(ProQuest: ... denotes non-US-ASCII text omitted.)
Xinquan Zhao 1 and Feng Jiang 1 and Zhigang Zhang 2 and Junhao Hu 3
Recommended by Xinzhi Liu
1, School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan 430073, China
2, Department of Statistics and Applied Mathematics, Hubei University of Economics, Wuhan 430205, China
3, School of Mathematics and Statistics, South-Central University for Nationalities, Wuhan 430074, China
Received 31 October 2012; Accepted 25 December 2012
This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
Since Lorenz discovered the well-known Lorenz chaotic system, many other chaotic systems have been found, including the well-known Rössler system and Chua's circuit, which serve as models of the study of chaos [ 1- 12].
The Lorenz system plays an important role in the study of nonlinear science and chaotic dynamics [ 13- 18]. We know that it is extremely difficult to obtain the information of chaotic attractor directly from system. Most of the results in the literature are based on computer simulations. When calculating the Lyapunov exponents of the system, one needs to assume that the system is bounded in order to conclude chaos. Therefore, the study of the globally attractive set of the Lorenz system is not only theoretically significant but also practically important. Moreover, Liao et al. [ 19, 20] gave globally exponentially attractive set and positive invariant set for the classical Lorenz system and the generalized system by constructive proofs. In addition, Yu et al. [ 21] studied the problem of invariant set of systems, which was considered as a more generalized Lorenz system.
In this paper, we consider the following three-dimensional autonomous systems with cross-product nonlinearities: [figure omitted; refer to PDF] where x T = ( x 1 , x 2 , x 3 ) and [figure omitted; refer to PDF] with a ij , b ijk , c i ∈R , i ,j ,k =1,2 ,3 . This second-order dynamic system may be regarded as the most general Lorenz system. For such system, we can choose Lyapunov function: [figure omitted; refer to PDF] which is obviously positive definite and radially unbounded, where d i , λ i , i =1,2 ,3 are undetermined parameters. In this paper, we will study this more general Lorenz system ( 1) than the classical system and the generalized Lorenz system. The result obtained contains earlier results as its special cases.
This paper is organized as follows. In Section 2, we define the globally exponentially attractive set and positive invariant set and the globally conditional exponentially attractive set and positive invariant set of the three-dimensional chaotic systems with cross-product nonlinearities. In Section 3, the qualitative analysis of the exponentially attractive set and positive invariant set of the chaotic systems has been done. In Section 4, we also suggest an idea to construct chaotic systems, and some new chaotic systems and switched chaotic systems are illustrated.
2. Preliminaries
In this section, we present some basic definitions which are needed for proving all theorems in the next section. For convenience, denote X : = ( x 1 , x 2 , x 3 ) and X (t ) : =X (t , t 0 , X 0 ) .
Definition 1.
For the three-dimensional autonomous systems with cross-product nonlinearities ( 1), if there exists compact (bounded and closed) set Ω ∈ R 3 such that for all X 0 ∈ R 3 , the following condition: ρ (X (t ) , Ω ) : = inf Y ∈ Ω || X (t ) -Y || [arrow right]0 as t [arrow right] + ∞ , holds, then the set Ω is said to be globally attractive. That is, system ( 1) is ultimately bounded; namely, system ( 1) is globally stable in the sense of Lagrange or dissipative with ultimate bound.
Furthermore, if for all X 0 ∈ Ω 0 ⊆ Ω ⊂ R 3 , X (t , t 0 , X 0 ) ⊆ Ω 0 , then Ω 0 for t ...5;0 is called the positive invariant set of the system ( 1).
Definition 2.
For the three-dimensional autonomous systems with cross-product nonlinearities ( 1), if there exist compact set Ω ⊂ R 3 such that for all X 0 ∈ R 3 and constants M >0 , α >0 such that ρ (X (t ) , Ω ) ...4;M e - α (t - t 0 ) , then the three-dimensional autonomous systems with cross-product nonlinearities system ( 1) are said to have globally exponentially attractive set, or the system ( 1) is globally exponentially stable in the sense of Lagrange, and Ω is called the globally exponentially attractive set.
Definition 3.
For the three-dimensional autonomous systems with cross-product nonlinearities ( 1), if there exist compact set Ω ⊂ R 3 , a constant α >0 , and a bounded function M ( x 0 ,t ) >0 on t , as t ...5; t 0 , such that ρ (X (t ) , Ω ) ...4;M ( x 0 ) e - α (t - t 0 ) , where M ( x 0 ) =sup M ( x 0 ,t ) , t ...5; t 0 , then the system ( 1) is said to have globally conditional exponentially attractive set, or the system ( 1) is globally conditional exponentially stable in the sense of Lagrange, and Ω is called the globally conditional exponentially attractive set.
In general, from the definition we see that a globally exponential attractive set is not necessarily a positive invariant set. But our results obtained in the next section indeed show that a globally exponentially attractive set is a positive invariant set.
Note that it is difficult to verify the existence of Ω in Definition 2. Since the Lyapunov direct method is still a powerful tool in the study of asymptotic behaviour of nonlinear dynamical systems, the following definition is more useful in applications.
Definition 4.
For three-dimensional autonomous systems with cross-product nonlinearities ( 1), if there exist a positive definite and radially unbounded Lyapunov function V (X (t ) ) and positive numbers L >0 , α >0 such that the following inequality [figure omitted; refer to PDF] is valid for V (X (t ) ) >L (t ...5; t 0 ) , then the system ( 1) is said to be globally exponentially attractive or globally exponentially stable in the sense of Lagrange, and Ω : = { X |" V (X (t ) ) ...4;L , t ...5; t 0 } is called the globally exponentially attractive set.
Definition 5.
For the three-dimensional autonomous systems with cross-product nonlinearities ( 1), if there exist a positive definite and radially unbounded Lyapunov function V (X (t ) ) and a bounded function L ( x 0 ,t ) >0 , on t , as t ...5; t 0 , and α >0 such that the following inequality [figure omitted; refer to PDF] is valid for V (X (t ) ) >L ( x 0 ) , t ...5; t 0 , where L ( x 0 ) =sup L ( x 0 ,t ) , t ...5; t 0 , then the system ( 1) is said to be globally conditional exponentially attractive or globally conditional exponentially stable in the sense of Lagrange, and Ω : = { X |" V (X (t ) ) ...4;L ( x 0 ) , t ...5; t 0 } is called the globally conditional exponentially attractive set.
3. Qualitative Analysis
We call the dynamic system ( 1) the first class three-dimensional chaotic system with cross-product nonlinearities ( 1), if there are some nonzero numbers { λ 1 , λ 2 , λ 3 } so as to satisfy conditions [figure omitted; refer to PDF]
Condition ( 6) is satisfied by some known three-dimensional quadratic autonomous chaotic systems, the well-known Lorenz system [ 1- 3], the Rössler system [ 5], the Rucklidge system [ 6], and the Chen system [ 7, 8]. Lorenz systems are widely studied and the references therein [ 9- 12, 19- 21]. For example, consider the classical Lorenz system [figure omitted; refer to PDF] and the general Lorenz systems [figure omitted; refer to PDF]
Thus it can be seen that condition ( 6) is very important in qualitative analysis of the exponentially attractive set and positive invariant set of Lorenz systems.
We will research this dynamic system in two cases.
First, supposing b 111 = b 122 = b 133 = b 211 = b 222 = b 233 = b 311 = b 322 = b 333 =0 , b iij = - b iji , i ,j =1,2 ,3 , ∃ b ijk ...0; - b ikj , i ...0;j ...0;k , the dynamic system ( 1) can be rewritten as [figure omitted; refer to PDF]
The construction techniques of this kind of Lorenz systems are to pay attention to satisfing formula [figure omitted; refer to PDF] where λ i , i =1,2 ,3 are parameters and [figure omitted; refer to PDF] where μ i , i =1,2 ,3 are undetermined parameters. And we always assume that the supremum f ( μ ,X ) < + ∞ in the paper.
Lemma 6.
Suppose λ i >0 , i =1,2 ,3 , [figure omitted; refer to PDF] The function ( 12) has maximum.
Proof.
Consider [figure omitted; refer to PDF]
Thus the Hesse matrix H f of the f ( μ ,X ) is a negative definite matrix; furthermore max X ∈ R 3 f ( μ ,X ) exists.
These parameters d i , μ i , λ i , i =1,2 ,3 will be determined by solving the maximum of f 1 ( μ ,X ) and formula ( 12), and let [figure omitted; refer to PDF]
Theorem 7.
If condition ( 6) exists, η =min { μ 1 , μ 2 , μ 3 } , M = max X ∈ R 3 f ( μ ,X ) , λ i >0 , i =1,2 ,3 , then the estimation [figure omitted; refer to PDF] holds, and the set [figure omitted; refer to PDF] is the globally exponentially attractive set and positive invariant set of system ( 10); that is, [figure omitted; refer to PDF]
Proof.
Differentiating the Lyapunov function V (X (t ) ) in ( 3) with respect to time t along the trajectory of system ( 10) yields [figure omitted; refer to PDF] where μ j >0 , j =1,2 ,3 . Integrating both sides of ( 19) yields ( 16) and ( 17). By the definition, taking into account limit on both sides of the above inequality ( 16) as t [arrow right] + ∞ results in inequality ( 18).
Now, the characters of some of the chaotic systems known are analysed by condition ( 6). When a 11 = - σ , a 12 = σ , a 21 = ρ , a 22 = - γ , a 33 = - β , b 213 = -1 , b 312 =1 , else a ij =0 , b ijk =0 , c 1 = c 2 = c 3 =0 , and λ 1 = λ , λ 2 = λ 3 =1 , d 1 = d 2 =0 , d 3 = λ σ + ρ , μ 1 = σ , μ 2 = γ , μ 3 =min { σ , γ } , η = η 1 = μ 3 , β > η 1 , system ( 10) can be rewritten as system ( 7): [figure omitted; refer to PDF] We have M = β 2 ( λ σ + ρ ) 2 / 4 ( β - η 1 ) . Thus [figure omitted; refer to PDF] is the globally exponentially attractive set and positive invariant set of system ( 7).
Example 1.
Further, taking ito accout μ 1 = σ , μ 2 = γ , μ 3 = β /2 , η = η 2 =min { σ , γ , β /2 } , the estimate [figure omitted; refer to PDF] holds and that [figure omitted; refer to PDF] is the globally uniform exponentially attractive set and positive invariant set of system ( 7).
Proof.
Again applying Lyapunov function given in ( 19) and evaluating the derivative of d V 1 /dt along the trajectory of system ( 16) lead to [figure omitted; refer to PDF] The conclusion of Example 2is obtained.
Example 2.
Furthermore, choose μ 1 = σ , μ 2 = γ , μ 3 = β , 0 < ξ 1 < β , η = η 3 =min { σ , γ , ξ 1 } . Get [figure omitted; refer to PDF] Then, the estimate [figure omitted; refer to PDF] holds and that [figure omitted; refer to PDF] is the globally exponentially attractive set and positive invariant set of system ( 7).
Example 3.
Taking a 11 = -a , a 12 =b , a 21 =c , a 22 = -1 , a 32 =d , a 33 = -1 , b 123 = b 312 =1 , b 213 = -1 else a ij =0 , b ijk =0 , c 1 = c 2 = c 3 =0 , and λ 1 = λ 3 =1 , λ 2 = 2 , d 1 =d , d 2 =0 , d 3 =b +2c , system ( 6), V (X (t ) ) , and f (u ,X ) can be rewritten as system ( 8): [figure omitted; refer to PDF] Thus [figure omitted; refer to PDF] We have [figure omitted; refer to PDF] then [figure omitted; refer to PDF] is the estimation of the globally exponentially attractive and positive invariant sets of system ( 8).
If b 111 = b 222 = b 333 =0 , ∃ b ijj ...0;0 , i ,j =1,2 ,3 , the dynamic system ( 1) is shown as [figure omitted; refer to PDF] In this case, we can take into account [figure omitted; refer to PDF] where [ · ] 3 denotes modulo-3.
Theorem 8.
Suppose that G 0 = ( x 1 0 , x 2 0 , ... , x n 0 ) is the stable point of the f ( μ ,X ) defined by ( 33). If the Hesse matrix of the f ( μ ,X ) is a negative definite matrix, the f ( μ ,X ) has maximum M and the estimation [figure omitted; refer to PDF] holds; that is, [figure omitted; refer to PDF] and the set [figure omitted; refer to PDF] is the globally exponentially attractive set and positive invariant set of system ( 32).
Proof.
If G 0 is the stable point of the f ( μ ,X ) , that is, [figure omitted; refer to PDF] and the Hesse matrix H f of the f ( μ ,X ) is a negative definite matrix, namely, [figure omitted; refer to PDF] The f ( μ ,X ) has the maximum M . Differentiating the Lyapunov function V (X (t ) ) in ( 3) with respect to time t along the trajectory of system ( 32) yields [figure omitted; refer to PDF]
The proof is complete.
4. Switched Chaotic Systems
Condition ( 6) has helpfully provided us with instructions on how to find the new chaotic systems. We construct a series of new chaotic systems that the condition ( 6) is fulfilled and study the switching system between them.
Example 4.
Consider a Lorenz system shown in Figure 1: [figure omitted; refer to PDF]
Solution . Here [figure omitted; refer to PDF] The Hesse matrix of the f ( μ ,X ) is a negative definite matrix, max f ( μ ,X ) [approximate]164045.42 . The set [figure omitted; refer to PDF] is the globally exponentially attractive set and positive invariant set of system ( 40).
Note . (a) If the Hesse matrix of the f ( μ ,X ) is not a negative definite matrix, the f ( μ ,X ) has no maximum M .
(b) If ∃ a i 0 i 0 ...5;0 , lim x i 0 [arrow right] ∞ f ( μ ,X ) = + ∞ , this type of chaotic system needs further research.
(c) We call the dynamic system ( 1) the second class three-dimensional chaotic system with cross-product nonlinearities, if it does not satisfy condition ( 6). For this class of chaotic systems, f ( μ ,X ) is a cubic polynomial and there is not maximum if we choose energy function ( 3) differentiating this Lyapunov function with respect to t along the trajectory of system ( 1). It is very useful to research these problems.
Figure 1: Simulation of system ( 40).
[figure omitted; refer to PDF]
Example 5.
The new chaotic system shown in Figure 2is [figure omitted; refer to PDF]
Figure 2: Simulation of system ( 43).
[figure omitted; refer to PDF]
Example 6.
The chaotic system shown in Figure 3is [figure omitted; refer to PDF]
Figure 3: Simulation of system ( 44).
[figure omitted; refer to PDF]
Example 7.
The chaotic system shown in Figure 4is [figure omitted; refer to PDF]
Figure 4: Simulation of system ( 45).
[figure omitted; refer to PDF]
Example 8.
The chaotic system shown in Figure 5is [figure omitted; refer to PDF]
Figure 5: Simulation of system ( 46).
[figure omitted; refer to PDF]
Example 9.
The chaotic system shown in Figure 6is [figure omitted; refer to PDF]
Figure 6: Simulation of system ( 47).
[figure omitted; refer to PDF]
Note . When we analyse Examples 5to 9by the previous means, for sup X ∈ R 3 f ( μ ,X ) = + ∞ , the globally exponentially attractive set and positive invariant set of them have not been obtained. The globally exponentially attractive set and positive invariant set really exist by their trajectories. Particularly, by Lü et al. chaotic system [ 11] and Example 9we conjecture that they have globally conditional exponentially attractive set and positive invariant set, according to preliminary study. These are waiting for us to do further research. Meanwhile, we can compute that the maximum Lyapunov exponents of Examples 4- 9are 1.06, 0.02, 1.84, 0.01, 0.92, and 0.95, respectively.
5. Simulation of Switched System
In this section, we will show some simulation results of the following switching system [figure omitted; refer to PDF] where x T = ( x 1 , x 2 , x 3 ) , σ is the switching law, and [figure omitted; refer to PDF] with a ij σ , b ijk σ , c i σ ∈R , i ,j ,k =1,2 ,3 . Each pair of ( A σ , C σ , B 1 σ , B 2 σ , B 3 σ ) takes the form from Example 1to Example 6. The switching law is that the system will stay in each subsystem for a constant time. In the following, we assume that (a, b) denotes a switched system which switches between system (a) and system (b). It can be seen from Figures 7to 12that the switched systems (18,20), (18,22), (18,23), (20,18), (20,22), (20,23), (22,18), (22,20), (22,23), (23,18), (23,20), and (23,22) can also yield chaotic systems.
Switched system between system ( 40) and others.
(a) System ( 40)-( 43)
[figure omitted; refer to PDF]
(b) System ( 40)-( 44)
[figure omitted; refer to PDF]
(c) System ( 40)-( 45)
[figure omitted; refer to PDF]
(d) System ( 40)-( 46)
[figure omitted; refer to PDF]
(e) System ( 40)-( 47)
[figure omitted; refer to PDF]
Switched system between system ( 43) and others.
(a) System ( 43)-( 40)
[figure omitted; refer to PDF]
(b) System ( 43)-( 44)
[figure omitted; refer to PDF]
(c) System ( 43)-( 45)
[figure omitted; refer to PDF]
(d) System ( 43)-( 46)
[figure omitted; refer to PDF]
(e) System ( 43)-( 47)
[figure omitted; refer to PDF]
Switched system between Example ( 44) and others.
(a) System ( 44)-( 40)
[figure omitted; refer to PDF]
(b) System ( 44)-( 43)
[figure omitted; refer to PDF]
(c) System ( 44)-( 45)
[figure omitted; refer to PDF]
(d) System ( 44)-( 46)
[figure omitted; refer to PDF]
(e) System ( 44)-( 47)
[figure omitted; refer to PDF]
Switched system between Example ( 45) and others.
(a) System ( 45)-( 40)
[figure omitted; refer to PDF]
(b) System ( 45)-( 43)
[figure omitted; refer to PDF]
(c) System ( 45)-( 44)
[figure omitted; refer to PDF]
(d) System ( 45)-( 46)
[figure omitted; refer to PDF]
(e) System ( 45)-( 47)
[figure omitted; refer to PDF]
Switched system between Example ( 46) and others.
(a) System ( 46)-( 40)
[figure omitted; refer to PDF]
(b) System ( 46)-( 40)
[figure omitted; refer to PDF]
(c) System ( 46)-( 44)
[figure omitted; refer to PDF]
(d) System ( 46)-( 45)
[figure omitted; refer to PDF]
(e) System ( 46)-( 47)
[figure omitted; refer to PDF]
Switched system between Example ( 47) and others.
(a) System ( 47)-( 40)
[figure omitted; refer to PDF]
(b) System ( 47)-( 43)
[figure omitted; refer to PDF]
(c) System ( 47)-( 44)
[figure omitted; refer to PDF]
(d) System ( 47)-( 45)
[figure omitted; refer to PDF]
(e) System ( 47)-( 46)
[figure omitted; refer to PDF]
6. Conclusion
In this paper, the methods in [ 19- 21] have been extended to study the globally exponentially or globally conditional exponentially attractive set and positive invariant set of the three-dimensional chaotic system family with cross-product nonlinearities. We have given two theorems for studying this question and given some examples to show that such system indeed has the globally exponentially or globally conditional exponentially attractive set and positive invariant set, and the exponential estimation is explicitly derived. We have also suggested an idea to construct the chaotic systems, and some new chaotic systems have been illustrated. The simulation results are given for switched system between these new chaotic systems. It is very interesting to further research that the Hesse matrix of the f ( μ ,X ) is not a negative definite matrix, and the dynamic system ( 1) is a second class three-dimensional chaotic system with cross-product nonlinearities.
Acknowledgments
This work is supported by the Fundamental Research Funds for the Central Universities, China Postdoctoral Science Foundation funded project under Grant 2012M511615, National Natural Science Foundation of China under Grants 60474011 and 60904005, and the Hubei Provincial Natural Science Foundation of China under Grant 2009CDB026.
[1] L. O. Chua, "Chua's circuit: an overview ten years later," Journal of Circuits, Systems, and Computers , vol. 4, pp. 117-159, 1994.
[2] E. N. Lorenz, "Deterministic nonperiodic flow," Journal of the Atmospheric Sciences , vol. 20, pp. 130-141, 1963.
[3] T. Lofaro, "A model of the dynamics of the Newton-Leipnik attractor," International Journal of Bifurcation and Chaos , vol. 7, no. 12, pp. 2723-2733, 1997.
[4] N. A. Magnitskii, S. V. Sidorov, "A new view of the Lorenz attractor," Differential Equations , vol. 37, no. 11, pp. 1568-1579, 2001.
[5] O. E. Rossler, "An equation for continuous chaos," Physics Letters A , vol. 15, pp. 397-398, 1976.
[6] A. M. Rucklidge, "Chaos in models of double convection," Journal of Fluid Mechanics , vol. 237, pp. 209-229, 1992.
[7] G. Chen, T. Ueta, "Yet another chaotic attractor," International Journal of Bifurcation and Chaos , vol. 9, no. 7, pp. 1465-1466, 1999.
[8] S. Celikovský, G. Chen, "On a generalized Lorenz canonical form of chaotic systems," International Journal of Bifurcation and Chaos , vol. 12, no. 8, pp. 1789-1812, 2002.
[9] T. Ueta, G. Chen, "Bifurcation analysis of Chen's equation," International Journal of Bifurcation and Chaos , vol. 10, no. 8, pp. 1917-1931, 2000.
[10] W. Liu, G. Chen, "A new chaotic system and its generation," International Journal of Bifurcation and Chaos , vol. 13, no. 1, pp. 261-267, 2003.
[11] J. Lü, G. Chen, D. Cheng, "A new chaotic system and beyond: the generalized Lorenz-like system," International Journal of Bifurcation and Chaos , vol. 14, no. 5, pp. 1507-1537, 2004.
[12] P. Wang, D. Li, X. Wu, J. Lü, X. Yu, "Ultimate bound estimation of a class of high dimensional quadratic autonomous dynamical systems," International Journal of Bifurcation and Chaos , vol. 21, no. 9, pp. 2679-2694, 2011.
[13] P. Wang, D. Li, Q. Hu, "Bounds of the hyper-chaotic Lorenz-Stenflo system," Communications in Nonlinear Science and Numerical Simulation , vol. 15, no. 9, pp. 2514-2520, 2010.
[14] J. M. Sarabia, E. Gómez-Déniz, M. Sarabia, F. Prieto, "A general method for generating parametric Lorenz and Leimkuhler curves," Journal of Informetrics , vol. 4, no. 4, pp. 524-539, 2010.
[15] W. G. Hoover, "Remark on "some simple chaotic flows"," Physical Review E , vol. 51, no. 1, pp. 759-760, 1995.
[16] R. Genesio, A. Tesi, "Harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems," Automatica , vol. 28, no. 3, pp. 531-548, 1992.
[17] A. L. Shil'nikov, "On bifurcations of the Lorenz attractor in the Shimizu-Morioka model," Physica D , vol. 62, no. 1-4, pp. 338-346, 1993.
[18] A. Shilnikov, G. Nicolis, C. Nicolis, "Bifurcation and predictability analysis of a low-order atmospheric circulation model," International Journal of Bifurcation and Chaos , vol. 5, pp. 17011-17711, 1995.
[19] X. Liao, H. Luo, Y. Fu, S. Xie, P. Yu, "On the globally exponentially attractive sets of the family of Lorenz systems," Science in China. Series E , vol. 37, pp. 715-769, 2007.
[20] X. Liao, Y. Fu, S. Xie, P. Yu, "Globally exponentially attractive sets of the family of Lorenz systems," Science in China. Series F , vol. 51, no. 3, pp. 283-292, 2008.
[21] P. Yu, X. X. Liao, S. L. Xie, Y. L. Fu, "A constructive proof on the existence of globally exponentially attractive set and positive invariant set of general Lorenz family," Communications in Nonlinear Science and Numerical Simulation , vol. 14, no. 7, pp. 2886-2896, 2009.
[]
You have requested "on-the-fly" machine translation of selected content from our databases. This functionality is provided solely for your convenience and is in no way intended to replace human translation. Show full disclaimer
Neither ProQuest nor its licensors make any representations or warranties with respect to the translations. The translations are automatically generated "AS IS" and "AS AVAILABLE" and are not retained in our systems. PROQUEST AND ITS LICENSORS SPECIFICALLY DISCLAIM ANY AND ALL EXPRESS OR IMPLIED WARRANTIES, INCLUDING WITHOUT LIMITATION, ANY WARRANTIES FOR AVAILABILITY, ACCURACY, TIMELINESS, COMPLETENESS, NON-INFRINGMENT, MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. Your use of the translations is subject to all use restrictions contained in your Electronic Products License Agreement and by using the translation functionality you agree to forgo any and all claims against ProQuest or its licensors for your use of the translation functionality and any output derived there from. Hide full disclaimer
Copyright © 2013 Xinquan Zhao et al. Xinquan Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
This paper introduces a new series of three-dimensional chaotic systems with cross-product nonlinearities. Based on some conditions, we analyze the globally exponentially or globally conditional exponentially attractive set and positive invariant set of these chaotic systems. Moreover, we give some known examples to show our results, and the exponential estimation is explicitly derived. Finally, we construct some three-dimensional chaotic systems with cross-product nonlinearities and study the switching system between them.
You have requested "on-the-fly" machine translation of selected content from our databases. This functionality is provided solely for your convenience and is in no way intended to replace human translation. Show full disclaimer
Neither ProQuest nor its licensors make any representations or warranties with respect to the translations. The translations are automatically generated "AS IS" and "AS AVAILABLE" and are not retained in our systems. PROQUEST AND ITS LICENSORS SPECIFICALLY DISCLAIM ANY AND ALL EXPRESS OR IMPLIED WARRANTIES, INCLUDING WITHOUT LIMITATION, ANY WARRANTIES FOR AVAILABILITY, ACCURACY, TIMELINESS, COMPLETENESS, NON-INFRINGMENT, MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. Your use of the translations is subject to all use restrictions contained in your Electronic Products License Agreement and by using the translation functionality you agree to forgo any and all claims against ProQuest or its licensors for your use of the translation functionality and any output derived there from. Hide full disclaimer