in 20th century”. Since then although
the fundamental and complex behavior of chaotic systems has been extensively studied and many results have been published, only until 1999 was the existence of the Lorenz attractor rigorously proved [Tucker, 1999, 2002; Stewart, 2002]. It was pointed out by Tucker [1999, 2002] that it was very difficult to obtain information of the attractor directly from the differential equation itself. To study the qualitative behavior of a chaotic system, the property of ultimate boundary plays a very important role in the study. This is because
∗
Author for correspondence. Currently, the author is also with School of Automatics, Wuhan University of Technology, Wuhan, Hubei 430070, P. R. China.
†
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the existence of such an ultimate boundary rules out the possibility of equilibrium points, periodic or quasi-periodic solutions, or other chaotic attractors existing outside the attractive set. The Russian scholar, G. Leonov was the first one to investigate the estimation of the global attractive set of the Lorenz system and obtained two important results [Leonov, 1988, 2001; Leonov et al., 1987a, 1987b]. Recently, Li et al. [2005] applied the method of Lagrange multiplier to obtain an estimation of the global attractive set for the Lorenz system which allows b taking the values 1 ≤ b ≤ 2. This improved Leonov’s estimation [Leonov, 1987a] which becomes singular when b → 1+ . We also obtained [Liao, 2004; Yu & Liao, 2006] some new results on the global attractive set of the Lorenz system, which improved Leonov’s results. Our methods simplified Leonov’s proofs. As a continuation of our previous work [Liao, 2004; Yu & Liao, 2006], in this paper, we want to further consider the estimation of the ultimate boundary (or the estimation of the global attractive set) of the Lorenz system for two types of parameter intervals. Our results improve and generalize the results obtained in [Leonov, 1988, 2001; Leonov et al., 1987a, 1987b]. We will give elementary but rigorous proofs for the results presented in this paper. Our method may provide a useful technique in further studying the ultimate boundary of chaotic systems, showing that the generalized Lyapunov function is still a powerful tool in the study of the qualitative behavior of chaotic systems.
a = 10, b = 8/3 and c = 28 [Lorenz, 1963]. The Lorenz system (1) is chaotic when the three parameters take certain values, and regular for other values. Regardless whether the Lorenz system is chaotic or not, we shall uniformly consider the ultimate boundary of the Lorenz system (1) (or the Lagrange asymptotic stability), and derive the estimation of the ultimate boundary. Theorem 1. For the first type of Lorenz system (1), we have the following estimations for the ultimate boundary:
(1) when b ≥ 2, ∀ a ∈ (0, +∞) and ∀ c ∈ [1, +∞), b2 c2 2 (t) + (z(t) − c)2 ≤ := R2 , lim y t→+∞ 4(b − 1) b2 c2 := R2 ; lim x2 (t) ≤ t→+∞ 4(b − 1) (2) (2) when 0 < b ≤ 2, ∀ a ∈ (0, +∞) and ∀ c ∈ [1, +∞), ˜ 2, lim y 2 (t) + (z(t) − c)2 ≤ c2 := R t→∞ (3) lim x2 (t) ≤ c2 := R ˜2. t→∞
The first estimation in Eq. (2) was obtained in [Leonov, 2001], and the second estima˜ 2 = σc2 (σ > 1) was given in tion in Eq. (3) with R [Liao, 2004]. Remark.
For system (1), with respect to the partial variables y and z, we construct the following generalized positive definite and radially unbounded Lyapunov function: V1 = y 2 +(z−c)2 . Differentiating V1 with respect to time along the trajectory of system (1) yields dV1 = −y 2 − bz 2 + bcz ≤ 0 dt (1) Proof.
2. The First Type of Lorenz System Consider the first type of Lorenz system, given by [Lorenz, 1963] x˙ = a(y − x), y˙ = cx − xz − y, z˙ = xy − bz,
(1)
where the dot denotes differentiation with respect to time t, the parameters a, b and c are assumed positive real numbers, satisfying a, b ∈ (0, +∞) and c ∈ [1, +∞), because when c ∈ (0, 1) it is easy to show that the origin (0, 0, 0) is the unique equilibrium point of the system and is globally, exponentially stable. Therefore, the parameters a, b and c are varied in the specified intervals, and we may call this Lorenz system as an interval Lorenz system. The typical set of parameter values for this Lorenz system to exhibit a chaotic attractor is:
when y 2 + bz 2 − bcz ≥ 0. Define an ellipse Q : y 2 + bz 2 − bcz = 0, then dV1 /dt = 0 on the ellipse Q, while dV1 /dt < 0 outside the ellipse Q. Thus, on any circle: V1 = l2 which encloses the ellipse Q, we have dV1 /dt ≤ 0. That is, the direction of any trajectory of system (1) starting from the circle V1 = l2 must move to inside of the circle. Now, we want to select the smallest circle which encloses the ellipse Q, i.e. to find the smallest radius,
Globally Attractive Set of the Lorenz Systems
l0 , of the circle V1 = l2 . To achieve this, first note that l0 ≥ c and z0 ≥ 0, where z0 is the coordinate of the intersection point. Thus, consider the following system of equations: 2 y + (z − c)2 = l2 , (4) y 2 + bz 2 − bcz = 0,
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from which it is easy to obtain l2 = (1 − b)z 2 + (b − 2)cz + c2 (b − 2)c 2 b2 c2 = −(b − 1) z − + 2(b − 1) 4(b − 1) 2 b2 c2 (2 − b)c + = −(b − 1) z + 2(b − 1) 4(b − 1) 2 b2 c2 (2 − b)c . − = (1 − b) z − 2(1 − b) 4(1 − b)
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3. The Second Type of Lorenz System Chen and L¨ u [2003] proposed the following family of Lorenz systems (or called unified system): x˙ = (25α + 10)(y − x), y˙ = (28 − 35α)x − xz − (1 − 29α)y,
(6)
8+α z, z˙ = xy − 3 where 0 ≤ α ≤ 1. Then,
(5)
Then, based on Eq. (5), we have the following results. (i) When b ≥ 2, the second line of Eq. (5) clearly shows that the unique real solution for z0 is obtained when l2 = b2 c2 /(4(b − 1)) ≥ c2 at which z = z0 = (b − 2)c/(2(b − 1)). Thus, l0 = b2 c2 /(4(b − 1)) for b ≥ 2. (ii) When 1 < b ≤ 2, it is easy to see from the third line of Eq. (5) that l2 ≤ c2 for any values of c and z ≥ 0, and l2 = c2 at z = z0 = 0. Thus, l0 = c for 1 < b ≤ 2. (iii) When 0 < b < 1, the fourth line of Eq. (5) indicates that the minimum value of l2 (≥ 0) is obtained when z = z0 = 0 at which l2 = c2 . Hence, l0 = c for 0 < b < 1. (iv) When b = 1, it is straightforward to obtain l0 = c, since l2 = c2 − cz ≤ c2 , and l2 = c2 at z = z0 = 0. The above results prove the first statement in Eqs. (2) and (3). Next, for system (1), construct the positive definite and radially unbounded Lyapunov function for the variable x : V2 = x2 /2. Differentiating V2 along the trajectory of system (1) results in dV2 = −a(x2 − xy) ≤ −a|x|(|x| − |y|) ≤ 0 dt (1) when |x| ≥ |y|, which shows that the second statement in Eqs. (2) and (3) are true. This completes the proof of Theorem 1.
(1) when α = 0, system (6) is the classical Lorenz system, and when 0 ≤ α < 0.8, it is called generalized Lorenz system [Chen & L¨ u, 2003]; (2) when α = 1, system (6) is the Chen system, and when 0.8 < α ≤ 1, it is called generalized Chen system [Chen & L¨ u, 2003]; (3) when α = 0.8, system (6) is the L¨ u system, which is also called generalized L¨ u system. For Chen and L¨ u systems, no correct results have been reported on the estimations of their global attractive and positive invariant sets. In fact, the ultimate boundedness of these two systems has not been proved. For the family of generalized Lorenz systems, no proof has been given for its global attractive and positive invariant set. However, for a subset of the family of generalized Lorenz systems (i.e. when 0 < α < 1/29), Li et al. [2003] have investigated the estimation on the ultimate boundary of the trajectories. In order to distinct this subset from the Lorenz system and the generalized Lorenz system, we call this subset of the family of generalized Lorenz system as the second type of Lorenz system with finite parameter intervals. Although all the parameters in the second type of Lorenz system are bounded, it has one more coefficient for the y variable in the second equation, compared to the classical Lorenz system. Therefore, the second type and the first type (which only contains three parameters) of Lorenz systems do not overlap. They need to be studied separately. For convenience, introduce the following notations: a(α) := 25α + 10, c(α) := 28 − 35α
b(α) :=
α+8 , 3
and d(α) := 1 − 29α.
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lim y 2 (t) + [z(t) − c(α)]2 t→∞ b2 (α)c2 (α) := R 2 , ≤ 4[b(α) − d(α)]d(α) b2 (α)c2 (α) lim x2 (t) ≤ := R 2 . t→∞ 4[b(α) − d(α)]d(α)
Then, for 0 < α < 1/29, we have 10 ≤ a(α) ≤
315 , 29
777 ≤ c(α) ≤ 28 29
and
8 ≤ b(α) ≤ 3, 3 0 < d(α) ≤ 1,
(8)
and thus system (6) can be rewritten as
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x˙ = a(α)(y − x), y˙ = c(α)x − xz − d(α)y, z˙ = xy − b(α)z.
For system (7), with respect to the partial variables y and z, construct the following generalized positive definite and radially unbounded Lyapunov function: 1 (9) V1 = {y 2 + [z − c(α)]2 }. 2 Differentiating V1 with respect to time, with the aid of the second and third equations of (7), yields Proof.
(7)
Theorem 2. For the second type of Lorenz system (7), let X ∗ = (0, 0, c(α)). We have the following estimations for the ultimate boundary:
dV1 = y y˙ + [z − c(α)]z˙ = c(α)xy − xyz − d(α)y 2 + [z − c(α)][xy − b(α)]z dt (7)
= −d(α)y 2 − d(α)z 2 + 2d(α)c(α)z − [b(α) − d(α)]z 2 + [b(α)c(α) − 2d(α)c(α)]z = −d(α)y 2 − d(α)[z − c(α)]2 + d(α)c2 (α) − [b(α) − d(α)] [b(α) − 2d(α)]c(α) 2 {[b(α) − 2d(α)]c(α)}2 + × z− 2[b(α) − d(α)] 4[b(α) − d(α)] ≤ −d(α)y 2 − d(α)[z − c(α)]2 + d(α)c2 (α) + = −d(α)y 2 − d(α)[z − c(α)]2 +
×
{[b(α) − 2d(α)]c(α)}2 4[b(α) − d(α)]
b2 (α)c2 (α) 4[b(α) − d(α)]
< 0
for y 2 + [z − c(α)]2 >
= 0
b2 (α)c2 (α) . for y 2 + [z − c(α)]2 = 4[b(α) − d(α)]d(α)
Therefore, we obtain the first estimation of (8), which implies that |y(t)| ≤ R. Finally, for the first equation of (7), similarly construct the positive definite and radially unbounded Lyapunov function x : V2 = x2 /2. Then, along the trajectory of system (7), by noting that y(t)| ≤ R, we have dV2 = xx ˙ = a(α)(yx − x2 ) ≤ a(α)(|y||x| − |x|2 ) dt (11) ≤ a(α)(R − |x|)|x| ≤ 0 for |x| ≥ R. Hence, the second estimation in (8) holds. The proof of Theorem 2 is complete. In the following, we give a more general result.
b2 (α)c2 (α) , 4[b(α) − d(α)]d(α)
(10)
Theorem 3. For the second type of Lorenz system (7), and for an arbitrary λ ≥ 0, let X ∗ = (0, 0, λa(α) + c(α)). Then, we have the following estimation for the ultimate boundary:
lim λx2 (t) + y 2 (t) + [z(t) − λa(α) − c(α)]2
t→∞
≤ Proof.
b2 (α)[λa(α) + c(α)]2 . 4[b(α) − d(α)]d(α)
(11)
Let
f (z) = −[b(α) − d(α)]2 z 2 + [b(α) − 2d(α)] × [λa(α) + c(α)]z + d(α)[λa(α) + c(α)]2 . Then, df /dz = −2[b(α) − d(α)]z + [b(α) − 2d(α)] × [λa(α) + c(α)] = 0 leads to z0 = ([b(α) −
Globally Attractive Set of the Lorenz Systems
2d(α)][λa(α) + c(α)])/(2[b(α) − d(α)]) at which (df /dz)(z0 ) = 0. Since b(α) > 1 > d(α), we have z0 > 0. Further, noticing that d2 f/dz 2 = −2[b(α) − d(α)] > 0, we obtain sup f (z) = max f (z) = z∈R
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z∈R
b2 (α)[λa(α) + c(α)]2 . 4[b(α) − d(α)]
Next, for system (7), construct the following generalized positive definite and radially unbounded Lyapunov function: 1 Vλ = {λx2 + y 2 + [z − λa(α) − c(α)]2 }, 2 where λ ≥ 0. At λ = 0, Vλ becomes the generalized positive definite and radially unbounded Lyapunov function V1 for the variables y and z. Computing dVλ /dt along the trajectory of system (7) and noticing that a(α) > 1, 0 < d(α) ≤ 1 results in dVλ dt
× [λa(α) + c(α)]z − [b(α) − d(α)]z 2 + [b(α) − 2d(α)][λa(α) + c(α)]z ≤ −λd(α)x2 − d(α)y 2 − d(α)[z − λa(α) − c(α)]2 + d(α)[λa(α) + c(α)]2 + f (z) ≤ −λd(α)x2 − d(α)y 2 − d(α)[z − λa(α) − c(α)]2 + d(α)[λa(α) + c(α)]2 + f (z0 ) = −λd(α)x2 − d(α)y 2 − d(α)[z − λa(α) − c(α)]2 +
b2 (α)[λa(α) + c(α)]2 4[b(α) − d(α)]
≤ 0 when λx2 + y 2 + [z − λa(α) − c(α)]2 ≥
2
2
(12)
lim λx2 (t) + y 2 (t) + [z(t) − λa(α) − c(α)]2
t→+∞
b2 (α)[λa(α) + c(α)]2 , 4[b(α) − d(α)]d(α)
≤
which is Eq. (11). This finishes the proof of Theorem 3.
2
= −λd(α)x − d(α)y − d(α)z + 2d(α)
z
z 50 40 30 20 10 0
-20-15
b2 (α)[λa(α) + c(α)]2 . 4[b(α) − d(α)]d(α)
Hence, we obtain
(7)
= λxx˙ + y y˙ + [z − λa(α) − c(α)]z˙
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60 40 20 0
-10 -5 x
-5 -15 0 5 10 15 -25 20
5
15 y
25
40 -25 -15
-5 x
(a)
5
-20 15
25
20 0 y
-40
(b)
z
z
250 200 150 100 50 0
300 200 100 0
-40 -20
0 x
-50 20 40 60 -100 (c)
150 100 50 y 0
200 -60 -40
-20 0 -100 20 40 60 80 -150 x
0
100 y
(d)
Fig. 1. Simulated phase portraits of the first type of Lorenz system (1) with the initial conditions x(0) = y(0) = z(0) = 10 for parameter values: (a) a = 10, b = 8/3, c = 28; (b) a = 10, b = 8/3, c = 40; (c) a = 10, b = 8/3, c = 130; (d) a = 10, b = 8/3, c = 180; (e) a = 20, b = 12, c = 130; (f) a = 0.5, b = 1.1, c = 1.0.
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z 250 200 150 100 50 0
5 y 0 -5
150 100
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-80 -60
0 -50 0 20 -100 40 60 x 80 -150
-40 -20
50 y
-10
-2
0
2
(e)
4
x
6
8
10
12
(f) Fig. 1.
(Continued )
Theorem 3 is a general result, and Eq. (11) covers many special cases considered so far, for example, as listed below.
Remarks.
(1) Take α = 0, λ ≥ 0 in Theorem 3, then Eq. (11) is the result given in the main theorem of [Liao, 2004].
(2) Take α = 0, λ = 1 in Theorem 3, then Eq. (11) gives the estimation obtained by Leonov [1987a]. (3) Take α = 0, λ = 0 in Theorem 3, then Eq. (11) becomes the estimation obtained by Leonov [1988]. z
z 90 70 50 30 10
40 30 20 10 20 -15 -10
-5 0 -10 5 10 15 20 -20 x
10 0 y
40 20 y
-15-10 -5 0 0 5 10 -20 15 20 25 x 30 -30
(a)
(b)
10
z 250 200 150 100 50 0
5 y 0
-25 -15 -5
5 x
15 25 (c)
-50 -100
150 100 50 y 0
-5 -10
-2
0
2
4
x
6
8
10
12
(d)
Fig. 2. Simulated phase portraits of the first type of Lorenz system (1) with the initial conditions x(0) = y(0) = z(0) = 10 for parameter values: (a) a = 10, b = 1.0, c = 28; (b) a = 10, b = 0.5, c = 50; (c) a = 2, b = 0.1, c = 130; (d) a = 0.1, b = 1.0, c = 1.0.
Globally Attractive Set of the Lorenz Systems
(4) Take 0 ≤ α ≤ 1, λ = 1 in Theorem 3, then Eq. (11) leads to the result given in Theorem 1 of [Li et al., 2005].
4. Numerical Simulation Results
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In this section, we present some numerical simulation results under the assumptions that a > 0, b > 0 and c ≥ 1. For the first type of Lorenz systems, a total of ten cases for choosing different values of
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the parameters, a, b and c are given. The 3-D phase portraits of the simulated trajectories are depicted in Fig. 1 (for the first six cases) and Fig. 2 (for the last four cases). For the first four cases, the typical values for a and b are chosen as a = 10 and b = 8/3, and the values of c are varied from 28 to 180. For other cases all the three parameters are varied. Note that the sixth and the last cases take the critical value c = 1. This type of Lorenz system exhibits chaotic motions in eight cases, except the sixth and
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 3. Simulated phase portraits of the second type of Lorenz system (6) with the initial conditions x(0) = y(0) = z(0) = 10 for α to equal: (a) 0.03; (b) 0.034; (c) 0.3; (d) 0.5; (e) 0.9; and (f) 0.95.
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the last cases which show convergence to stable points. However, it should be pointed out that these stable points, as shown in Figs 1(f) and 2(d), are not equilibrium points. For the second type of Lorenz system, a total of six cases are presented for different values of α. The simulated phase portraits are shown in Fig. 3. For the first two cases: α = 0.03 and α = 0.034, both of them satisfy α ∈ (0, 1/29). Formula (8) given in Theorem 2 can provide estimations for the boundaries of the attractive sets. The numerical results indicate that except for the case α = 0.9, all other cases exhibit either generalized Lorenz attractor or generalized Chen attractor. When α = 0.9, the trajectory [see Fig. 3(e)] shows a quasi-periodic motion. When α is further increased from 1, it is found that for the values of α > 1, but very close to 1, the motion is chaotic; while when α is slightly greater than 1, the motion is quasi-periodic, and then quickly becomes periodic.
5. Conclusion In this paper, we have given a survey on the results obtained so far on the estimations of the global attractive and positive invariant set of the Lorenz attractor. We have considered two types of Lorenz systems and obtained new estimations. We apply generalized Lyapunov functions to give simple proofs based on elementary algebra. The simple approach shows that Lyapunov function is still a useful tool in the study of complex nonlinear systems. Numerical simulations are presented to support our theoretical results.
Acknowledgments This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC, No. R2686A02)), and the National Natural Science Foundation of China (NNSFC, No. 60274007, 60474011).
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