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Dynamics of the stochastic Lorenz-Haken system
Chaos, Solitons and Fractals 91 (2016) 670–678
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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Dynamics of the stochastic Lorenz-Haken system Lijie Li, Yu Feng, Yongjian Liu∗ Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing, Yulin Normal University, Yulin 537000, PR China
a r t i c l e
i n f o
Article history: Received 29 May 2016 Revised 7 August 2016 Accepted 1 September 2016
Keywords: Stochastic asymptotic behavior Exponential attractive set Stochastic Lorenz-Haken system Random attractor
a b s t r a c t In this paper, the dynamics of the stochastic Lorenz-Haken system are discussed, and some new results are presented. Firstly, the asymptotic behavior of the stochastic Lorenz-Haken system is analyzed. The interesting thing is that all of solutions of the system can tend to zero under some parameters conditions and never go through the hyper-plane x = 0 as the large time. Secondly, the globally exponential attractive set and a four-dimensional ellipsoidal ultimate boundary are derived. The two-dimensional parabolic ultimate bound with respect to x − u is also established. The numerical results to estimate the ultimate boundary are also presented for verification. Finally, the random attractor set and the bifurcation phenomenon for the system are analyzed. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction The stability of lasers and related systems has been the focus of much attention [1–4], even since the earlier stage of the experimental realizations and of the founding of the theory of lasers. To analyze the dynamical behavior of lasers systems is a challenging task. Even for the dynamical behavior of the single-mode lasers system with detuning, at present, it is far from being fully understood. The analogy establishes a link between two quite different kinds of systems, and allows one to directly transfer the results obtained in one kind of system to the other. In fact, in the earlier stage, people found that single-mode laser equations are eligible analogues to the well-studied Lorenz equations. The abundant research results of Lorenz equations contribute greatly to our understanding of the dynamical behavior of lasers. And that motivates people to study the related differential equations of laser which can be analogize. The literature [1] pointed out that detuned laser can be described by complex Lorenz equations and establish a close analogy between detuned laser and baroclinic instability. Moreover, by coordinate transformation, the original complex equation can be reduced to the following four-dimensional autonomous nonlinear dynamics
⎧ ⎪ ⎨x˙ = k(y − x ),
y˙ = r1 x − y − ez + kz2 /x − xu, ⎪ ⎩z˙ = r2 x + ey − z − kyz/x, u˙ = −bu + xy,
∗
Corresponding author. Fax: 8607752661766. E-mail address: [email protected] (Y. Liu).
http://dx.doi.org/10.1016/j.chaos.2016.09.003 0960-0779/© 2016 Elsevier Ltd. All rights reserved.
(1.1)
where k, r1 , r2 , b and e are real parameters. When k < 1 + b, the lasers steadily oscillate, while k > 1 + b, the laser system goes through Hopf bifurcation into chaos and hyperchaos. System (1.1) displays a typical hyperchaotic attractor when k = 6, r1 = 91, r2 = −1.5, b = 1.2, e = 2.5, the corresponding three-dimensional phase diagrams are shown in Fig. 1. These results were obtained only for some specific parameters. It is very difficult to answer, in a systematic way, how each parameter influences higher-order instabilities, or really suppresses or destroys the complicated or chaotic behavior [5,6]. A large amount of numerica1 evidence is still needed before a satisfactory answer can be given [7]. All this may motivate further studies of these systems. On the one hand, laser will remain stable only under the condition without external influence, so the scientists work in the deep underground to develop the stability of laser. But when the laser is installed in the satellite, airplane, submarine and other high speed running body, it will be affected by the mechanical noise, the barometric pressure and temperature or some other random factors of external environment. So it is more realistic to consider the random factors in the model. We focus on the case that parameters r1 , r2 and b of Lorenz-Hake system are perturbed by environmental noises in this paper. Ideally we would also like to introduce stochastic environmental variation into the other parameters such as the transmission coefficient k and e, but to do this would make the analysis much too difficult. Hence modeling stochastic Lorenz-Haken system by replacing the parameters r1 , r2 and b, by r1 → r1 + r10 dB(t ), r2 → r1 + r20 dB(t ) and b → b − b0 dB(t ), one
L. Li et al. / Chaos, Solitons and Fractals 91 (2016) 670–678
u
z
0 −10 −20 40
20
0
20
0
120 100 80 60
20
10 x
y −20 0 120 u
u
y −20 0
30
20
10 x
120 100 80 60 40
671
100 80
20
10 x
0 −20
0
20
0 z
−10 0 z −20 −20 y
40
20
Fig. 1. Phase diagrams for the hyaperchaotic Lorenz-Haken system with system parameters (k = 6, r1 = 91, r2 = −1.5, b = 1.2, e = 2.5) and initial values (x0 , y0 , z0 , u0 ) = (5, 10, −3, 70 ).
50
x0<0
40
x >0 0
30
x(t)
20 10 0 −10 −20 −30 −40 0
5
10
15
20
25 t
30
35
40
45
50
Fig. 2. Time series diagram of the stochastic Lorenz-Haken system with k = 6, r1 = 91, r2 = −1.5, b = 1.2, e = 2.5, r10 = 1, r20 = 1, b0 = 1, and different initial values (x0 , y0 , z0 , u0 ) = (5, 10, −3, 70 ) (blue), (x0 , y0 , z0 , u0 ) = (−5, 10, −3, 70 ) (red), respectively.
gets the following system of stochastic differential equations
⎧ dx = (−kx + ky )dt, ⎪ ⎨ 2 dy = (r1 x − y − ez + kzx − xu )dt + r10 xdB(t ), kyz ⎪ ⎩dz = (r2 x + ey − z − x )dt + r20 xdB(t ), du = (−bu + xy )dt + b0 udB(t ).
(1.2)
Since there are fractions on the right side of (1.2), the system orbits can never go through the hyper-plane x = 0 and the zero solution isn’t the solution of the system. While in laser systems, x > 0, we always set x0 > 0. Numerical simulations also show that, when x0 > 0, the orbits are kept in the section where x > 0, and if x0 < 0, the system orbits are in the section where x < 0. These consist with the results of the deterministic Lorenz-Haken system [4]. These are verified in Fig. 2 by the time series diagrams with two different initial values. The interesting thing is that all of so-
lutions of the system (1.2) can tend to zero and never go through the hyper-plane x = 0 as the large time. According to the physical meaning of equation, we are interested in system long-time dynamic behavior, such as stability, attractor set and other bifurcation phenomenon in this paper. On the other hand, to locate and estimate of the attractors’ bound is important in qualitative analysis of differential equations [8–10], technically, this is a difficult work even in a deterministic system [11,12]. For the deterministic system, Yu and Liao [13] give the concept of the exponential attractive set and estimate the globally attractive and positive invariant set of the typical Lorenz system. For the stochastic system, the estimation global attractive set of system is obtained [14–16]. By using a technique combining the generalized Lyapunov function theory and optimization, globally exponential attractive set and a four-dimensional ellipsoidal
672
L. Li et al. / Chaos, Solitons and Fractals 91 (2016) 670–678
ultimate bound are derived in this paper. The two-dimensional parabolic ultimate bound with respect to x − u also is established. And numerical results to estimate the ultimate bound are also presented for verification. Furthermore, from the dynamical system viewpoint, the random attractor and bifurcation phenomenon of Lorenz-Haken system with parameters perturbed by white noise are studied. We hope that the investigation of this paper can help understanding the rich dynamic of the stochastic Lorenz-Haken system and offer some enlightenments for the study of the detuned single-mode laser. This paper is organized as follows. In Section 2, the asymptotical behavior of the system (1.2) is studied. In Section 3, globally exponential attractive set and the boundedness of the stochastic system are derived. In Section 4, the random attractor and the bifurcation phenomenon for the stochastic system are analyzed. The conclusions are given in Section 5. For convenience, some preliminaries and notions are given in the following literature [15,16], and the related marks are consistent with the above references.
x(t)
10
5
0 0
5
Proof. Consider V (X ) =
+
+
+ 2cxz
−(1+k )r1 −r2 e r , c = 1+2k . When the parameters k(1+k ) |r |(e−|r | ) 2 + r 2 ) + r , the function 0, 2 1+k 2 > 12 (r10 1 20
5
y(t) z(t)
−10 0
V is positive-definite.
u(t) |r2 |(e−|r2 | )
>
|r |(e−|r | )
2 + r 2 ) + r = 20.625. All solutions of 1 and 2 1+k 2 = 21 > 12 (r10 1 20 the system (1.2) tend to zero in probability as t → ∞.
Theorem 2. Let X(t; t0 , X0 ) be the solution of the system (1.2), where X0 = (x(t0 ), y(t0 ), z(t0 ), u(t0 )), x(t0 ) = 0. Suppose that the parame|r |(e−|r | ) 2 + r 2 ) + r and 2b > b2 . ters r1 > 0, k > 0, r2 < 0, 2 1+k 2 > 12 (r10 1 20 0 log |X (t; t0 , X0 )| ≤ 0 a.s..
c=
1 2 2 (ax
+ y2 + z2 + 2cxz + u2 ),
a=
When the supposes are satisfied, the function
V is positive-definite. Hence ε /2|X|2 ≤V(X, t)≤ξ /2|X|2 , where ε and
5
t
ξ are the smallest (positive) and the biggest eigenvalue of the positive-definite matrix T, respectively,
⎛
a ⎜0 T =⎝ c 0
0 1 0 0
c 0 1 0
⎞
0 0⎟ . 0⎠ 1
From the proof of Theorem 1, LV(X, t ) ≤ − 2ξλ V (X, t ), where
2 + r 2 ), 1, b − 1 b2 }. Similar to the proof of λ = min{ka − cr2 − 12 (r10 2 0 20
Corollary 4.3.4 of [17], we can deduce
lim sup
where
−10
Fig. 3. Time series diagrams of the stochastic Lorenz-Haken system with k = 5, r1 = 20, r2 = −2, b = 6, e = 65, r10 = 1, r20 = 0.5, b0 = 1, and initial values (x0 , y0 , z0 , u0 ) = (10, 10, −3, −20 ).
t→∞
r2 . 1+k
15
0
−30 0
Let k = 5, r1 = 20, r2 = −2, b = 6, e = 65, r10 = 1, r20 = 0.5, b0 = 1, and initial values (x0 , y0 , z0 , u0 ) = (10, 10, −3, −20 ). In Fig. 3, the number results show that all solutions X(t; t0 , X0 ) of the system (1.2) tend to zero for large t. In this case, 2b = 12 > b20 =
V (X ) =
t
−20
1 2 1 2 2 2 2 (r10 + r20 ) + r1 , ka − cr2 − 2 (r10 + r20 ) > 0. Thus, LV is negativedefinite. Similar to the proof of Theorem 4.2.3 of [17], one can prove that the solutions X(t; t0 , X0 ) (x(t0 ) = 0) of the system (1.2) tend to zero in probability as t → +∞.
−(1+k )r1 −r2 e , k(1+k )
5
10
+z(r2 x + ey − z − kyz/x ) + cx(r2 x + ey − z − kyz/x ) 1 2 2 +czk(y − x ) + u(−bu + xy ) + ((r10 + r20 )x2 + b20 u2 ) 2 1 2 1 2 = − ka − cr2 − (r10 + r20 ) x2 − y2 − z2 − b − b20 u2 . 2 2
Proof. Let
10
0
LV = axk(y − x ) + y(r1 x − y − ez + kz2 /x − xu )
1 t
15
10
Then,
Then limt→∞ sup
10
20
where a =
1+k
t
30
r1 > 0, k > 0, r2 <
Notice that the parameters r1 > 0, k > 0, r2 < 0,
15
−10
Theorem 1. Let X (t; t0 , X0 ) be the solution of the system (1.2), where X0 = (x(t0 ), y(t0 ), z(t0 ), u(t0 )), x(t0 ) = 0. Suppose that the parame|r |(e−|r | ) 2 + r 2 ) + r and 2b > b2 . ters r1 > 0, k > 0, r2 < 0, 2 1+k 2 > 12 (r10 1 20 0 Then all solutions X (t; t0 , X0 ) of the system (1.2) tend to zero in probability as t → ∞. + u2 ),
10
0
The purpose of this section is to study the asymptotic behavior of the system (1.2). Some results are the following.
z2
15
10
2. Stochastic asymptotic behavior
y2
10
20
−20 0
1 2 2 (ax
t
1 λ log (|x(t )| + |y(t )| + |z (t )| + |u(t )| ) < − ≤ 0 a.s. t ξ
This means that Theorem 2 holds. The demonstration in detail is omitted.
L. Li et al. / Chaos, Solitons and Fractals 91 (2016) 670–678
G(u ) ≤ sup G(u ) = 4r12 |b0 | = L2 ,
3. Exponential attractive set and bounedness In this section, we consider the exponential attractive set and ultimate boundary of the four dimension stochastic Lorenz-Haken system (1.2), by using a technique combining the generalized Lyapunov function theory and optimization. Furthermore, the twodimensional parabolic ultimate boundary with respect to x − u is also established. The study of the bundedness for other systems can be found in [14–16]. Theorem 3. Suppose that the parameters k > 0, r1 > 2 ) and 2b > b2 . Let L = r20 1 0
2r12 [b40 +(2b−b20 )2 ] 2b−b20
and l =
r22 4
L1 , l0
where l0 is
k
⎜ r1 ⎜ M=⎜ ⎝
2 2 (2r1 − r10 − r20 )
0
−
−
0
k r r1 2
0
k r r1 2
⎞
0
0
0
0
2
0
0
0
2b − b20
dV (X ) ≤ (−l0V + L1 )dt + (l1V + L2 )dB(t ), where l0 is smallest (positive) eigenvalue of the positive-definite matrix M,
l1 = max (|r10 | + |r20 | )
k , |r10 |, |r20 |, 3|b0 | . r1
From (3.2) and the calculating the expectation, one gets
EV (X ) ≤ V (X0 ) +
t
t0
[−l0 EV (X ) + L1 ]ds.
EV (X ) ≤ V (X0 ) exp{−l0 (t − t0 )} + L1 = V (X0 ) exp{−l0 (t − t0 )} +
Then the following estimate holds Let l =
E[V (X ) − l] ≤ [V (X0 ) − l] exp{−l0 (t − t0 )}.
is globally exponential attractive set of system (1.2), where V (X ) = r1 2 x + y2 + z 2 + ( u − 2r1 )2 . k
r1 V ( X ) = x2 + y2 + z 2 + ( u − 2r1 )2 . k
(3.1)
dV (X ) 2 2 = [−(2r1 − r10 − r20 )x2 − 2y2 − 2z2 − (2b − b20 )u2
+2r xz + 4br1 u]dt + [2r10 xy + 2r20 xz + 2b0 u(u − 2r1 )]dB(t ) ⎡ 2 ⎤ T r1 r1 ≤ ⎣− x, y, z, u−2r1 M x, y, z, u − 2r1 +F (u )⎦dt k
k
+[(|r10 | + |r20 | )x2 + |r10 |y2 + |r20 |z2 +3|b0 |(u − 2r1 )2 + G(u )]dB(t ), where
(3.2)
2 2 (2r1 − r10 − r20 )
0
0
k r r1 2
0
−
k r r1 2
0
2
0
0
0
2
0
0
2b −
0
⎞ ⎟ ⎟ ⎟, ⎠
When E(V (X ) − l ) > 0, E(V (X0 ) − l ) > 0, the follow-
lim EV (X ) ≤ l.
2r1 b20 1 = − (2b − b20 ) u − 2 2b − b20
2
+
= −|b0 |(u − 4r1 ) + Notice that k > 0, r1 > M is positive-definite. Since
r22 4
+ y2 + z 2 + ( u − 2r1 )2 ] ≤
2r12 [b40 + (2b − b20 )2 ] 2b − b20
,
2 + r 2 ) and 2b > b2 , the matrix + 12 (r10 20 0
2r12 [b40 + (2b − b20 )2 ]
= L1 ,
6
This is the globally exponential attractive set and positive invariant set of the stochastic Lorenz-Haken system. Then, we have following results of the ultimate boundary about x − y − z, x − y − u, x − z − u and y − z − u, which are the exponentially attractive sets of the stochastic Lorenz-Haken system.
E
91 6
91 6
91 6
x2 + y2 + z2 ≤ 33571.8637,
x2 + y2 + (u − 182 )2 ≤ 33571.8637,
x2 + z2 + (u − 182 )2 ≤ 33571.8637,
The corresponding three-dimensional projections of exponentially attractive sets are shown in Fig. 4. And we also have the following results,
|b0 |.
2b − b20
r1 2 x k
91 = X |E x2 + y2 + z2 + (u − 182 )2 ≤ 33571.8637 .
G(u ) = −3|b0 |(u − 2r1 ) + 2|b0 |u(u − 2r1 ) 4r12
That is, = {X |EV (X ) ≤ l } = {X |E[
E y2 + z2 + (u − 182 )2 ≤ 33571.8637.
b20
2
2
l }.
E
1 (2b − b20 )(u − 2r1 ) − (2b − b20 )u2 + 4br1 u 2
F (u ) ≤ sup F (u ) =
L1 {1 − exp{−l0 (t − t0 )}}. l0
t→∞
E
and
F (u ) =
exp{−l0 (s − t0 )}ds
Let k = 6, r1 = 91, r2 = −1.5, b = 1.2, e = 2.5, r10 = 0.5, r20 = 0.5 and b0 = 0.5. Calculate l = 33571.8637. We give the following estimate of the ultimate boundary
Applying Itô’s formula to (3.1), one has
−
t0
Thus,
Proof. Define the Lyapunov function
⎜ ⎜ M=⎜ ⎝
t
E[V (X ) − l] ≤ [V (X0 ) − l] exp{−l0 − (t − t0 )}.
= {X |EV (X ) ≤ l }
k r1
L1 . l0
ing estimate holds,
In particular,
⎛
(3.4)
From above inequation, one can obtain
⎟ ⎟ ⎟. ⎠
2
(3.3)
one obtains
2 + + 12 (r10
smallest (positive) eigenvalue of the positive-definite matrix
⎛
673
|Ex| ≤ 12.0809, |Ey| ≤ 183.2262, |Ez| ≤ 183.2262, −1.2262 ≤ Eu ≤ 365.2262.
(3.5)
The numerical solutions, which are stochastic processes, of stochastic Lorenz-Haken system (1.2) are obtained by Euler2 Maruyama method. The simulated time series about r(r = 91 6 x + y2 + z2 + (u − 182 )2 ) is displayed in Fig. 5. In addition, all the stochastic processes’ scopes and the ultimate boundary of the corresponding expectations are listed in Table 1. From the Table 1, we are pleased to see that the simulation results and the theoretical results of (3.5) are consistent.
674
L. Li et al. / Chaos, Solitons and Fractals 91 (2016) 670–678 Table 1 The ultimate boundary for stochastic Lorenz-Haken system. Scopes of stochastic processes
Simulated results of expectation
Theoretical estimates of expectation
0.0158 ≤ x ≤ 29.5772 −67.8171 ≤ y ≤ 66.8146 −37.1425 ≤ z ≤ 71.5711 35.6241 ≤ u ≤ 245.9096 139.4273 ≤ r ≤ 24022.1240
Ex = 8.323298 Ey = 8.341710 Ez = 1.522135 Eu = 94.742224 Er = 10115.3207
|Ex| ≤ 12.0809 |Ey| ≤ 183.2262 |Ez| ≤ 183.2262 −1.2262 ≤ Eu ≤ 365.2262 0 ≤ Er ≤ 33571.8637
r=
91 2 x 6
+ y2 + z2 + (u − 182 )2 .
Applying the Itô’s formula to (3.6), one has
dV = [2x(−kx + ky ) − 2k(−bu + xy )]dt − 2kb0 udB(t ) = −2k(V + (2k − b)u )dt + b0 (V − x2 )dB(t ) < −bV dt + b0V dB(t ).
(3.7)
Calculating the expectation, one gets
EV (t ) ≤ V (t0 ) −
t
t0
bEV (s )ds.
From above inequation, one can obtain
EV (t ) ≤ V (t0 )e−b(t−t0 ) . Thus
lim EVxu = lim E(x2 − 2ku ) ≤ 0.
t→+∞
t→+∞
This completes the proof.
Corollary 1. When 0 < b < 2k, b0 > 0, for every a ∈ R, the system (1.2) has the following two-dimensional parabolic ultimate boundary lim E(x2 − 2ku + a ) ≤ a.
t→+∞
Theorem 5. Let the parameters r1 < 0, 0 < 2k < b or r1 > 0, 0 < k < b < 2k. If b20 < 2b − 2k, the system (1.2) has the following twodimensional parabolic ultimate boundary
E[(x2 − 2k(u − 2r1 ))2 ] ≤ l, where l =
8kr12 (b2 −2kb20 ) 2b−2k−b20
.
Proof. Construct the following Lyapunov function
Vxu = (x2 − 2k(u − 2r1 ))2 .
(3.8)
Applying the Itô’s formula to (3.8), one has
dVxu = [2(x2 − 2k(u − 2r1 ))(2x(−kx + ky ) − 2k(−bu + xy )) +4k2 b20 u2 ]dt + 4kb20 (−x2 + 2k(u − 2r1 ))udB(t ) = [4k(x2 − 2k(u − 2r1 ))(−x2 + bu ) + 4k2 b20 u2 ]dt +4kb20 (−x2 + 2k(u − 2r1 ))udB(t ) = [−2k(x2 − 2k(u − 2r1 ))2 + H (x, u )]dt Fig. 4. The three-dimensional projection of exponentially attractive set of the stochastic Lorenz-Haken system with k = 6, r1 = 91, r2 = −1.5, b = 1.2, e = 2.5, r10 = 0.5, r20 = 0.5, b0 = 0.5, and initial values (x0 , y0 , z0 , u0 ) = (5, 10, −3, 70 ).
Theorem 4. When 0 < b < 2k, b0 > 0, the system (1.2) has the following two-dimensional parabolic ultimate bound
lim E(x2 − 2ku ) ≤ 0.
Proof. Construct the following Lyapunov function
(3.6)
(3.9)
where
H (x, u ) = 2k(x2 −2k(u−2r1 ))2 +4k(x2 −2k(u − 2r1 ))(−x2 + bu ) +4k2 b20 u2 . Let
t→+∞
Vxu = x2 − 2ku.
+4kb20 (−x2 + 2k(u − 2r1 ))udB(t ),
⎧ ⎪ ⎨ ∂ H = 8kx(−x2 + bu ) = 0, ∂x ∂ ⎪ ⎩ H = 4k[2k(2k − 2b + b20 )u − 4kr1 (2k − b) + bx2 ] = 0. ∂u
L. Li et al. / Chaos, Solitons and Fractals 91 (2016) 670–678
2.5
x 10
675
4
2
r
1.5
1
0.5
0 0
5
10
15
20
25 t
30
35
40
45
50
x2 + y2 + z2 + (u − 182 )2 ) of the stochastic Lorenz-Haken system with k = 6, r1 = 91, r2 = −1.5, b = 1.2, e = 2.5, r10 = 0.5, Fig. 5. The diagram of time series r (t )(r = 91 6 r20 = 0.5, b0 = 0.5, and initial values (x0 , y0 , z0 , u0 ) = (5, 10, −3, 70 ).
2
if b20 < min{ 2bk , 2b − 2k}, function H(x, u) has the global maximum value,
Then the solutions for the above equations are
⎧ ⎨x = 0,
⎧ 2 ⎨x = bu,
⎩u =
⎩u =
2r1 ( 2k − b ) , 2k − 2b + b20
4kr1 (2k − b) . (b − 2k )2 + 4kb20
The Hessian matrix of the function H(x, u) is
Hxx Hxu
Hxu Huu
=
8k(−3x2 + bu ) 8bkx
H (x, u ) ≤
8bkx 8k2 (2k − 2b + b20 ) .
lim
2r (2k−b) xε →0,u= 1 2k−2b+b2 0
H
xε xε
= 8k(−3xε + bu ), 2
H
2 3 2 2 2 2 2 xε xε Huu − (Hxε u ) = 64k (−3xε + bu )(2k − 2b + b0 ) − 64b k xε .
For ε be arbitrary, one can deduce that, when the parameters r1 < 2 0, 0 < 2k < b (or r1 > 0, 0 < k < b < 2k) and b20 < min{ 2bk , 2b − 2k},
lim xε →0,u=
2r1 (2k−b) 2k−2b+b2 0
lim xε →0,u=
2r1 (2k−b) 2k−2b+b2 0
Hxε xε = 16r1 bk
2k − b < 0, 2k − 2b + b20
− (H )2 = 128r bk3 (2k − b) > 0. Hxε xε Huu 1 xε u
That means the corresponding Hessian matrix is strictly negative. 4kr1 (2k−b) Case 2. x2 = bu, u = 2. 2 (b−2k ) +4kb0
In this case, when k > 0,
H − (H )2 = 8k (−3x2 + bu ) · 8k2 (2k − 2b + b2 ) − 64b2 k2 x2 Hxx uu xu 0
= −64k2 x2 [(b − 2k )2 + 2kb20 ] < 0. That means function H(x, u) couldn’t get the local extremum value in this case. Obviously, the Hessian matrix of the function H(x, u) is continuous. When the parameters r1 < 0, 0 < 2k < b (or r1 > 0, 0 < k < b < 2k), from case 1 and case 2, one can deduce that
16k2 r12 (b2 − 2kb20 ) 2b − 2k − b20
= lxu .
dVxu ≤ (−2kVxu + lxu )dt + 4kb20 (−x2 + 2k(u − 2r1 ))udB(t ). Similar to the proof of Theorem 3, one can obtain
E Vxu (x, t ) −
(2k−2b+b0 )
Notice that X = (0, y(t ), z(t ), u(t )) is not the solution of system (1.2). One can’t rudely set x = 0. Here take the limit x → 0. For every ε > 0, let −ε xε ε (xε = 0 ). One has
(xε ,u )
∈R2
Then
!
Next, one discusses the above H(x, u). 2r1 (2k−b) Case 1. x = 0, u = 2 .
sup H (xε , u ) =
lxu 2k
"
Thus, lim EVxu ≤ t→+∞
!
"
≤ Vxu (x(t0 ), u(t0 )) − lxu . 2k
lxu exp(−2k(t − t0 )). 2k
That is
E[(x2 − 2k(u − 2r1 ))2 ] ≤ l. Thus we complete the proof of the Theorem 5. √ Using the properties of expectations Ex ≤ Ex2 , one gets the following conclusion. Corollary 2. Let the parameters r1 < 0, 0 < 2k < b or r1 > 0, 0 < k < b < 2k. If b20 < 2b − 2k, the system (1.2) has the following twodimensional parabolic ultimate bound
E [ x 2 − 2 k ( u − 2 r 1 )] ≤ where l =
8kr12 (b2 −2kb20 ) 2b−2k−b20
#
l,
.
Let k = 6, r1 = 91, r2 = −1.5, b = 8, e = 2.5, r10 = 0.5, r20 = 0.5 and b0 = 0.5. One can calculate l = 6465804.80 and give the following estimate of the ultimate bound
$ % = X |E[x2 − 12(u − 182 )]2 ≤ 6465804.80 ,
which is the two-dimensional parabolic ultimate bound with respect to x − u of the stochastic Lorenz-Haken system. The simulated time series is displayed in Fig. 6. And the simulated results show that E[x2 − 12(u − 182 )]2 = 3353891.87 ≤ 6465800, which is consistent with the theoretical analysis result. In addition, according to Corollary 2, one can obtain
E[x2 − 12(u − 182 )] ≤
√ 6465804.80 = 2542.80.
(3.10)
676
L. Li et al. / Chaos, Solitons and Fractals 91 (2016) 670–678
5.5
x 10
6
5 4.5
r
4 3.5 3 2.5 2 1.5 0
5
10
15
20
25 t
30
35
40
45
50
Fig. 6. The diagram of time series r (t )(r = (x2 + 12(u − 182 ))2 of the stochastic Lorenz-Haken system with k = 6, r1 = 91, r2 = −1.5, b = 8, e = 2.5, r10 = 0.5, r20 = 0.5, b0 = 0.5, and initial values (x0 , y0 , z0 , u0 ) = (5, 10, −3, 70 ).
By using the properties of expectations and Theorem 3, one has
E [ x 2 − 2 k ( u − 2 r 1 )] ≤ E ( x 2 + ( u − 2 r 1 ) 2 + k 2 ) 91 ≤ E ( x2 + y2 + z 2 + ( u − 2r1 )2 ) + k2 6 L1 ≤ + k2 , l0
V (X ) =
r1 2 x + y2 + z 2 + ( u − 2r1 )2 . k
From the proof of Theorem 4 and Itô’s formula, one has
V (X ) ≤ V (X0 ) +
t
0
[−l0V (X ) + L1 ]ds +
t 0
( l 1 V ( X ) + L 2 )d B ( s ) , (4.1)
where L1 and l0 are defined in Theorem 3. Take the same parameters as in the above, we have
E[x2 − 12(u − 182 )] ≤ 131471.32. We also have the following result according to Corollary 1,
E[x2 − 12(u − 182 )] ≤ 2184.
Proof. Let
(3.11)
The simulated results show that E[x2 − 12(u − 182 )] = 1827.18, which is consistent with the theoretical analysis result from above. We know that if the higher order moments are bounded, the lower order moments are bounded. But if the lower order moments are bounded, the higher order moments are not always bounded, even that they do not exist. Theorem 4 estimates the boundary of E(x2 − 2ku ). In Theorem 5, the boundary of E[(x2 − 2k(u − 2r1 ))2 ] is derived, which is the higher order moment ultimate boundary about x − u. To our best knowledge, the result about ultimate boundary of second-order moment is very few for the stochastic systems.
where L1 and L2 are defined in (3.3), l0 and l1 are defined in (3.4), respectively. Suppose that Z satisfies the following equation
Z ≤ V (X0 ) +
t 0
[−l0 Zs + L1 ]ds +
t
0
( l 1 Z s + L 2 )d B ( s ) .
(4.2)
Then Z is finite on any finite interval [0, T] for any T > 0. Hence, (4.1) and (4.2) yield that the process V(X) is finite on [0, T] for any 0 < T < ∞. That is to say, the random dynamical system ϕ (t, ω, x) is global to the forward. r2
2 + Theorem 7. Suppose that the parameters k > 0, r1 > 42 + 12 (r10 2 ) and 2b > b2 . Then the stochastic Lorenz-Haken system generate r20 0 by (1.2) possesses the unique parameter dependent tempered random attractor A with domain of attraction D (A ) containing the universe of sets Cl (U ), generated by
U = {(D(ω ))ω∈ |D(ω ) ⊂ R4 is a tempered random set}. Proof. Let
r1 2 x + y2 + z 2 + ( u − 2r1 )2 . k
4. Random attractors and bifurcation
V (X ) =
In this section, we consider the random attractor generating by the system (1.2) and bifurcation phenomenon from the dynamical system viewpoint. The study of the existence of random attractor and stochastic bifurcation behavior for other stochastic systems can be found in [18–25].
From the proof of Theorem 4 and Itô’s formula, one obtains
r2
2 + Theorem 6. Suppose that the parameters k > 0, r1 > 42 + 12 (r10 2 2 r20 ) and 2b > b0 . Then the stochastic Lorenz-Haken system (1.2) generates a smooth RDS ϕ which is global to forward. That is Dt = R4 for all t > 0. Moreover, ϕ (t, θ t ω, x) is Ft0 -measurable.
V (X ) ≤ V (X0 ) +
0
t
[−l0V (X ) + L1 ]ds +
t 0
( l 1 V ( X ) + L 2 )d B ( s ) , (4.3)
where L1 and L2 are defined in (3.3), l0 and l1 are defined in (3.4), respectively. Denote ϕ (t, ω, (x, y, z)) and ψ (t, ω, V(X)) the random dynamical system which are generated by (1.2) and
Zt = V (X0 ) +
t 0
[−l0 Zs + L1 ]ds +
t 0
( l 1 Z s + L 2 )d B ( s ) ,
(4.4)
L. Li et al. / Chaos, Solitons and Fractals 91 (2016) 670–678
"
respectively. The random dynamical system generated by (4.4) is
ψ (t, ω )x = exp{−l0t + l1 B(t )} t × x + L1 exp{l0 s − l1 B(s )}d(s ) .
+czk(y−x ) + u(−bu+xy ) +
0
ξ ( ω ) = L1
−∞
− 1−
U = {I(ω ) ⊂ R+ is a tempered random set}.
Next, we study bifurcation phenomenon of the system (1.2). Some the basic conceptions and notions can be found in the literature [14,16,26] and the related marks are consistent with the above references. Proposition 1. Suppose that the conditions of Theorem 5 are satisfied. Denote A(ω) the random attractor of the random dynamical system ϕ generated by stochastic Lorenz-Haken Eq. (1.2). Then (i) The random attractor A supports all invariant measures; (ii) The unstable set of any invariant measure μ is contained in + random attractor, that is, Mμ ( ω ) ⊂ A ( ω ); (iii) There exists an invariant Markov measure μ supported by 0 -measurable. ∂ A(ω), that is, μω is F−∞ Proof. Similar to the proof in Proposition 7.5 of [27]. 2. Let
k|r2 | 2 (1+k ) , 2b − b0 }/ξ ,
l2 = min{
2|r2 | (e−|r2 | ) 1+k
2 + r 2 ), 2− − 2r1 − (r10 20
where ξ is defined in Theorem 2. Suppose that the
parameters k > 0,
k) − 2(1+ k
< r2 < 0,
|r2 |(e−|r2 | ) 1+k
>
1 2 2 (r10
+
2 ) r20
+ r1
and 2b > b20 . Then the top Lyapunov exponents L3 of the linear stochastic equation
⎧ dx = (−kx + ky )dt, ⎪ ⎪ ⎪ ⎨dy = (r x − y − ez )dt + r xdB(t ), 1 10 ⎪ d z = r x + ey − z d t + r ( ) 2 20 xdB (t ), ⎪ ⎪ ⎩ du = −budt + b0 udB(t ),
is bounded by L3 ≤ −l2 . Proof. Consider V (X ) =
1 2 2 2 2 2 (ax + y + z + 2cxz + u ), where a = −(1+k )r1 −r2 e r2 , c = 1+k . When the parameters r1 > 0, k > 0, r2 < k(1+k ) |r |(e−|r | ) 2 + r 2 ) + r , the function V is positive-definite. 0, 2 1+k 2 > 12 (r10 1 20
Then,
dV (X ) =
! axk(y − x ) + y(r1 x − y − ez ) + z(r2 x + ey − z ) +cx(r2 x + ey − z )
where
k|c | 2 z − 2
b−
b20 2
1−
"
k|c | 2 y 2
u2 dt
$
&
'
%
2 2 l2 = min 2ka − 2cr2 − r10 + r20 , 2 − k|c|, 2b − b20 /ξ
= min
That is, for any I ∈ U , the random variable η (ω ) = supx∈I(ω ) x + satisfies limt→∞ log [η (θt ω )/t] = 0. Then, for any ε > 0, − ε t limt→∞ e η (θt ω ) = 0. It is obvious that U is closed under inclusion and × {x} ∈ U for all x ∈ R+ . The rest of the proof is the same as that for Theorem 5.4 from the references Schenk-Hoppé [27], we omit it here.
1 2 2 (r + r20 ) x2 − 2 10
1 + [(|r10 | + |r20 | )x2 + |r10 |y2 + |r20 |z2 + |b0 |u2 ]dB(t ) 2 ≤ −l2V (X )dt + l3V (X )dB(t ) (4.7)
exp{l0 s − l1W (s )}d(s ) > 0.
It is easy to prove that ψ (t, θ−t θ x(θ−t ω )) → ξ (ω ) as t → ∞ for initial value x(ω ) ∈ R+ such that exp(−l0 )x(θ−t ω ) → 0 for some l0 > 0. Hence, one may definite the universe of set
Proposition
− ka − cr2 −
(4.6)
Thus ψ (t, ω) has the unique invariant measure which is a Dirac measure supported by 0
!
≤
From (4.3), one can obtain
1 2 2 ((r + r20 )x2 + b20 u2 ) dt 2 10
+[|r10 |xy + |r20 |xz + |b0 |u2 ]dB(t )
(4.5)
V (ϕ (t, ω, (x, y, z ))) ≤ ψ (t, ω )V (X ).
677
&2 ' k|r2 | 2|r2 |(e−| r2 | ) 2 −2r1 − r10 + r20 , 2− , 2b − b20 /ξ , 1+k (1+k )
l3 = max {|r10 | + |r20 |, |b0 |}. From (4.7) and Doléans-Dade’s exponent formula, one obtains
V (X ) ≤ exp{−l2 t + l3 B(t )}. One has
log V (X ) t −l2 t + l3 B(t ) ≤ lim t→∞ t = −l2 .
L3 = lim
t→∞
k) Theorem 8. Suppose that the parameters k > 0, − 2(1+ < r2 < k
|r |(e−|r | )
r2
2 + r 2 ) + r , r > 2 + 1 (r 2 + r 2 ) and 2b > b2 . 0, 2 1+k 2 > 12 (r10 1 1 4 2 10 20 20 0 Then the random dynamical system generate by the stochastic LorenzHaken system (1.2) possesses the unique tempered random attractors A(ω ) = {0} with domain of attraction D(A) containing the universe of Cl (U ) given by
U = {D(ω )ω∈ |D(ω ) ⊂ R3 is a tempered random set}, where Cl (U ) is the completion universe of U under inclusion. In particular, for any x ∈ R3 , the solutions ϕ (t, θ−t ω )x and ϕ (t, ω)x tend to zero exponentially fast as t → ∞. Proof. Proposition 2 and l3 > 0 yield that the random dynamical system generated (1.2) is stability. Define the Lyapunov function on R4
V (X ) =
r1 2 x + y2 + z 2 + ( u − 2r1 )2 , k
then V(X) ≥ 0 and V (X ) = 0 if and only x = y = z = 0, u = 2r1 . Applying Itô’s formula to the above equation, one has
dV (X ) ≤ [−l0V (X ) + L1 ]dt + (l1V (X ) + L2 )dB(t ), where L1 and L2 are defined in (3.3), l0 and l1 are defined in (3.4), respectively. Corollary 5.6.3. of [24] yields that the random dynamical system generated by the stochastic differential equation
dZt = [−l0 Zt + L1 ]dt + (l1 Zt + L2 )dB(t ), has the unique invariant measure δ 0 which is the Dirac measure at 0. As the proof in Theorem 6, one has that A(ω ) = {0} = V −1 ({0} ) is the unique random attractor attracting any set form U. Hence, it means that, for any X ∈ R3 , ϕ (t, θ−t ω )x and ϕ (t, ω)x tend to zero as t → ∞ exponentially, thus complete the proof of the theorem.
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L. Li et al. / Chaos, Solitons and Fractals 91 (2016) 670–678
5. Conclusions
References
The interest in the stability of lasers and related systems has been ever increasing, especially since the discovery of the fundamental analogy between single-mode laser equations and the wellstudied Lorenz equations. In laser physics and in particular in laser dynamics the Lorenz-Haken model is generally recognized as the most fundamental model from the conceptual point of view. On the one hand, when the laser is installed in the satellite, airplane, submarine and other high speed running body, it will be affected by the mechanical noise, the barometric pressure and temperature or some other random factors of external environment. On the other hand, a complete knowledge about the stability of stationary states as well as oscillations, especially a systematic understanding of the influence of the parameter variations on the nature of the dynamics, is still lacking. A large amount of numerica1 evidence is still needed before a satisfactory answer can be given. This paper is devoted to investigating the stochastic LorenzHaken system. The globally dynamics, such as asymptotic behavior, exponential attractive set and random attractor are rigorously studied. Firstly, the interesting thing is that all of solutions of the system (1.2) can tend to zero and never go through the hyper-plane x = 0 as the large time. These consist with the results of the deterministic Lorenz-Haken system. Secondly, we estimate the globally exponential attractive set and a four-dimensional ellipsoidal ultimate bound. The two-dimensional parabolic ultimate boundary with respect to x − u also is established. And numerical results to estimate the ultimate bound are also presented for verification. Finally, random attractors and the bifurcation phenomenon for the system is analyzed. It is hoped that the investigation of this paper can help understanding the rich dynamic of the stochastic LorenzHaken system and offer some enlightenments for the study of the detuned single-mode laser.
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Acknowledgments The authors would like to express their sincere thanks to the editor and anonymous referees for their careful and pertinent suggestions for improving the presentation of the work. This work was supported by the National Natural Science Foundation of China (Grant No. 11561069), the Guangxi Natural Science Foundation of China (Grant No. 2015GXNSFAA139002), the Scientific Research Foundation of the Higher Education Institutions of Guangxi of China (Grant No. YB2014309, KY2016YB364), the Opening Foundation of Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing (Grant No. 2015CSOBDP0202), the University Key Teachers Training Program from Guangxi Higher Education Institutions of China (Document No. [2014]39) and the special Foundation of Excellent Talents in Yulin Normal University (Grant NO.G2014003).
Contents lists available at ScienceDirect
Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Dynamics of the stochastic Lorenz-Haken system Lijie Li, Yu Feng, Yongjian Liu∗ Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing, Yulin Normal University, Yulin 537000, PR China
a r t i c l e
i n f o
Article history: Received 29 May 2016 Revised 7 August 2016 Accepted 1 September 2016
Keywords: Stochastic asymptotic behavior Exponential attractive set Stochastic Lorenz-Haken system Random attractor
a b s t r a c t In this paper, the dynamics of the stochastic Lorenz-Haken system are discussed, and some new results are presented. Firstly, the asymptotic behavior of the stochastic Lorenz-Haken system is analyzed. The interesting thing is that all of solutions of the system can tend to zero under some parameters conditions and never go through the hyper-plane x = 0 as the large time. Secondly, the globally exponential attractive set and a four-dimensional ellipsoidal ultimate boundary are derived. The two-dimensional parabolic ultimate bound with respect to x − u is also established. The numerical results to estimate the ultimate boundary are also presented for verification. Finally, the random attractor set and the bifurcation phenomenon for the system are analyzed. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction The stability of lasers and related systems has been the focus of much attention [1–4], even since the earlier stage of the experimental realizations and of the founding of the theory of lasers. To analyze the dynamical behavior of lasers systems is a challenging task. Even for the dynamical behavior of the single-mode lasers system with detuning, at present, it is far from being fully understood. The analogy establishes a link between two quite different kinds of systems, and allows one to directly transfer the results obtained in one kind of system to the other. In fact, in the earlier stage, people found that single-mode laser equations are eligible analogues to the well-studied Lorenz equations. The abundant research results of Lorenz equations contribute greatly to our understanding of the dynamical behavior of lasers. And that motivates people to study the related differential equations of laser which can be analogize. The literature [1] pointed out that detuned laser can be described by complex Lorenz equations and establish a close analogy between detuned laser and baroclinic instability. Moreover, by coordinate transformation, the original complex equation can be reduced to the following four-dimensional autonomous nonlinear dynamics
⎧ ⎪ ⎨x˙ = k(y − x ),
y˙ = r1 x − y − ez + kz2 /x − xu, ⎪ ⎩z˙ = r2 x + ey − z − kyz/x, u˙ = −bu + xy,
∗
Corresponding author. Fax: 8607752661766. E-mail address: [email protected] (Y. Liu).
http://dx.doi.org/10.1016/j.chaos.2016.09.003 0960-0779/© 2016 Elsevier Ltd. All rights reserved.
(1.1)
where k, r1 , r2 , b and e are real parameters. When k < 1 + b, the lasers steadily oscillate, while k > 1 + b, the laser system goes through Hopf bifurcation into chaos and hyperchaos. System (1.1) displays a typical hyperchaotic attractor when k = 6, r1 = 91, r2 = −1.5, b = 1.2, e = 2.5, the corresponding three-dimensional phase diagrams are shown in Fig. 1. These results were obtained only for some specific parameters. It is very difficult to answer, in a systematic way, how each parameter influences higher-order instabilities, or really suppresses or destroys the complicated or chaotic behavior [5,6]. A large amount of numerica1 evidence is still needed before a satisfactory answer can be given [7]. All this may motivate further studies of these systems. On the one hand, laser will remain stable only under the condition without external influence, so the scientists work in the deep underground to develop the stability of laser. But when the laser is installed in the satellite, airplane, submarine and other high speed running body, it will be affected by the mechanical noise, the barometric pressure and temperature or some other random factors of external environment. So it is more realistic to consider the random factors in the model. We focus on the case that parameters r1 , r2 and b of Lorenz-Hake system are perturbed by environmental noises in this paper. Ideally we would also like to introduce stochastic environmental variation into the other parameters such as the transmission coefficient k and e, but to do this would make the analysis much too difficult. Hence modeling stochastic Lorenz-Haken system by replacing the parameters r1 , r2 and b, by r1 → r1 + r10 dB(t ), r2 → r1 + r20 dB(t ) and b → b − b0 dB(t ), one
L. Li et al. / Chaos, Solitons and Fractals 91 (2016) 670–678
u
z
0 −10 −20 40
20
0
20
0
120 100 80 60
20
10 x
y −20 0 120 u
u
y −20 0
30
20
10 x
120 100 80 60 40
671
100 80
20
10 x
0 −20
0
20
0 z
−10 0 z −20 −20 y
40
20
Fig. 1. Phase diagrams for the hyaperchaotic Lorenz-Haken system with system parameters (k = 6, r1 = 91, r2 = −1.5, b = 1.2, e = 2.5) and initial values (x0 , y0 , z0 , u0 ) = (5, 10, −3, 70 ).
50
x0<0
40
x >0 0
30
x(t)
20 10 0 −10 −20 −30 −40 0
5
10
15
20
25 t
30
35
40
45
50
Fig. 2. Time series diagram of the stochastic Lorenz-Haken system with k = 6, r1 = 91, r2 = −1.5, b = 1.2, e = 2.5, r10 = 1, r20 = 1, b0 = 1, and different initial values (x0 , y0 , z0 , u0 ) = (5, 10, −3, 70 ) (blue), (x0 , y0 , z0 , u0 ) = (−5, 10, −3, 70 ) (red), respectively.
gets the following system of stochastic differential equations
⎧ dx = (−kx + ky )dt, ⎪ ⎨ 2 dy = (r1 x − y − ez + kzx − xu )dt + r10 xdB(t ), kyz ⎪ ⎩dz = (r2 x + ey − z − x )dt + r20 xdB(t ), du = (−bu + xy )dt + b0 udB(t ).
(1.2)
Since there are fractions on the right side of (1.2), the system orbits can never go through the hyper-plane x = 0 and the zero solution isn’t the solution of the system. While in laser systems, x > 0, we always set x0 > 0. Numerical simulations also show that, when x0 > 0, the orbits are kept in the section where x > 0, and if x0 < 0, the system orbits are in the section where x < 0. These consist with the results of the deterministic Lorenz-Haken system [4]. These are verified in Fig. 2 by the time series diagrams with two different initial values. The interesting thing is that all of so-
lutions of the system (1.2) can tend to zero and never go through the hyper-plane x = 0 as the large time. According to the physical meaning of equation, we are interested in system long-time dynamic behavior, such as stability, attractor set and other bifurcation phenomenon in this paper. On the other hand, to locate and estimate of the attractors’ bound is important in qualitative analysis of differential equations [8–10], technically, this is a difficult work even in a deterministic system [11,12]. For the deterministic system, Yu and Liao [13] give the concept of the exponential attractive set and estimate the globally attractive and positive invariant set of the typical Lorenz system. For the stochastic system, the estimation global attractive set of system is obtained [14–16]. By using a technique combining the generalized Lyapunov function theory and optimization, globally exponential attractive set and a four-dimensional ellipsoidal
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L. Li et al. / Chaos, Solitons and Fractals 91 (2016) 670–678
ultimate bound are derived in this paper. The two-dimensional parabolic ultimate bound with respect to x − u also is established. And numerical results to estimate the ultimate bound are also presented for verification. Furthermore, from the dynamical system viewpoint, the random attractor and bifurcation phenomenon of Lorenz-Haken system with parameters perturbed by white noise are studied. We hope that the investigation of this paper can help understanding the rich dynamic of the stochastic Lorenz-Haken system and offer some enlightenments for the study of the detuned single-mode laser. This paper is organized as follows. In Section 2, the asymptotical behavior of the system (1.2) is studied. In Section 3, globally exponential attractive set and the boundedness of the stochastic system are derived. In Section 4, the random attractor and the bifurcation phenomenon for the stochastic system are analyzed. The conclusions are given in Section 5. For convenience, some preliminaries and notions are given in the following literature [15,16], and the related marks are consistent with the above references.
x(t)
10
5
0 0
5
Proof. Consider V (X ) =
+
+
+ 2cxz
−(1+k )r1 −r2 e r , c = 1+2k . When the parameters k(1+k ) |r |(e−|r | ) 2 + r 2 ) + r , the function 0, 2 1+k 2 > 12 (r10 1 20
5
y(t) z(t)
−10 0
V is positive-definite.
u(t) |r2 |(e−|r2 | )
>
|r |(e−|r | )
2 + r 2 ) + r = 20.625. All solutions of 1 and 2 1+k 2 = 21 > 12 (r10 1 20 the system (1.2) tend to zero in probability as t → ∞.
Theorem 2. Let X(t; t0 , X0 ) be the solution of the system (1.2), where X0 = (x(t0 ), y(t0 ), z(t0 ), u(t0 )), x(t0 ) = 0. Suppose that the parame|r |(e−|r | ) 2 + r 2 ) + r and 2b > b2 . ters r1 > 0, k > 0, r2 < 0, 2 1+k 2 > 12 (r10 1 20 0 log |X (t; t0 , X0 )| ≤ 0 a.s..
c=
1 2 2 (ax
+ y2 + z2 + 2cxz + u2 ),
a=
When the supposes are satisfied, the function
V is positive-definite. Hence ε /2|X|2 ≤V(X, t)≤ξ /2|X|2 , where ε and
5
t
ξ are the smallest (positive) and the biggest eigenvalue of the positive-definite matrix T, respectively,
⎛
a ⎜0 T =⎝ c 0
0 1 0 0
c 0 1 0
⎞
0 0⎟ . 0⎠ 1
From the proof of Theorem 1, LV(X, t ) ≤ − 2ξλ V (X, t ), where
2 + r 2 ), 1, b − 1 b2 }. Similar to the proof of λ = min{ka − cr2 − 12 (r10 2 0 20
Corollary 4.3.4 of [17], we can deduce
lim sup
where
−10
Fig. 3. Time series diagrams of the stochastic Lorenz-Haken system with k = 5, r1 = 20, r2 = −2, b = 6, e = 65, r10 = 1, r20 = 0.5, b0 = 1, and initial values (x0 , y0 , z0 , u0 ) = (10, 10, −3, −20 ).
t→∞
r2 . 1+k
15
0
−30 0
Let k = 5, r1 = 20, r2 = −2, b = 6, e = 65, r10 = 1, r20 = 0.5, b0 = 1, and initial values (x0 , y0 , z0 , u0 ) = (10, 10, −3, −20 ). In Fig. 3, the number results show that all solutions X(t; t0 , X0 ) of the system (1.2) tend to zero for large t. In this case, 2b = 12 > b20 =
V (X ) =
t
−20
1 2 1 2 2 2 2 (r10 + r20 ) + r1 , ka − cr2 − 2 (r10 + r20 ) > 0. Thus, LV is negativedefinite. Similar to the proof of Theorem 4.2.3 of [17], one can prove that the solutions X(t; t0 , X0 ) (x(t0 ) = 0) of the system (1.2) tend to zero in probability as t → +∞.
−(1+k )r1 −r2 e , k(1+k )
5
10
+z(r2 x + ey − z − kyz/x ) + cx(r2 x + ey − z − kyz/x ) 1 2 2 +czk(y − x ) + u(−bu + xy ) + ((r10 + r20 )x2 + b20 u2 ) 2 1 2 1 2 = − ka − cr2 − (r10 + r20 ) x2 − y2 − z2 − b − b20 u2 . 2 2
Proof. Let
10
0
LV = axk(y − x ) + y(r1 x − y − ez + kz2 /x − xu )
1 t
15
10
Then,
Then limt→∞ sup
10
20
where a =
1+k
t
30
r1 > 0, k > 0, r2 <
Notice that the parameters r1 > 0, k > 0, r2 < 0,
15
−10
Theorem 1. Let X (t; t0 , X0 ) be the solution of the system (1.2), where X0 = (x(t0 ), y(t0 ), z(t0 ), u(t0 )), x(t0 ) = 0. Suppose that the parame|r |(e−|r | ) 2 + r 2 ) + r and 2b > b2 . ters r1 > 0, k > 0, r2 < 0, 2 1+k 2 > 12 (r10 1 20 0 Then all solutions X (t; t0 , X0 ) of the system (1.2) tend to zero in probability as t → ∞. + u2 ),
10
0
The purpose of this section is to study the asymptotic behavior of the system (1.2). Some results are the following.
z2
15
10
2. Stochastic asymptotic behavior
y2
10
20
−20 0
1 2 2 (ax
t
1 λ log (|x(t )| + |y(t )| + |z (t )| + |u(t )| ) < − ≤ 0 a.s. t ξ
This means that Theorem 2 holds. The demonstration in detail is omitted.
L. Li et al. / Chaos, Solitons and Fractals 91 (2016) 670–678
G(u ) ≤ sup G(u ) = 4r12 |b0 | = L2 ,
3. Exponential attractive set and bounedness In this section, we consider the exponential attractive set and ultimate boundary of the four dimension stochastic Lorenz-Haken system (1.2), by using a technique combining the generalized Lyapunov function theory and optimization. Furthermore, the twodimensional parabolic ultimate boundary with respect to x − u is also established. The study of the bundedness for other systems can be found in [14–16]. Theorem 3. Suppose that the parameters k > 0, r1 > 2 ) and 2b > b2 . Let L = r20 1 0
2r12 [b40 +(2b−b20 )2 ] 2b−b20
and l =
r22 4
L1 , l0
where l0 is
k
⎜ r1 ⎜ M=⎜ ⎝
2 2 (2r1 − r10 − r20 )
0
−
−
0
k r r1 2
0
k r r1 2
⎞
0
0
0
0
2
0
0
0
2b − b20
dV (X ) ≤ (−l0V + L1 )dt + (l1V + L2 )dB(t ), where l0 is smallest (positive) eigenvalue of the positive-definite matrix M,
l1 = max (|r10 | + |r20 | )
k , |r10 |, |r20 |, 3|b0 | . r1
From (3.2) and the calculating the expectation, one gets
EV (X ) ≤ V (X0 ) +
t
t0
[−l0 EV (X ) + L1 ]ds.
EV (X ) ≤ V (X0 ) exp{−l0 (t − t0 )} + L1 = V (X0 ) exp{−l0 (t − t0 )} +
Then the following estimate holds Let l =
E[V (X ) − l] ≤ [V (X0 ) − l] exp{−l0 (t − t0 )}.
is globally exponential attractive set of system (1.2), where V (X ) = r1 2 x + y2 + z 2 + ( u − 2r1 )2 . k
r1 V ( X ) = x2 + y2 + z 2 + ( u − 2r1 )2 . k
(3.1)
dV (X ) 2 2 = [−(2r1 − r10 − r20 )x2 − 2y2 − 2z2 − (2b − b20 )u2
+2r xz + 4br1 u]dt + [2r10 xy + 2r20 xz + 2b0 u(u − 2r1 )]dB(t ) ⎡ 2 ⎤ T r1 r1 ≤ ⎣− x, y, z, u−2r1 M x, y, z, u − 2r1 +F (u )⎦dt k
k
+[(|r10 | + |r20 | )x2 + |r10 |y2 + |r20 |z2 +3|b0 |(u − 2r1 )2 + G(u )]dB(t ), where
(3.2)
2 2 (2r1 − r10 − r20 )
0
0
k r r1 2
0
−
k r r1 2
0
2
0
0
0
2
0
0
2b −
0
⎞ ⎟ ⎟ ⎟, ⎠
When E(V (X ) − l ) > 0, E(V (X0 ) − l ) > 0, the follow-
lim EV (X ) ≤ l.
2r1 b20 1 = − (2b − b20 ) u − 2 2b − b20
2
+
= −|b0 |(u − 4r1 ) + Notice that k > 0, r1 > M is positive-definite. Since
r22 4
+ y2 + z 2 + ( u − 2r1 )2 ] ≤
2r12 [b40 + (2b − b20 )2 ] 2b − b20
,
2 + r 2 ) and 2b > b2 , the matrix + 12 (r10 20 0
2r12 [b40 + (2b − b20 )2 ]
= L1 ,
6
This is the globally exponential attractive set and positive invariant set of the stochastic Lorenz-Haken system. Then, we have following results of the ultimate boundary about x − y − z, x − y − u, x − z − u and y − z − u, which are the exponentially attractive sets of the stochastic Lorenz-Haken system.
E
91 6
91 6
91 6
x2 + y2 + z2 ≤ 33571.8637,
x2 + y2 + (u − 182 )2 ≤ 33571.8637,
x2 + z2 + (u − 182 )2 ≤ 33571.8637,
The corresponding three-dimensional projections of exponentially attractive sets are shown in Fig. 4. And we also have the following results,
|b0 |.
2b − b20
r1 2 x k
91 = X |E x2 + y2 + z2 + (u − 182 )2 ≤ 33571.8637 .
G(u ) = −3|b0 |(u − 2r1 ) + 2|b0 |u(u − 2r1 ) 4r12
That is, = {X |EV (X ) ≤ l } = {X |E[
E y2 + z2 + (u − 182 )2 ≤ 33571.8637.
b20
2
2
l }.
E
1 (2b − b20 )(u − 2r1 ) − (2b − b20 )u2 + 4br1 u 2
F (u ) ≤ sup F (u ) =
L1 {1 − exp{−l0 (t − t0 )}}. l0
t→∞
E
and
F (u ) =
exp{−l0 (s − t0 )}ds
Let k = 6, r1 = 91, r2 = −1.5, b = 1.2, e = 2.5, r10 = 0.5, r20 = 0.5 and b0 = 0.5. Calculate l = 33571.8637. We give the following estimate of the ultimate boundary
Applying Itô’s formula to (3.1), one has
−
t0
Thus,
Proof. Define the Lyapunov function
⎜ ⎜ M=⎜ ⎝
t
E[V (X ) − l] ≤ [V (X0 ) − l] exp{−l0 − (t − t0 )}.
= {X |EV (X ) ≤ l }
k r1
L1 . l0
ing estimate holds,
In particular,
⎛
(3.4)
From above inequation, one can obtain
⎟ ⎟ ⎟. ⎠
2
(3.3)
one obtains
2 + + 12 (r10
smallest (positive) eigenvalue of the positive-definite matrix
⎛
673
|Ex| ≤ 12.0809, |Ey| ≤ 183.2262, |Ez| ≤ 183.2262, −1.2262 ≤ Eu ≤ 365.2262.
(3.5)
The numerical solutions, which are stochastic processes, of stochastic Lorenz-Haken system (1.2) are obtained by Euler2 Maruyama method. The simulated time series about r(r = 91 6 x + y2 + z2 + (u − 182 )2 ) is displayed in Fig. 5. In addition, all the stochastic processes’ scopes and the ultimate boundary of the corresponding expectations are listed in Table 1. From the Table 1, we are pleased to see that the simulation results and the theoretical results of (3.5) are consistent.
674
L. Li et al. / Chaos, Solitons and Fractals 91 (2016) 670–678 Table 1 The ultimate boundary for stochastic Lorenz-Haken system. Scopes of stochastic processes
Simulated results of expectation
Theoretical estimates of expectation
0.0158 ≤ x ≤ 29.5772 −67.8171 ≤ y ≤ 66.8146 −37.1425 ≤ z ≤ 71.5711 35.6241 ≤ u ≤ 245.9096 139.4273 ≤ r ≤ 24022.1240
Ex = 8.323298 Ey = 8.341710 Ez = 1.522135 Eu = 94.742224 Er = 10115.3207
|Ex| ≤ 12.0809 |Ey| ≤ 183.2262 |Ez| ≤ 183.2262 −1.2262 ≤ Eu ≤ 365.2262 0 ≤ Er ≤ 33571.8637
r=
91 2 x 6
+ y2 + z2 + (u − 182 )2 .
Applying the Itô’s formula to (3.6), one has
dV = [2x(−kx + ky ) − 2k(−bu + xy )]dt − 2kb0 udB(t ) = −2k(V + (2k − b)u )dt + b0 (V − x2 )dB(t ) < −bV dt + b0V dB(t ).
(3.7)
Calculating the expectation, one gets
EV (t ) ≤ V (t0 ) −
t
t0
bEV (s )ds.
From above inequation, one can obtain
EV (t ) ≤ V (t0 )e−b(t−t0 ) . Thus
lim EVxu = lim E(x2 − 2ku ) ≤ 0.
t→+∞
t→+∞
This completes the proof.
Corollary 1. When 0 < b < 2k, b0 > 0, for every a ∈ R, the system (1.2) has the following two-dimensional parabolic ultimate boundary lim E(x2 − 2ku + a ) ≤ a.
t→+∞
Theorem 5. Let the parameters r1 < 0, 0 < 2k < b or r1 > 0, 0 < k < b < 2k. If b20 < 2b − 2k, the system (1.2) has the following twodimensional parabolic ultimate boundary
E[(x2 − 2k(u − 2r1 ))2 ] ≤ l, where l =
8kr12 (b2 −2kb20 ) 2b−2k−b20
.
Proof. Construct the following Lyapunov function
Vxu = (x2 − 2k(u − 2r1 ))2 .
(3.8)
Applying the Itô’s formula to (3.8), one has
dVxu = [2(x2 − 2k(u − 2r1 ))(2x(−kx + ky ) − 2k(−bu + xy )) +4k2 b20 u2 ]dt + 4kb20 (−x2 + 2k(u − 2r1 ))udB(t ) = [4k(x2 − 2k(u − 2r1 ))(−x2 + bu ) + 4k2 b20 u2 ]dt +4kb20 (−x2 + 2k(u − 2r1 ))udB(t ) = [−2k(x2 − 2k(u − 2r1 ))2 + H (x, u )]dt Fig. 4. The three-dimensional projection of exponentially attractive set of the stochastic Lorenz-Haken system with k = 6, r1 = 91, r2 = −1.5, b = 1.2, e = 2.5, r10 = 0.5, r20 = 0.5, b0 = 0.5, and initial values (x0 , y0 , z0 , u0 ) = (5, 10, −3, 70 ).
Theorem 4. When 0 < b < 2k, b0 > 0, the system (1.2) has the following two-dimensional parabolic ultimate bound
lim E(x2 − 2ku ) ≤ 0.
Proof. Construct the following Lyapunov function
(3.6)
(3.9)
where
H (x, u ) = 2k(x2 −2k(u−2r1 ))2 +4k(x2 −2k(u − 2r1 ))(−x2 + bu ) +4k2 b20 u2 . Let
t→+∞
Vxu = x2 − 2ku.
+4kb20 (−x2 + 2k(u − 2r1 ))udB(t ),
⎧ ⎪ ⎨ ∂ H = 8kx(−x2 + bu ) = 0, ∂x ∂ ⎪ ⎩ H = 4k[2k(2k − 2b + b20 )u − 4kr1 (2k − b) + bx2 ] = 0. ∂u
L. Li et al. / Chaos, Solitons and Fractals 91 (2016) 670–678
2.5
x 10
675
4
2
r
1.5
1
0.5
0 0
5
10
15
20
25 t
30
35
40
45
50
x2 + y2 + z2 + (u − 182 )2 ) of the stochastic Lorenz-Haken system with k = 6, r1 = 91, r2 = −1.5, b = 1.2, e = 2.5, r10 = 0.5, Fig. 5. The diagram of time series r (t )(r = 91 6 r20 = 0.5, b0 = 0.5, and initial values (x0 , y0 , z0 , u0 ) = (5, 10, −3, 70 ).
2
if b20 < min{ 2bk , 2b − 2k}, function H(x, u) has the global maximum value,
Then the solutions for the above equations are
⎧ ⎨x = 0,
⎧ 2 ⎨x = bu,
⎩u =
⎩u =
2r1 ( 2k − b ) , 2k − 2b + b20
4kr1 (2k − b) . (b − 2k )2 + 4kb20
The Hessian matrix of the function H(x, u) is
Hxx Hxu
Hxu Huu
=
8k(−3x2 + bu ) 8bkx
H (x, u ) ≤
8bkx 8k2 (2k − 2b + b20 ) .
lim
2r (2k−b) xε →0,u= 1 2k−2b+b2 0
H
xε xε
= 8k(−3xε + bu ), 2
H
2 3 2 2 2 2 2 xε xε Huu − (Hxε u ) = 64k (−3xε + bu )(2k − 2b + b0 ) − 64b k xε .
For ε be arbitrary, one can deduce that, when the parameters r1 < 2 0, 0 < 2k < b (or r1 > 0, 0 < k < b < 2k) and b20 < min{ 2bk , 2b − 2k},
lim xε →0,u=
2r1 (2k−b) 2k−2b+b2 0
lim xε →0,u=
2r1 (2k−b) 2k−2b+b2 0
Hxε xε = 16r1 bk
2k − b < 0, 2k − 2b + b20
− (H )2 = 128r bk3 (2k − b) > 0. Hxε xε Huu 1 xε u
That means the corresponding Hessian matrix is strictly negative. 4kr1 (2k−b) Case 2. x2 = bu, u = 2. 2 (b−2k ) +4kb0
In this case, when k > 0,
H − (H )2 = 8k (−3x2 + bu ) · 8k2 (2k − 2b + b2 ) − 64b2 k2 x2 Hxx uu xu 0
= −64k2 x2 [(b − 2k )2 + 2kb20 ] < 0. That means function H(x, u) couldn’t get the local extremum value in this case. Obviously, the Hessian matrix of the function H(x, u) is continuous. When the parameters r1 < 0, 0 < 2k < b (or r1 > 0, 0 < k < b < 2k), from case 1 and case 2, one can deduce that
16k2 r12 (b2 − 2kb20 ) 2b − 2k − b20
= lxu .
dVxu ≤ (−2kVxu + lxu )dt + 4kb20 (−x2 + 2k(u − 2r1 ))udB(t ). Similar to the proof of Theorem 3, one can obtain
E Vxu (x, t ) −
(2k−2b+b0 )
Notice that X = (0, y(t ), z(t ), u(t )) is not the solution of system (1.2). One can’t rudely set x = 0. Here take the limit x → 0. For every ε > 0, let −ε xε ε (xε = 0 ). One has
(xε ,u )
∈R2
Then
!
Next, one discusses the above H(x, u). 2r1 (2k−b) Case 1. x = 0, u = 2 .
sup H (xε , u ) =
lxu 2k
"
Thus, lim EVxu ≤ t→+∞
!
"
≤ Vxu (x(t0 ), u(t0 )) − lxu . 2k
lxu exp(−2k(t − t0 )). 2k
That is
E[(x2 − 2k(u − 2r1 ))2 ] ≤ l. Thus we complete the proof of the Theorem 5. √ Using the properties of expectations Ex ≤ Ex2 , one gets the following conclusion. Corollary 2. Let the parameters r1 < 0, 0 < 2k < b or r1 > 0, 0 < k < b < 2k. If b20 < 2b − 2k, the system (1.2) has the following twodimensional parabolic ultimate bound
E [ x 2 − 2 k ( u − 2 r 1 )] ≤ where l =
8kr12 (b2 −2kb20 ) 2b−2k−b20
#
l,
.
Let k = 6, r1 = 91, r2 = −1.5, b = 8, e = 2.5, r10 = 0.5, r20 = 0.5 and b0 = 0.5. One can calculate l = 6465804.80 and give the following estimate of the ultimate bound
$ % = X |E[x2 − 12(u − 182 )]2 ≤ 6465804.80 ,
which is the two-dimensional parabolic ultimate bound with respect to x − u of the stochastic Lorenz-Haken system. The simulated time series is displayed in Fig. 6. And the simulated results show that E[x2 − 12(u − 182 )]2 = 3353891.87 ≤ 6465800, which is consistent with the theoretical analysis result. In addition, according to Corollary 2, one can obtain
E[x2 − 12(u − 182 )] ≤
√ 6465804.80 = 2542.80.
(3.10)
676
L. Li et al. / Chaos, Solitons and Fractals 91 (2016) 670–678
5.5
x 10
6
5 4.5
r
4 3.5 3 2.5 2 1.5 0
5
10
15
20
25 t
30
35
40
45
50
Fig. 6. The diagram of time series r (t )(r = (x2 + 12(u − 182 ))2 of the stochastic Lorenz-Haken system with k = 6, r1 = 91, r2 = −1.5, b = 8, e = 2.5, r10 = 0.5, r20 = 0.5, b0 = 0.5, and initial values (x0 , y0 , z0 , u0 ) = (5, 10, −3, 70 ).
By using the properties of expectations and Theorem 3, one has
E [ x 2 − 2 k ( u − 2 r 1 )] ≤ E ( x 2 + ( u − 2 r 1 ) 2 + k 2 ) 91 ≤ E ( x2 + y2 + z 2 + ( u − 2r1 )2 ) + k2 6 L1 ≤ + k2 , l0
V (X ) =
r1 2 x + y2 + z 2 + ( u − 2r1 )2 . k
From the proof of Theorem 4 and Itô’s formula, one has
V (X ) ≤ V (X0 ) +
t
0
[−l0V (X ) + L1 ]ds +
t 0
( l 1 V ( X ) + L 2 )d B ( s ) , (4.1)
where L1 and l0 are defined in Theorem 3. Take the same parameters as in the above, we have
E[x2 − 12(u − 182 )] ≤ 131471.32. We also have the following result according to Corollary 1,
E[x2 − 12(u − 182 )] ≤ 2184.
Proof. Let
(3.11)
The simulated results show that E[x2 − 12(u − 182 )] = 1827.18, which is consistent with the theoretical analysis result from above. We know that if the higher order moments are bounded, the lower order moments are bounded. But if the lower order moments are bounded, the higher order moments are not always bounded, even that they do not exist. Theorem 4 estimates the boundary of E(x2 − 2ku ). In Theorem 5, the boundary of E[(x2 − 2k(u − 2r1 ))2 ] is derived, which is the higher order moment ultimate boundary about x − u. To our best knowledge, the result about ultimate boundary of second-order moment is very few for the stochastic systems.
where L1 and L2 are defined in (3.3), l0 and l1 are defined in (3.4), respectively. Suppose that Z satisfies the following equation
Z ≤ V (X0 ) +
t 0
[−l0 Zs + L1 ]ds +
t
0
( l 1 Z s + L 2 )d B ( s ) .
(4.2)
Then Z is finite on any finite interval [0, T] for any T > 0. Hence, (4.1) and (4.2) yield that the process V(X) is finite on [0, T] for any 0 < T < ∞. That is to say, the random dynamical system ϕ (t, ω, x) is global to the forward. r2
2 + Theorem 7. Suppose that the parameters k > 0, r1 > 42 + 12 (r10 2 ) and 2b > b2 . Then the stochastic Lorenz-Haken system generate r20 0 by (1.2) possesses the unique parameter dependent tempered random attractor A with domain of attraction D (A ) containing the universe of sets Cl (U ), generated by
U = {(D(ω ))ω∈ |D(ω ) ⊂ R4 is a tempered random set}. Proof. Let
r1 2 x + y2 + z 2 + ( u − 2r1 )2 . k
4. Random attractors and bifurcation
V (X ) =
In this section, we consider the random attractor generating by the system (1.2) and bifurcation phenomenon from the dynamical system viewpoint. The study of the existence of random attractor and stochastic bifurcation behavior for other stochastic systems can be found in [18–25].
From the proof of Theorem 4 and Itô’s formula, one obtains
r2
2 + Theorem 6. Suppose that the parameters k > 0, r1 > 42 + 12 (r10 2 2 r20 ) and 2b > b0 . Then the stochastic Lorenz-Haken system (1.2) generates a smooth RDS ϕ which is global to forward. That is Dt = R4 for all t > 0. Moreover, ϕ (t, θ t ω, x) is Ft0 -measurable.
V (X ) ≤ V (X0 ) +
0
t
[−l0V (X ) + L1 ]ds +
t 0
( l 1 V ( X ) + L 2 )d B ( s ) , (4.3)
where L1 and L2 are defined in (3.3), l0 and l1 are defined in (3.4), respectively. Denote ϕ (t, ω, (x, y, z)) and ψ (t, ω, V(X)) the random dynamical system which are generated by (1.2) and
Zt = V (X0 ) +
t 0
[−l0 Zs + L1 ]ds +
t 0
( l 1 Z s + L 2 )d B ( s ) ,
(4.4)
L. Li et al. / Chaos, Solitons and Fractals 91 (2016) 670–678
"
respectively. The random dynamical system generated by (4.4) is
ψ (t, ω )x = exp{−l0t + l1 B(t )} t × x + L1 exp{l0 s − l1 B(s )}d(s ) .
+czk(y−x ) + u(−bu+xy ) +
0
ξ ( ω ) = L1
−∞
− 1−
U = {I(ω ) ⊂ R+ is a tempered random set}.
Next, we study bifurcation phenomenon of the system (1.2). Some the basic conceptions and notions can be found in the literature [14,16,26] and the related marks are consistent with the above references. Proposition 1. Suppose that the conditions of Theorem 5 are satisfied. Denote A(ω) the random attractor of the random dynamical system ϕ generated by stochastic Lorenz-Haken Eq. (1.2). Then (i) The random attractor A supports all invariant measures; (ii) The unstable set of any invariant measure μ is contained in + random attractor, that is, Mμ ( ω ) ⊂ A ( ω ); (iii) There exists an invariant Markov measure μ supported by 0 -measurable. ∂ A(ω), that is, μω is F−∞ Proof. Similar to the proof in Proposition 7.5 of [27]. 2. Let
k|r2 | 2 (1+k ) , 2b − b0 }/ξ ,
l2 = min{
2|r2 | (e−|r2 | ) 1+k
2 + r 2 ), 2− − 2r1 − (r10 20
where ξ is defined in Theorem 2. Suppose that the
parameters k > 0,
k) − 2(1+ k
< r2 < 0,
|r2 |(e−|r2 | ) 1+k
>
1 2 2 (r10
+
2 ) r20
+ r1
and 2b > b20 . Then the top Lyapunov exponents L3 of the linear stochastic equation
⎧ dx = (−kx + ky )dt, ⎪ ⎪ ⎪ ⎨dy = (r x − y − ez )dt + r xdB(t ), 1 10 ⎪ d z = r x + ey − z d t + r ( ) 2 20 xdB (t ), ⎪ ⎪ ⎩ du = −budt + b0 udB(t ),
is bounded by L3 ≤ −l2 . Proof. Consider V (X ) =
1 2 2 2 2 2 (ax + y + z + 2cxz + u ), where a = −(1+k )r1 −r2 e r2 , c = 1+k . When the parameters r1 > 0, k > 0, r2 < k(1+k ) |r |(e−|r | ) 2 + r 2 ) + r , the function V is positive-definite. 0, 2 1+k 2 > 12 (r10 1 20
Then,
dV (X ) =
! axk(y − x ) + y(r1 x − y − ez ) + z(r2 x + ey − z ) +cx(r2 x + ey − z )
where
k|c | 2 z − 2
b−
b20 2
1−
"
k|c | 2 y 2
u2 dt
$
&
'
%
2 2 l2 = min 2ka − 2cr2 − r10 + r20 , 2 − k|c|, 2b − b20 /ξ
= min
That is, for any I ∈ U , the random variable η (ω ) = supx∈I(ω ) x + satisfies limt→∞ log [η (θt ω )/t] = 0. Then, for any ε > 0, − ε t limt→∞ e η (θt ω ) = 0. It is obvious that U is closed under inclusion and × {x} ∈ U for all x ∈ R+ . The rest of the proof is the same as that for Theorem 5.4 from the references Schenk-Hoppé [27], we omit it here.
1 2 2 (r + r20 ) x2 − 2 10
1 + [(|r10 | + |r20 | )x2 + |r10 |y2 + |r20 |z2 + |b0 |u2 ]dB(t ) 2 ≤ −l2V (X )dt + l3V (X )dB(t ) (4.7)
exp{l0 s − l1W (s )}d(s ) > 0.
It is easy to prove that ψ (t, θ−t θ x(θ−t ω )) → ξ (ω ) as t → ∞ for initial value x(ω ) ∈ R+ such that exp(−l0 )x(θ−t ω ) → 0 for some l0 > 0. Hence, one may definite the universe of set
Proposition
− ka − cr2 −
(4.6)
Thus ψ (t, ω) has the unique invariant measure which is a Dirac measure supported by 0
!
≤
From (4.3), one can obtain
1 2 2 ((r + r20 )x2 + b20 u2 ) dt 2 10
+[|r10 |xy + |r20 |xz + |b0 |u2 ]dB(t )
(4.5)
V (ϕ (t, ω, (x, y, z ))) ≤ ψ (t, ω )V (X ).
677
&2 ' k|r2 | 2|r2 |(e−| r2 | ) 2 −2r1 − r10 + r20 , 2− , 2b − b20 /ξ , 1+k (1+k )
l3 = max {|r10 | + |r20 |, |b0 |}. From (4.7) and Doléans-Dade’s exponent formula, one obtains
V (X ) ≤ exp{−l2 t + l3 B(t )}. One has
log V (X ) t −l2 t + l3 B(t ) ≤ lim t→∞ t = −l2 .
L3 = lim
t→∞
k) Theorem 8. Suppose that the parameters k > 0, − 2(1+ < r2 < k
|r |(e−|r | )
r2
2 + r 2 ) + r , r > 2 + 1 (r 2 + r 2 ) and 2b > b2 . 0, 2 1+k 2 > 12 (r10 1 1 4 2 10 20 20 0 Then the random dynamical system generate by the stochastic LorenzHaken system (1.2) possesses the unique tempered random attractors A(ω ) = {0} with domain of attraction D(A) containing the universe of Cl (U ) given by
U = {D(ω )ω∈ |D(ω ) ⊂ R3 is a tempered random set}, where Cl (U ) is the completion universe of U under inclusion. In particular, for any x ∈ R3 , the solutions ϕ (t, θ−t ω )x and ϕ (t, ω)x tend to zero exponentially fast as t → ∞. Proof. Proposition 2 and l3 > 0 yield that the random dynamical system generated (1.2) is stability. Define the Lyapunov function on R4
V (X ) =
r1 2 x + y2 + z 2 + ( u − 2r1 )2 , k
then V(X) ≥ 0 and V (X ) = 0 if and only x = y = z = 0, u = 2r1 . Applying Itô’s formula to the above equation, one has
dV (X ) ≤ [−l0V (X ) + L1 ]dt + (l1V (X ) + L2 )dB(t ), where L1 and L2 are defined in (3.3), l0 and l1 are defined in (3.4), respectively. Corollary 5.6.3. of [24] yields that the random dynamical system generated by the stochastic differential equation
dZt = [−l0 Zt + L1 ]dt + (l1 Zt + L2 )dB(t ), has the unique invariant measure δ 0 which is the Dirac measure at 0. As the proof in Theorem 6, one has that A(ω ) = {0} = V −1 ({0} ) is the unique random attractor attracting any set form U. Hence, it means that, for any X ∈ R3 , ϕ (t, θ−t ω )x and ϕ (t, ω)x tend to zero as t → ∞ exponentially, thus complete the proof of the theorem.
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L. Li et al. / Chaos, Solitons and Fractals 91 (2016) 670–678
5. Conclusions
References
The interest in the stability of lasers and related systems has been ever increasing, especially since the discovery of the fundamental analogy between single-mode laser equations and the wellstudied Lorenz equations. In laser physics and in particular in laser dynamics the Lorenz-Haken model is generally recognized as the most fundamental model from the conceptual point of view. On the one hand, when the laser is installed in the satellite, airplane, submarine and other high speed running body, it will be affected by the mechanical noise, the barometric pressure and temperature or some other random factors of external environment. On the other hand, a complete knowledge about the stability of stationary states as well as oscillations, especially a systematic understanding of the influence of the parameter variations on the nature of the dynamics, is still lacking. A large amount of numerica1 evidence is still needed before a satisfactory answer can be given. This paper is devoted to investigating the stochastic LorenzHaken system. The globally dynamics, such as asymptotic behavior, exponential attractive set and random attractor are rigorously studied. Firstly, the interesting thing is that all of solutions of the system (1.2) can tend to zero and never go through the hyper-plane x = 0 as the large time. These consist with the results of the deterministic Lorenz-Haken system. Secondly, we estimate the globally exponential attractive set and a four-dimensional ellipsoidal ultimate bound. The two-dimensional parabolic ultimate boundary with respect to x − u also is established. And numerical results to estimate the ultimate bound are also presented for verification. Finally, random attractors and the bifurcation phenomenon for the system is analyzed. It is hoped that the investigation of this paper can help understanding the rich dynamic of the stochastic LorenzHaken system and offer some enlightenments for the study of the detuned single-mode laser.
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Acknowledgments The authors would like to express their sincere thanks to the editor and anonymous referees for their careful and pertinent suggestions for improving the presentation of the work. This work was supported by the National Natural Science Foundation of China (Grant No. 11561069), the Guangxi Natural Science Foundation of China (Grant No. 2015GXNSFAA139002), the Scientific Research Foundation of the Higher Education Institutions of Guangxi of China (Grant No. YB2014309, KY2016YB364), the Opening Foundation of Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing (Grant No. 2015CSOBDP0202), the University Key Teachers Training Program from Guangxi Higher Education Institutions of China (Document No. [2014]39) and the special Foundation of Excellent Talents in Yulin Normal University (Grant NO.G2014003).