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Estimating the ultimate bound and positively invariant set for the Lorenz and a unified chaotic system

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Abstract

To estimate the ultimate bound and positively invariant set for a dynamic system is an important but quite challenging task in general. In this paper, we attempt to investigate the ultimate bound and positively invariant set for two specific systems, the Lorenz system and a unified chaotic system. We derive an ellipsoidal estimate of the ultimate bound and positively invariant set for the Lorenz system, for all the positive values of its parameters a, b and c, and obtain the minimum value of volume for the ellipsoid. Comparing with the best results in the current literature [D. Li, J. Lu, X. Wu, G. Chen, Estimating the bounds for the Lorenz family of chaotic systems, Chaos Solitons Fractals 23 (2005) 529–534; X. Liao, On the global basin of attraction and positively invariant set for the Lorenz chaotic system and its application in chaos control and synchronization, Sci. China Ser. E 34 (2004) 1404–1419], our new results fill up the gap of the estimate for the cases of 0<a<1 and 0<b<2 [X. Liao, On the global basin of attraction and positively invariant set for the Lorenz chaotic system and its application in chaos control and synchronization, Sci. China Ser. E 34 (2004) 1404–1419]. Furthermore, the estimation derived here contains the results given in [D. Li, J. Lu, X. Wu, G. Chen, Estimating the bounds for the Lorenz family of chaotic systems, Chaos Solitons Fractals 23 (2005) 529–534] and [X. Liao, On the global basin of attraction and positively invariant set for the Lorenz chaotic system and its application in chaos control and synchronization, Sci. China Ser. E 34 (2004) 1404–1419] as special cases. Along the same line, we also provide estimates of cylindrical and ellipsoidal bounds for a unified chaotic system, for its parameter range , and obtain the minimum value of volume for the ellipsoid. The estimate is more accurate than and also extends the result of [D. Li, J. Lu, X. Wu, G. Chen, Estimating the bounds for the Lorenz family of chaotic systems, Chaos Solitons Fractals 23 (2005) 529–534] and [X. Liao, On the global basin of attraction and positively invariant set for the Lorenz chaotic system and its application in chaos control and synchronization, Sci. China Ser. E 34 (2004) 1404–1419].

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... 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. ...
... 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 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). ...
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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 < 1/29. The ultimate bound set and globally exponential attractive set of this generalized Lorenz system are still unknown for α ∉ [0, 1/29). Comparing with the best results in the current literature [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], our new results fill up the gap of the estimate for the case of 1/29 ≤ α < 14/173. Furthermore, the estimation derived here contains the results given 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, J. Math. Anal. Appl. 323 (2006) 844-853] as special case for the case of 0 ≤ α < 1/29.
... The control complex networks with coupling delays and delays in the dynamical nodes can be described as follows: 11 1 When the delayed dynamical network (1) achieves synchronization, namely, the states 12 ...
... conditions. If there is a nonempty subset , then the error system can be described by 11  is positive definite, the above condition is satisfied. So, it includes many well-known systems, such as the Lorenz system, Chen system, Lü system, recurrent neural networks, Chua's circuit, and so on. ...
... Then, combining (10})-(11) and Hypothesis 1, we have By Schur complement and 0   , we can get the follow matrix inequality. ...
... If there is a bounded domain such that all trajectories of the system eventually enter into it, then it is much easier to understand the global behavior of the system, and the numerical simulations and theoretical analysis can be applied with confidence. Further, the existence of global boundedness implies that all equilibrium solutions, periodic solutions, and chaotic solutions can only be found in this bounded region, which is useful for some engineering applications such as chaos control and chaos synchronization [15,18,19]. There are some results on the estimates of the ultimate bound of the complex Lorenz system. ...
... In the Lorenz-like system, the sign of the coefficient is negative, which makes the construction of a Lyapunov function much easier, whereas the sign of the coefficient for the Chen system is positive, which is a technically difficult problem for constructing a Lyapunov function. For example, for the real system (3) with 0 ≤ α < 1 29 , the sign of the coefficient of the single variable y is negative, the ultimate bounds were obtained in [18]; for α > 1 29 , the sign of the coefficient of the single variable y is positive, the ultimate bounds for 1 29 ≤ α < 14 173 were obtained in [35] (system (3) is called generalized Lorenz system in [35]); for large positive α, some ultimate bounds were obtained in [3]. ...
... Here, one difference between real and complex systems is that one needs to consider the effects of the real and imaginary parts of the parameters, which cause new problems. For the complex system, results corresponding to the work for the real system in [18,35] are obtained in Lemma 3.3 and Theorem 3.4, and results corresponding to the work for the real system in [3] are obtained in Theorem 3.7. ...
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... Modeling with differential equations is a versatile tool in all natural sciences with a traditional history [2,3,4,8,9,10,11,12,13,14,15,16,17,18,19,21,22,23,24,25]. Especially in pharmacokinetics or system biology, one often uses the tool of compartmental models for modeling different types of systems [7]. ...
... holds for all z ∈ R d , then the solution of the aforementioned initial value problem (14) exists for all time t ∈ R and moreover, it holds ...
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... For α = 0 one obtains the Lorenz system, for α = 1 the Chen system. The system is chaotic for α ∈ [0, 1], see [2]. ...
... Then, the Lyapunov-like function (4) does not depend on the coordinate x 1 and the sublevel set (2) is a cylinder. Next, we want to determine the constant γ of the sublevel set (2). For the reason, we consider (5) again and omit the quantifier of γ. ...
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... t 0 , then X is said to be a positive invariant set for the system (2.1) [Ref. 16, ...
... By Ref. 16 [Theorem 1], the positive invariant set is contained in the set p 2 1 þ p 2 2 þ p 3 À 38 ð Þ 2 < 1600. Consequently, the attractor is contained in the region f p 1 ; p 2 ; p 3 ð Þ : jp 1 j < 40; jp 2 j < 40; À2 < p 3 < 80g: ...
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... Consider Chen system as node dynamics in drive layer, Lorenz system as the node dynamics in response layer and auxiliary layer. It is obvious that Assumption 2.1 is satisfied [27]. ...
... In general, quadratic chaotic systems are frequently used in synchronization schemes, because the ultimate bound region can be approximated simpler. Basically, the ultimate bound region of a quadratic chaotic system can be estimated by solving an optimization problem [28][29][30][31][32][33]. Memristorbased chaotic systems are a class of quadratic systems. ...
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... The unified chaotic system [44] can be described by ...
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... when a = 10, b = 8/3, c = 28, the Lorenz system shows the chaotic behavior. Due to the bounded chaotic attractors in a certain region [32,33], the assumption (7) is evidently satisfied in the Lorenz system. For brevity, we consider a complex dynamical network consisting of six identical nodes in order to validate the above theoretical results. ...
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... Since s(t) = [s 1 , s 2 , s 3 ] is a chaotic orbit and locates in a bounded region [7], i.e., |s 1 (t)|, |s 2 (t)| ≤ M 1 = 28.918, and |s 3 (t)| ≤ M 2 = 56.918, for all t, by some ...
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... Definition 3 [Li et al., 2006;Wang et al., 2008a]. Suppose there is a compact set Ω ⊂ R n satisfying lim t→∞ ρ(X(t), Ω) = 0, such that X 0 ∈ R n /Ω. ...
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... Ever since the Lorenz system was put forward, its ultimate bound has been investigated by Leonov et al. [6]. Li et al. [8] extended the results in the paper [7] and estimated the ultimate bounds for the Lorenz system family. Yu and Liao [36] estimated the ultimate bound and positively invariant set for a general chaotic system, which does not belong to the known Lorenz system or the Chen system, or the Lorenz family. ...
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... We have applied known approaches to find proofs of a bounded attracting set [Swinnerton-Dyer, 2001;Pogromsky et al., 2003;Li et al., 2006;Yu & Liao, 2008;Yu et al., 2009;Wang et al., 2011;Zhang et al., 2014;Wang et al., 2014;Zhou et al., 2017]; however, we have not been successful. Therefore, proofs of a bounded attracting set Table 1. ...
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... Chaotic systems are ultimately bounded [ 31,22,32,33] and the phase portraits of the systems will be ultimately trapped in some compact sets. When the parameter values are taken as a = 0.9, b = 0.2, c = 1.2 and d = 0.5, non-linear financial system model (2) exhibits chaotic behavior. ...
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... that is, if for any ε > 0, there exists T > t 0 such that ϕ(t, t 0 , x 0 ) ∈ ε for any t ≥ T , then the set (t) is named as an ultimate bound for system (1). If ϕ(t, t 0 , x 0 ) ∈ (t) for any x 0 ∈ (t 0 ), then (t) is called as positive invariant set for system (1) [28]. ...
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... Four-dimensional ellipsoidal ultimate bound and two-dimensional parabolic bound of Lorenz Haken system discussed in [7]. Ultimate bound and positively invariant set for the Lorenz system and the unified chaotic systems was studied in [6]. The discussion on ellipsoidal ultimate bound for unified chaotic system and two dimensional bound for the Chen system, Lu system, and unified system can be found in [5]. ...
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This paper introduces a unified chaotic system that contains the Lorenz and the Chen systems as two dual systems at the two extremes of its parameter spectrum. The new system represents the continued transition from the Lorenz to the Chen system and is chaotic over the entire spectrum of the key system parameter. Dynamical behaviors of the unified system are investigated in somewhat detail.
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The coining of a chaotic attractor in a simple three-dimensional autonomous system was reported. The chaotic attractor represented the transition between the Lorenz attractor and the Chen attractor, thereby connecting the two forms of attractors. The system was not diffeomorphic with the Lorenz and the Chen's systems as the eigenvalue structures of the corresponding equlibrium points were not equivalent.
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Attractor localization of the Lorenz system
  • Leonov