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June  2017, 22(4): 1673-1681. doi: 10.3934/dcdsb.2017080

On global boundedness of the Chen system

1. 

College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China

2. 

Mathematical post-doctoral station, College of Mathematics and Statistics, Southwest University, Chongqing 400716, China

3. 

College of Electronic and Information Engineering, Southwest University, Chongqing 400716, China

4. 

College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

5. 

College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China

Received  June 2016 Revised  September 2016 Published  February 2017

This paper deals with the open problem of the global boundedness of the Chen system based on Lyapunov stability theory, which was proposed by Qin and Chen (2007). The innovation of the paper is that this paper not only proves the Chen system is global bounded for a certain range of the parameters according to stability theory of dynamical systems but also gives a family of mathematical expressions of global exponential attractive sets for the Chen system with respect to the parameters of this system. Furthermore, the exponential rate of the trajectories is also obtained.

Citation: Fuchen Zhang, Xiaofeng Liao, Chunlai Mu, Guangyun Zhang, Yi-An Chen. On global boundedness of the Chen system. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1673-1681. doi: 10.3934/dcdsb.2017080
References:
[1]

V. BraginV. VagaitsevN. Kuznetsov and G. Leonov, Algorithms for finding hidden oscil lations in nonlinear systems. The Aizerman and Kalman conjectures and Chua's circuits, J. Comput. Syst. Sci. Int., 50 (2011), 511-543. doi: 10.1134/S106423071104006X. Google Scholar

[2] G. Chen and J. Lu, Dynamical Analysis, Control and Synchronization of the Lorenz Systems Family, Science Press, Beijing, 2003.
[3]

G. Chen and T. Ueta, Yet another chaotic attractor, Int. J. Bifurc. Chaos Appl. Sci.Eng., 9 (1999), 1465-1466. doi: 10.1142/S0218127499001024. Google Scholar

[4]

E. Elsayed, Solutions of rational difference system of order two, Math. Comput. Model., 55 (2012), 378-384. doi: 10.1016/j.mcm.2011.08.012. Google Scholar

[5]

N. KuznetsovT. Mokaev and P. Vasilyev, Numerical justification of Leonov conjecture on Lyapunov dimension of Rossler attractor, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 1027-1034. doi: 10.1016/j.cnsns.2013.07.026. Google Scholar

[6]

G. Leonov, Bounds for attractors and the existence of homoclinic orbits in the Lorenz system, J. Appl. Math. Mech., 65 (2001), 19-32. doi: 10.1016/S0021-8928(01)00004-1. Google Scholar

[7]

G. Leonov, General existence conditions of homoclinic trajectories in dissipative systems, Lorenz, Shimizu-Morioka, Lu and Chen systems. Phys. Lett. A, 376 (2012), 3045-3050. doi: 10.1016/j.physleta.2012.07.003. Google Scholar

[8]

G. A. Leonov, The Tricomi problem for the Shimizu-Morioka dynamical system, Dokl. Math., 86 (2012), 850-853. doi: 10.1134/S1064562412060324. Google Scholar

[9]

G. A. Leonov, Existence criterion of homoclinic trajectories in the Glukhovsky-Dolzhansky system, Phys. Lett. A, 379 (2015), 524-528. doi: 10.1016/j.physleta.2014.12.005. Google Scholar

[10]

G. Leonov and V. Boichenko, Lyapunov's direct method in the estimation of the Hausdorff dimension of attractors, Acta Appl. Math., 26 (1992), 1-60. doi: 10.1007/BF00046607. Google Scholar

[11]

G. LeonovA. Bunin and N. Koksch, Attractor localization of the Lorenz system, Z. Angew. Math. Mech., 67 (1987), 649-656. doi: 10.1002/zamm.19870671215. Google Scholar

[12]

X. LiaoY. FuS. Xie and P. Yu, Globally exponentially attractive sets of the family of Lorenz systems, Science in China Series F: Information Sciences, 51 (2008), 283-292. doi: 10.1007/s11432-008-0024-2. Google Scholar

[13]

G. Leonov and N. Kuznetsov, Hidden attractors in dynamical systems. From hidden oscilla tions in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits, Int. J. Bifurc. Chaos Appl. Sci. Eng., 2013 (23), 69pp. doi: 10.1142/S0218127413300024. Google Scholar

[14]

G. A. Leonov and N. V. Kuznetsov, On differences and similarities in the analysis of Lorenz, Chen and Lu systems, Appl. Math. Comput., 256 (2015), 334-343. doi: 10.1016/j.amc.2014.12.132. Google Scholar

[15]

G. LeonovN. Kuznetsov and V. Vagaitsev, Localization of hidden Chua's attractors, Phys. Lett. A, 375 (2011), 2230-2233. doi: 10.1016/j.physleta.2011.04.037. Google Scholar

[16]

X. LiaoP. YuS. Xie and Y. Fu, Study on the global property of the smooth Chua's system, Int. J. Bifurc. Chaos Appl. Sci. Eng., 16 (2006), 2815-2841. doi: 10.1142/S0218127406016483. Google Scholar

[17]

F. Zhang, On a model of the dynamical systems describing convective fluid motion in rotating cavity, Appl. Math. Comput., 268 (2015), 873-882. doi: 10.1016/j.amc.2015.06.120. Google Scholar

[18]

F. ZhangC. MuP. ZhengD. Lin and G. Zhang, The dynamical analysis of a new chaotic system and simulation, Math. Methods Appl. Sci., 37 (2014), 1838-1846. doi: 10.1002/mma.2939. Google Scholar

[19]

F. ZhangY. Shu and H. Yang, Bounds for a new chaotic system and its application in chaos synchronization, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 1501-1508. doi: 10.1016/j.cnsns.2010.05.032. Google Scholar

[20]

F. ZhangY. ShuH. Yang and X. Li, Estimating the ultimate bound and positively invariant set for a synchronous motor and its application in chaos synchronization, Chaos Solitons Fractals, 44 (2011), 137-144. Google Scholar

[21]

F. Zhang and G. Zhang, Dynamics of a low-order atmospheric circulation chaotic model, Optik, 127 (2016), 4105-4108. Google Scholar

[22]

F. Zhang and G. Zhang, Further results on ultimate bound on the trajectories of the Lorenz System, Qual. Theory Dyn. Syst., 15 (2016), 221-235. doi: 10.1007/s12346-015-0137-0. Google Scholar

[23]

F. ZhangG. ZhangD. Lin and X. Sun, Global attractive sets of a novel bounded chaotic system, Neural Comput. Appl., 79 (2015), 539-547. Google Scholar

show all references

References:
[1]

V. BraginV. VagaitsevN. Kuznetsov and G. Leonov, Algorithms for finding hidden oscil lations in nonlinear systems. The Aizerman and Kalman conjectures and Chua's circuits, J. Comput. Syst. Sci. Int., 50 (2011), 511-543. doi: 10.1134/S106423071104006X. Google Scholar

[2] G. Chen and J. Lu, Dynamical Analysis, Control and Synchronization of the Lorenz Systems Family, Science Press, Beijing, 2003.
[3]

G. Chen and T. Ueta, Yet another chaotic attractor, Int. J. Bifurc. Chaos Appl. Sci.Eng., 9 (1999), 1465-1466. doi: 10.1142/S0218127499001024. Google Scholar

[4]

E. Elsayed, Solutions of rational difference system of order two, Math. Comput. Model., 55 (2012), 378-384. doi: 10.1016/j.mcm.2011.08.012. Google Scholar

[5]

N. KuznetsovT. Mokaev and P. Vasilyev, Numerical justification of Leonov conjecture on Lyapunov dimension of Rossler attractor, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 1027-1034. doi: 10.1016/j.cnsns.2013.07.026. Google Scholar

[6]

G. Leonov, Bounds for attractors and the existence of homoclinic orbits in the Lorenz system, J. Appl. Math. Mech., 65 (2001), 19-32. doi: 10.1016/S0021-8928(01)00004-1. Google Scholar

[7]

G. Leonov, General existence conditions of homoclinic trajectories in dissipative systems, Lorenz, Shimizu-Morioka, Lu and Chen systems. Phys. Lett. A, 376 (2012), 3045-3050. doi: 10.1016/j.physleta.2012.07.003. Google Scholar

[8]

G. A. Leonov, The Tricomi problem for the Shimizu-Morioka dynamical system, Dokl. Math., 86 (2012), 850-853. doi: 10.1134/S1064562412060324. Google Scholar

[9]

G. A. Leonov, Existence criterion of homoclinic trajectories in the Glukhovsky-Dolzhansky system, Phys. Lett. A, 379 (2015), 524-528. doi: 10.1016/j.physleta.2014.12.005. Google Scholar

[10]

G. Leonov and V. Boichenko, Lyapunov's direct method in the estimation of the Hausdorff dimension of attractors, Acta Appl. Math., 26 (1992), 1-60. doi: 10.1007/BF00046607. Google Scholar

[11]

G. LeonovA. Bunin and N. Koksch, Attractor localization of the Lorenz system, Z. Angew. Math. Mech., 67 (1987), 649-656. doi: 10.1002/zamm.19870671215. Google Scholar

[12]

X. LiaoY. FuS. Xie and P. Yu, Globally exponentially attractive sets of the family of Lorenz systems, Science in China Series F: Information Sciences, 51 (2008), 283-292. doi: 10.1007/s11432-008-0024-2. Google Scholar

[13]

G. Leonov and N. Kuznetsov, Hidden attractors in dynamical systems. From hidden oscilla tions in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits, Int. J. Bifurc. Chaos Appl. Sci. Eng., 2013 (23), 69pp. doi: 10.1142/S0218127413300024. Google Scholar

[14]

G. A. Leonov and N. V. Kuznetsov, On differences and similarities in the analysis of Lorenz, Chen and Lu systems, Appl. Math. Comput., 256 (2015), 334-343. doi: 10.1016/j.amc.2014.12.132. Google Scholar

[15]

G. LeonovN. Kuznetsov and V. Vagaitsev, Localization of hidden Chua's attractors, Phys. Lett. A, 375 (2011), 2230-2233. doi: 10.1016/j.physleta.2011.04.037. Google Scholar

[16]

X. LiaoP. YuS. Xie and Y. Fu, Study on the global property of the smooth Chua's system, Int. J. Bifurc. Chaos Appl. Sci. Eng., 16 (2006), 2815-2841. doi: 10.1142/S0218127406016483. Google Scholar

[17]

F. Zhang, On a model of the dynamical systems describing convective fluid motion in rotating cavity, Appl. Math. Comput., 268 (2015), 873-882. doi: 10.1016/j.amc.2015.06.120. Google Scholar

[18]

F. ZhangC. MuP. ZhengD. Lin and G. Zhang, The dynamical analysis of a new chaotic system and simulation, Math. Methods Appl. Sci., 37 (2014), 1838-1846. doi: 10.1002/mma.2939. Google Scholar

[19]

F. ZhangY. Shu and H. Yang, Bounds for a new chaotic system and its application in chaos synchronization, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 1501-1508. doi: 10.1016/j.cnsns.2010.05.032. Google Scholar

[20]

F. ZhangY. ShuH. Yang and X. Li, Estimating the ultimate bound and positively invariant set for a synchronous motor and its application in chaos synchronization, Chaos Solitons Fractals, 44 (2011), 137-144. Google Scholar

[21]

F. Zhang and G. Zhang, Dynamics of a low-order atmospheric circulation chaotic model, Optik, 127 (2016), 4105-4108. Google Scholar

[22]

F. Zhang and G. Zhang, Further results on ultimate bound on the trajectories of the Lorenz System, Qual. Theory Dyn. Syst., 15 (2016), 221-235. doi: 10.1007/s12346-015-0137-0. Google Scholar

[23]

F. ZhangG. ZhangD. Lin and X. Sun, Global attractive sets of a novel bounded chaotic system, Neural Comput. Appl., 79 (2015), 539-547. Google Scholar

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