Dynamical behaviors of a generalized Lorenz family

In this paper, the ultimate bound set and globally exponentially attractive set of a generalized Lorenz system are studied according to Lyapunov stability theory and optimization theory. The method of constructing Lyapunov-like functions applied to the former Lorenz-type systems (see, e.g. Lorenz system, Rossler system, Chua system) isn't applicable to this generalized Lorenz system. We overcome this difficulty by adding a cross term to the Lyapunov-like functions that used for the Lorenz system to study this generalized Lorenz system. The authors in [D. Li, J. Lu, X. Wu, G. Chen, Estimating the ultimate bound and positively invariant set for the Lorenz system and a unified chaotic system, Journal of Mathematical Analysis and Applications 323 (2006) 844-853] obtained the ultimate bound set of this generalized Lorenz system but only for some cases with $0 ≤ α

Ultimate boundedness of chaotic dynamical systems is one of the fundamental concepts in dynamical systems, which plays an important role in investigating the stability of the equilibrium, estimating the Lyapunov dimension of attractors and the Hausdorff dimension of attractors, the existence of periodic solutions, chaos control, chaos synchronization [2,5,11]. Ultimate boundedness of the Lorenz system has been investigated by Leonov et al. in a series of articles [7,12]. Since then other studies have developed ultimate bounds of similar chaotic dynamical systems [22,24,[26][27][28][29][30]. However, the approach taken in each is only suitable for that particular system. It is very difficult to propose a universal approach to estimate ultimate bounds for an arbitrary chaotic dynamical system. To this end, we will study ultimate bounds of the new generalized Lorenz system in this paper.
The former Lorenz-type equations [22,24,26,[28][29][30] that we are searching for a global bounded region have a common characteristic: the elements of the main diagonal of the matrix A are all negative [22,24,26,[28][29][30], where the matrix A is the Jacobian matrix df dx of a continuous-time dynamical system defined by dx dt = f (x) , x ∈ R 3 , evaluated at the origin (0, 0, 0). However, there are positive numbers in the elements of main diagonal of matrix C, where matrix C is the Jacobian matrix of the generalized Lorenz system evaluated at the origin (0, 0, 0). The origin (0, 0, 0) is an equilibrium point of this generalized Lorenz system. The method of constructing Lyapunov-like functions applied to the former three-dimensional Lorenz-type dynamical systems [22,24,26,[28][29][30] isn't applicable to this generalized Lorenz system. We overcome this difficulty by adding a cross term xy to the Lyapunov-like function study this generalized Lorenz system. Using these Lyapunov-like functions, the present paper obtains the globally attractive sets for this generalized Lorenz system for several parameter ranges. The results obtained in this paper contain the existing results in [22] as special cases.
The rest of the paper is organized as follows. Section 2 gives the mathematical model of the generalized Lorenz system and the main results of this paper. Conclusions are drawn in Sect.3.
2. Dynamical systems model and main results. The unified system is described by [14,22]: where α ∈ R is the system parameter. The system (1) is considered as a transition system between the Lorenz system [6] and the Chen system [3]. The system (1) reduces to the original Lorenz system with α = 0, and system (1) is the original Chen system with α = 1 [22]. For simplicity, let us denote 10+25α = a α , Some dynamics of the generalized Lorenz systems (2) were studied in [14,22]. In the following, we will discuss the ultimate bound set and global attractive set of the generalized Lorenz systems (2).
Consider the system 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 (3) satisfying X (t 0 , t 0 , X 0 ) = X 0 which for simplicity is denoted as X (t) . Assume Ω ⊂ R n is a compact set. Define the distance between a point X (t, t 0 , X 0 ) and the set Ω as We will give the following definitions and introduce Lemma 1 which will be used in Theorem 1.
Definition 1. ( [15,16,22,23,25,26]). Suppose that there exists a compact set Q ⊆ R n such that  If, for any X 0 ∈ Q and all t ≥ t 0 , such that X (t, t 0 , X 0 ) ⊆ Q, then the set Q is called a positive invariant set of system (3).
In the following, we will study the ultimate bound set and global attractive sets of the generalized Lorenz system (2). We can get Theorem 1 for system (2) according to Lemma 1.
Theorem 1. Suppose that ∀λ > 0, ∀m > 0 and α ∈ −2 5 , 1 29 . Then the following set is an ultimate bound set and positively invariant set of generalized Lorenz system (2), where . Proof. Define the following generalized positively definite and radially unbounded Lyapunov-like function Differentiating the above Lyapunov-like function V λ,m (x, y, z) in (6) with respect to time t along the trajectory of system (2) yields Obviously, we can see that Γ 1 that defined by is an ellipsoid in Thus, the maximum of V λ,m (X) can only be reached on Γ 1 . Since the V λ,m (X) is a continuous function and Γ 1 is a bounded closed set, then the function (6) V λ,m (X) can reach its maximum value max V λ,m (X) = R 2 , (X ∈ Γ 1 ) on the surface Γ 1 defined in (7). Obviously, {(x, y, z)|V λ,m (X) ≤ maxV λ,m (X), X ∈ Γ 1 } contains the solutions of the generalized Lorenz system (2). By solving the following conditional extremum problem, one can get the maximum value of the function (6): Let us take By solving the following conditional extremum problem of V λ,m (X) in (8), one can easily get the conditional extremum problem According to Lemma 1, we can easily get the above conditional extremum problem (9) as: . This completes the proof. Remark 1. i) Let us take ∀λ > 0, ∀m > 0 in Theorem 1, then we can get a series of ultimate bounds sets and positively invariant sets of the generalized Lorenz system (2) according to Theorem 1. ii) Let us take m = 1 in Theorem 1, then we can get that is an ultimate bound set and positively invariant set of generalized Lorenz system (2), where . Although in [22], the authors construct the generalized Lyapunov-like function V (x, y, z) = λx 2 + y 2 + (z − b α − λa α ) 2 , ∀λ > 0 and prove that there exists the ultimate bound set and positively invariant set for the generalized Lorenz system (2) for 0 ≤ α < 1 29 . In particular, let us take m = 1 in Theorem 1, then we can get the conclusion that obtained in [22]. The results presented in Theorem 1 contain the existing results in [22] as special cases. iii) Let us take m = 1, λ = 1 in Theorem 1, then we can get that is an ultimate bound set and positively invariant set of generalized Lorenz system (2), where . Though Theorem 1 gives the ultimate bound set and positively invariant set of the generalized Lorenz system (2), it does not gives the global exponential attractive set of the generalized Lorenz system (2). The global exponential attractive set of the generalized Lorenz system (2) is described by the following Theorem 2.
Proof. When − 2 5 < α < 1 29 , we can get Define the following generalized positively definite and radially unbounded Lyapunovlike function , ∀λ > 0, ∀m > 0, Differentiating the above Lyapunov-like function V λ,m (X) with respect to time t along the trajectory of system (2) yields That is equivalent to say that Thus, we have which clearly shows that is a global exponential attractive set of system (2). This completes the proof. Let us introduce Lemma 2 which will be used in the following parts of this paper.
When α < 14 173 , we can get When α > − 52 149 , we can get Therefore, to summarize what has been mentioned above, we can get Lemma 2. This completes the proof.
Lemma 3. Suppose that α ∈ 1 29 , 14 173 . Then we can get the following inequality for system (2) lim Proof. Let us define Then, its derivative along the orbits of system (2) is and, dV dt When α ∈ 1 29 , 14 173 , we have the following inequality according to Lemma 2 Thus, we have dV dt For any initial value V (t 0 ) = V 0 , we have This completes the proof. Remark 2: The method to prove the inequality in Lemma 3 is using the method in [12]. As early as in 1987, G. A. Leonov et al. have given the method to prove this kind of inequalities for the Lorenz system in the excellent paper [12].
Lemma 4. Suppose α ∈ 1 29 , 14 173 , and let Then we can obtain Proof. When α ∈ 1 29 , 14 173 , we have the following inequality according to Lemma So, we have And we can also get Therefore, we can get This completes the proof.
For simplicity, let us simplify model (2) with the following reversible linear transform: x = x, y 1 = y − ηx, z = z. Then model (2) takes the form where The global attractive sets of system (2) for parameter α ∈ 1 29 , 14 173 is described by the following Theorem 3 according to the above lemmas.
Proof. Let us define where ∀λ > 0, ∀m > 0, c 2 = b α − c α η + a α η (1 − η) . According to Lemma 4, we can get Therefore, we can get Define the following generalized positively definite and radially unbounded Lyapunovlike function According to Lemma 3 and Lemma 4, we have Since lim t→+∞ x 2 − 2a α z ≤ 0, so there exists a positive constant T 0 > 0, when t > T 0 , combining with Lemma 3 and Lemma 4 we have Thus, we have By the definition, taking upper limit on both sides of the above inequality (15) as t → +∞ results in Namely, the set Φ λ,m = (x, y 1 , z)| λx 2 + my 2 is a global exponential attractive set of system (13). Thus, is a global exponential attractive set of the generalized Lorenz system (2). This completes the proof.
Remark 3. 1) Let us take ∀λ > 0, ∀m > 0, then we can get a series of global exponential attractive sets of system (2) according to Theorem 3.
This completes the proof.
3. Conclusions. In this paper, we have extended the method developed in [22], [24], [26], [28]- [30] to study the globally exponentially attractive set and positive invariant set for a more general Lorenz family. It has been shown that such a system indeed has globally exponentially attractive set and positive invariant set, and contains all the existing relative results as special cases. Exponential estimation is explicitly derived. The approach presented in this paper may be applied to study other dynamical systems in [13], [17]. The results that obtained in this paper offer theoretical support to study the Hausdorff dimension of attractors for this generalized Lorenz system. These theoretical results are also important and useful in chaos control, chaos synchronization.