New results of the ultimate bound on the trajectories of the family of the Lorenz systems

. In this paper, the global exponential attractive sets of a class of continuous-time dynamical systems deﬁned by ˙ x = f ( x ) , x ∈ R 3 , are studied. The elements of main diagonal of matrix A are both negative numbers and zero, where matrix A is the Jacobian matrix dfdx of a continuous-time dynamical system deﬁned by ˙ x = f ( x ) , x ∈ R 3 , evaluated at the origin x 0 = (0 , 0 , 0) . The former equations [1-6] that we are searching for a global bounded region have a common characteristic: The elements of main diagonal of matrix A are all negative, where matrix A is the Jacobian matrix dfdx of a continuous-time dynamical system deﬁned by ˙ x = f ( x ) , x ∈ R n , evaluated at the origin x 0 = (0 , 0 , ··· , 0) 1 × n . For the reason that the elements of main diagonal of matrix A are both negative numbers and zero for this class of dynamical systems , the method for constructing the Lyapunov functions that applied to the former dynamical systems does not work for this class of dynamical systems. We overcome this diﬃculty by adding a cross term xy to the Lyapunov functions of this class of dynamical systems and get a perfect result through many integral inequalities and the generalized Lyapunov functions.


1.
Introduction. Ultimate boundedness of a dynamical system is very important for us to study the qualitative behavior of the given dynamical system. If we can show that a continuous-time dynamical system has a global attractive set, then we know that the system cannot possess hidden attractors outside the global attractive set. This is very important for engineering applications [1][2][3][4][5][6][7][8], since it is very difficult to predict the existence of hidden attractors for the given dynamical system. The global attractive set also plays an important role in estimating the Lyapunov dimension of the chaotic attractors, the Hausdorff dimension of the chaotic attractor, chaos control, chaos synchronization, and many other applications [9][10][11]. Due to the significance of scientific and engineering background of the famous Lorenz system, the Russian scholar G.A. Leonov studied the global boundedness of the Lorenz system and obtained many important results [5][6]. Then Liao et al. obtained a new global attractive set of the Lorenz system by constructing a family of generalized Lyapunov functions, and applied their results to study chaos control and chaos synchronization of the Lorenz system [1]. Furthermore, Zhang et al. studied the boundedness for a new continuous-time dynamical system and obtained some useful results [4]. In searching for a global bounded region, one generally would like to choose a Lyapunov function and apply the Lyapunov stability criterion. However, note that the former equations [1][2][3][4][5][6] that we are searching for a global bounded region have a common characteristic: the elements of main diagonal of matrix A are all negative, where matrix A is the Jacobian matrix df dx of a continuous-time dynamical system defined byẋ = f (x) , x ∈ R n , evaluated at the origin x 0 = (0, 0, · · · , 0) 1×n . However, for the reason that the elements of main diagonal of matrix A are both negative numbers and zero, where matrix A is the Jacobian matrix of the continuous-time dynamical systems (1)- (4), evaluated at the origin x 0 = (0, 0, 0) . Therefore, the method for constructing the Lyapunov functions that applied to the former equations does not work for this class of dynamical systems (1)-(4). We overcome this difficulty by adding a cross term xy to the Lyapunov functions of this class of dynamical systems(1)- (4).
Despite the fact that many qualitative and quantitative results on dynamical systems (1)-(4) have been obtained, there is a fundamental question that has not been completely answered so far: is there a global trapping region where the system attractor exists? In this paper, we will investigate the global exponential attractive set for dynamical systems (1)-(4). By constructing a suitable Lyapunov function and through the proper integral inequalities, we have obtained the global exponential attractive sets for dynamical systems (1)-(4).

2.
Bounds for solutions of the family of Lorenz systems. It is well known that all trajectories of the Lorenz system are ultimately bounded for arbitrary positive parameters σ, r, b, that is to say, there exists a trapping region R (σ, r, b) ⊂ R 3 which ultimately bounds all trajectories. Many expressions on the boundedness of its trajectories exist (see, e.g., [1][2][5][6]).
The former equations [1][2][3][4][5][6] that we are searching for a global bounded region have a common characteristic: the elements of main diagonal of matrix A are all negative, where matrix A is the Jacobian matrix df dx of a continuous-time dynamical system defined byẋ = f (x) , x ∈ R n , evaluated at the origin x 0 = (0, 0, · · · , 0) 1×n . However, for the reason that the elements of main diagonal of matrix A are both negative numbers and zero for systems (1)-(4), where matrix A is the Jacobian matrix df dx of the dynamical systems (1)-(4), evaluated at the origin x 0 = (0, 0, 0). It seems that the method for constructing the Lyapunov functions that applied to the former dynamical systems does not work for this class of dynamical systems (1)-(4). We overcome this difficulty by adding a cross term xy to the Lyapunov functions of the dynamical systems (1)-(4).
In the following, we will discuss the global exponential attractive sets of dynamical systems (1)-(4). Firstly, we will give the following definition and introduce Lemma 2.2. Consider the systemẊ where X = (x 1 , x 2 , . . . , x n ) ∈ R n , f : R n → R n , t 0 ≥ 0 is the initial time, X 0 ∈ R n is a initial value and X (t, t 0 , X 0 ) is a solution to the system (5) satisfying X (t 0 , t 0 , X 0 ) = X 0 .
Lemma 2.2. When 2p ≥ s > 0, system (1) has the following estimate about x − z defined by Then, its derivative along the orbits of system (1) iṡ When 2p ≥ s > 0, we haveV + sV ≤ 0. For any initial value V (t 0 ) = V 0 , according to the comparison theorem, we have This completes the proof. (ii) Let p = a, q = c − a, k = a, h = 1, s = b, then we have the conclusions that when 2a > b > 0, system (3) has the estimate about x − z defined by (iii) Let p = a, q = ab, k = a, h = 1, s = c, then we have the conclusions that when 2a > c > 0, system (4) has the following estimate about x − z defined by In the following, we will study the global exponential attractive set of system (1) for the case of 2p > s > 0, q > 0, k > 0, h > 0. Before going into details, let us simplify model (1) with the following reversible transform: where a 1 , p, c 1 , k, h, s are positive parameters of system (6) and 0 < ∀n < s hk > 0. When 2p > s > 0, we have the following theorem.
When 2p > s > 0, we can get the following inequality for system (1), given by Especially, the set is a global exponential attractive set of the system (1).
Proof. Let us denote Let us take the generalized positive definite and radially unbounded Lyapunov function V m (X) as Computing the derivative of V m (X) along the trajectory of system (6), we havė By comparison theorem and integrating both sides of formula (7) yields By the definition, taking limit on both sides of the above inequality (8) Namely, the set is a global exponential attractive set of system (1). This completes the proof.
2) Let p = a, q = c − a, k = a, h = 1, s = b, then we get the following conclusions for system (3): When 2a > b > 0, we can get the inequality for the system (3), given by is a global exponential attractive set of system (3).
3) Let p = a, q = ab, k = a, h = 1, s = c, then we get the following conclusions for system(4): When 2a > c > 0, we can get the inequality for system (4), given by is a global exponential attractive set of the system (4).

Theorem 2.4. Let us denote
If 2p > s > 0, then we can get the inequality for system (1), given by

FUCHEN ZHANG, CHUNLAI MU, SHOUMING ZHOU AND PAN ZHENG
Especially, the set is a global exponential attractive set of the system (1).
Proof. Let us denote According tok > 0, h > 0, 1 > s 3p > n > 0, then we can obtain Let us take the generalized positive definite and radially unbounded Lyapunov function V 2 (X) as When V 2 (X) > L 0 , V 2 (X 0 ) > L 0 , Computing the derivative of V 2 (X) along the trajectory of system (6), we havė By comparison theorem and integrating both sides of formula (10) yields By the definition, taking limit on both sides of the above inequality (11) as t → +∞ results in lim Namely, the set is a global exponential attractive set of the system (1). This completes the proof.
Remark 4. We can get a series estimates of global exponential attractive sets of dynamical systems (2)-(4) according to Theorem 2.4. 1) Let p = a, q = b, k = 1, h = 1, s = c, then we have the following conclusions for dynamical system (2): When 2a > c > 0, we can get the inequality for system (2), given by > 0. Especially, the set is a global exponential attractive set of system (2).
2) Let p = a, q = c − a, k = a, h = 1, s = b, then we have the following conclusions for system (3): If 2a > b > 0, then we can get the inequality for system (3), given by > 0. Especially, the set is a global exponential attractive set of system (3).
3) Let us take p = a, q = ab, k = a, h = 1, s = c, then we have the following conclusions for system (4): If 2a > c > 0, then we can get the inequality for system (4), given by > 0. Especially, the set is a global exponential attractive set of system (4).
Theorem 2.5. Let us denote

FUCHEN ZHANG, CHUNLAI MU, SHOUMING ZHOU AND PAN ZHENG
If 2p > s > 0, then we can get the inequality for system (1), given by Especially, the set is a global exponential attractive set of the system (1).
Especially, let us take p = 10, q = 40, k = 16, h = 1, s = 2.5 for system (1) and we can conclude from the above formula that Proof. Let us take a special case for n = n 0 = s 4p , then 0 < n 0 < s 3p , so n 0 satisfies the conditions of Theorem 2.3 and Theorem 2.4. If 2p > s > 0, then we can obtain 0 < n 0 < 1 2 . According to Theorem 2.4, we can get According to Theorem 2.4, we can also get y 2 1 ≤ 2L1 h . Let us take V 3 (X) as (1−2n0)h , Computing the derivative of V 3 (X) along the trajectory of system (6), we havė By comparison theorem and integrating both sides of formula (13) yields By the definition, taking limit on both sides of the above inequality as t → +∞ results in This completes the proof.
Remark 5. We can get a series estimates of global exponential attractive sets for dynamical systems (2)-(4) according to Theorem 2.5. 1) Let us take p = a, q = b, k = 1, h = 1, s = c, then we have the following conclusions for system (2): If 2a > c > 0, then we can get the inequality for system (2), given by where n (2) 0 a c−3an Especially, the set is a global exponential attractive set of the system (2).
2) Let us take p = a, q = c − a, k = a, h = 1, s = b, then we have the following conclusions for dynamical system (3): If 2a > b > 0, then we can get the inequality for system (3), given by Especially, the set is a global exponential attractive set of system (3).
3) Let us take p = a, q = ab, k = a, h = 1, s = c, then we have the following conclusions for dynamical system (4): If 2a > c > 0, then we can get the inequality for system (4), given by Especially, the set is a global exponential attractive set of system (4).

Conclusions.
To estimate the boundedness of a dynamical system is a challenging but an interesting work in general. In this paper, we have investigated the global exponential attractive sets for a class of continuous-time dynamical systems defined byẋ = f (x) , x ∈ R 3 , with the elements of main diagonal of matrix A are both negative numbers and zero, where matrix A is the Jacobian matrix df dx of a continuous-time dynamical system defined byẋ = f (x) , x ∈ R 3 , evaluated at the origin x 0 = (0, 0, 0) . The global exponential attractive sets for dynamical systems (1)-(4) are obtained in this paper by many proper inequalities and the generalized Lyapunov functions. The global exponential attractive sets of dynamical systems (1)-(4) obtained in this paper also offer theoretical support for chaotic control and chaotic synchronization of systems (1)-(4). The new method for constructing the generalized Lyapunov function presented in this paper can be extended to consider other dynamical systems. At present, to study the global exponential attractive sets of many dynamical systems is still a difficult mathematical problem. How to get the global exponential attractive sets for these systems is still an unsolved question. So, it is very interesting for us to explore the global exponential attractive sets of these dynamical systems.