doi: 10.3934/dcdsb.2021291
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Boundedness of the complex Chen system

1. 

Department of Mathematics, Shandong University, Weihai, Shandong 264209, China

2. 

Department of Electrical Engineering, City University of Hong Kong, Hong Kong SAR, China

Received  May 2021 Revised  October 2021 Early access December 2021

Fund Project: This work was supported by the National Natural Science Foundation of China (No. 11701328) and Young Scholars Program of Shandong University, Weihai (No. 2017WHWLJH09)

Some ultimate bounds are derived for the complex Chen system.

Citation: Xu Zhang, Guanrong Chen. Boundedness of the complex Chen system. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021291
References:
[1] C. Bailly and G. Comte-Bellot, Turbulence, Springer International Publishing Switzerland, 2015.  doi: 10.1007/978-3-319-16160-0.
[2]

R. Barboza, On Lorenz and Chen systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 28 (2018), 1850018, 8 pp. doi: 10.1142/S0218127418500189.

[3]

R. Barboza and G. Chen, On the global boundedness of the Chen system, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 21 (2011), 3373-3385.  doi: 10.1142/S021812741103060X.

[4]

S. Celikovsky and G. Chen, Generalized Lorenz systems family revisited, Int. J. Bifurcation Chaos, 31 (2021), 2150079, 15 pp.

[5]

D. Cheban and J. Duan, Recurrent motions and global attractors of nonautonomous Lorenz systems, Dyn. Syst., 19 (2004), 41-59.  doi: 10.1080/14689360310001624132.

[6]

G. Chen, Generalized Lorenz systems family, preprint, arXiv: 2006.04066, 2020.

[7]

G. Chen and T. Ueta, Yet another chaotic attractor, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 9 (1999), 1465-1466.  doi: 10.1142/S0218127499001024.

[8] P. A. DavidsonY. Kaneda and K. R. Sreenivasan, Ten Chapters in Turbulence, Cambridge Univ. Press, 2013. 
[9]

A. C. FowlerJ. D. Gibbon and M. J. McGuinness, The real and complex Lorenz equations and their relevance to physical systems, Phys. D, 7 (1983), 126-134.  doi: 10.1016/0167-2789(83)90123-9.

[10]

A. C. FowlerM. J. McGuinness and J. D. Gibbon, The complex Lorenz equations, Phys. D, 4 (1981/82), 139-163.  doi: 10.1016/0167-2789(82)90057-4.

[11]

M. Franz and M. Zhang, Suppression and creation of chaos in a periodically forced Lorenz system, Phys. Rev. E, 52 (1995), 3558-3565.  doi: 10.1103/PhysRevE.52.3558.

[12]

H. Haken, Analogy between higher instabilities in fluids and lasers, Phys. Lett. A, 53 (1975), 77-78.  doi: 10.1016/0375-9601(75)90353-9.

[13]

C. Lainscsek, A class of Lorenz-like systems, Chaos, 22 (2012), 013126, 5 pp. doi: 10.1063/1.3689438.

[14] L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Pergamon, Oxford, 1959. 
[15]

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

[16]

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.

[17]

C. Letellier, G. F. V. Amaral and L. A. Aguirre, Insights into the algebraic structure of Lorenz-like systems using feedback circuit analysis and piecewise affine models, Chaos, 17 (2007), 023104, 11 pp. doi: 10.1063/1.2645725.

[18]

D. LiJ. LuX. Wu and 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.  doi: 10.1016/j.jmaa.2005.11.008.

[19]

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 (in Chinese), Sci. China Ser. E, 34 (2004), 1404-1419. 

[20]

E. N. Lorenz, Deterministic non-periodic flow, J. Atmospheric Sci., 20 (1963), 130-141.  doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.

[21]

J. LüG. ChenD. Cheng and S. Celikovsky, Bridge the gap between the Lorenz system and the Chen system, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 12 (2002), 2917-2926.  doi: 10.1142/S021812740200631X.

[22]

G. M. MahmoudT. Bountis and E. E. Mahmoud, Active control and global synchronization of the complex Chen and Lü systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 4295-4308.  doi: 10.1142/S0218127407019962.

[23]

J. Pedlosky, Finite-amplitude baroclinic waves with small dissipation, J. Atmos. Sci., 28 (1971), 587-597.  doi: 10.1175/1520-0469(1971)028<0587:FABWWS>2.0.CO;2.

[24]

J. Pedlosky, The effect of $\beta$ on the chaotic behavior of unstable baroclinic wave, J. Atmos. Sci., 38 (1981), 717-731. 

[25]

W.-X. Qin and G. Chen, On the boundedness of solutions of the Chen system, J. Math. Anal. Appl., 329 (2007), 445-451.  doi: 10.1016/j.jmaa.2006.06.091.

[26]

D. Ruelle and F. Takens, On the nature of turbulence, Commun. Math. Phys., 20 (1971), 167-192.  doi: 10.1007/BF01646553.

[27]

H. Saberi NikS. Effati and J. Saberi-Nadjafi, New ultimate bound sets and exponential finite-time synchronization for the complex Lorenz system, J. Complexity, 31 (2015), 715-730.  doi: 10.1016/j.jco.2015.03.001.

[28]

R. SaravananO. NarayanK. Banerjee and J. K. Bhattacharjee, Chaos in a periodically forced Lorenz system, Phys. Rev. A, 31 (1985), 520-522.  doi: 10.1103/PhysRevA.31.520.

[29]

E. A. Sataev, Non-existence of stable trajectories in non-autonomous perturbations of systems of Lorenz type, Mat. Sb., 196 (2005), 99-134.  doi: 10.1070/SM2005v196n04ABEH000892.

[30]

P. Sooraksa and G. Chen, Chen system as a controlled weather model –physical principle, engineering design and real applications, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 28 (2018), 1830009, 12 pp. doi: 10.1142/S0218127418300094.

[31] C. Sparrow, The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors, New York, Oxford, 1982. 
[32] A. Tsinober, The Essence of Turbulence as a Physical Phenomenon with Emphasis on Issues of Paradigmatic Nature, Springer Nature Switzerland AG, 2019.  doi: 10.1007/978-3-319-99531-1.
[33]

V. Yu. Toronov and V. L. Derbov, Boundedness of attractors in the complex Lorenz model, Phys. Rev. E, 55 (1997), 3689-3692.  doi: 10.1103/PhysRevE.55.3689.

[34]

F. ZhangX. LiaoC. MuG. Zhang and Y.-A. Chen, On global boundedness of the Chen system, Discrete Contin. Dyna. Syst.-B, 22 (2017), 1673-1681.  doi: 10.3934/dcdsb.2017080.

[35]

F. ZhangX. LiaoG. ZhangC. MuM. Xiao and P. Zhou, Dynamical behaviors of a generalized Lorenz system, Discrete Contin. Dyna. Syst.-B, 22 (2017), 3707-3720.  doi: 10.3934/dcdsb.2017184.

[36]

F. Zhang and G. Zhang, Boundedness solutions of the complex Lorenz chaotic system, Appl. Math. Comput., 243 (2014), 12-23.  doi: 10.1016/j.amc.2014.05.102.

[37]

X. Zhang, Dynamics of a class of nonautonomous Lorenz-type systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 26 (2016), 1650208, 12 pp. doi: 10.1142/S0218127416502084.

[38]

X. Zhang, Dynamics of a class of fractional-order nonautonomous Lorenz-type systems, Chaos, 27 (2017), 041104, 7 pp. doi: 10.1063/1.4981909.

[39]

X. Zhang, Boundedness of a class of complex Lorenz systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 31 (2021), 2150101, 22 pp. doi: 10.1142/S0218127421501017.

[40]

Q. ZhaoS. Zhou and X. Li, Synchronization slaved by partial-states in lattices of non-autonomous coupled Lorenz equation, Commun. Nonlinear Sci. Numer. Simul., 13 (2008), 928-938.  doi: 10.1016/j.cnsns.2006.09.001.

show all references

References:
[1] C. Bailly and G. Comte-Bellot, Turbulence, Springer International Publishing Switzerland, 2015.  doi: 10.1007/978-3-319-16160-0.
[2]

R. Barboza, On Lorenz and Chen systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 28 (2018), 1850018, 8 pp. doi: 10.1142/S0218127418500189.

[3]

R. Barboza and G. Chen, On the global boundedness of the Chen system, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 21 (2011), 3373-3385.  doi: 10.1142/S021812741103060X.

[4]

S. Celikovsky and G. Chen, Generalized Lorenz systems family revisited, Int. J. Bifurcation Chaos, 31 (2021), 2150079, 15 pp.

[5]

D. Cheban and J. Duan, Recurrent motions and global attractors of nonautonomous Lorenz systems, Dyn. Syst., 19 (2004), 41-59.  doi: 10.1080/14689360310001624132.

[6]

G. Chen, Generalized Lorenz systems family, preprint, arXiv: 2006.04066, 2020.

[7]

G. Chen and T. Ueta, Yet another chaotic attractor, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 9 (1999), 1465-1466.  doi: 10.1142/S0218127499001024.

[8] P. A. DavidsonY. Kaneda and K. R. Sreenivasan, Ten Chapters in Turbulence, Cambridge Univ. Press, 2013. 
[9]

A. C. FowlerJ. D. Gibbon and M. J. McGuinness, The real and complex Lorenz equations and their relevance to physical systems, Phys. D, 7 (1983), 126-134.  doi: 10.1016/0167-2789(83)90123-9.

[10]

A. C. FowlerM. J. McGuinness and J. D. Gibbon, The complex Lorenz equations, Phys. D, 4 (1981/82), 139-163.  doi: 10.1016/0167-2789(82)90057-4.

[11]

M. Franz and M. Zhang, Suppression and creation of chaos in a periodically forced Lorenz system, Phys. Rev. E, 52 (1995), 3558-3565.  doi: 10.1103/PhysRevE.52.3558.

[12]

H. Haken, Analogy between higher instabilities in fluids and lasers, Phys. Lett. A, 53 (1975), 77-78.  doi: 10.1016/0375-9601(75)90353-9.

[13]

C. Lainscsek, A class of Lorenz-like systems, Chaos, 22 (2012), 013126, 5 pp. doi: 10.1063/1.3689438.

[14] L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Pergamon, Oxford, 1959. 
[15]

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

[16]

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.

[17]

C. Letellier, G. F. V. Amaral and L. A. Aguirre, Insights into the algebraic structure of Lorenz-like systems using feedback circuit analysis and piecewise affine models, Chaos, 17 (2007), 023104, 11 pp. doi: 10.1063/1.2645725.

[18]

D. LiJ. LuX. Wu and 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.  doi: 10.1016/j.jmaa.2005.11.008.

[19]

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 (in Chinese), Sci. China Ser. E, 34 (2004), 1404-1419. 

[20]

E. N. Lorenz, Deterministic non-periodic flow, J. Atmospheric Sci., 20 (1963), 130-141.  doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.

[21]

J. LüG. ChenD. Cheng and S. Celikovsky, Bridge the gap between the Lorenz system and the Chen system, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 12 (2002), 2917-2926.  doi: 10.1142/S021812740200631X.

[22]

G. M. MahmoudT. Bountis and E. E. Mahmoud, Active control and global synchronization of the complex Chen and Lü systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 4295-4308.  doi: 10.1142/S0218127407019962.

[23]

J. Pedlosky, Finite-amplitude baroclinic waves with small dissipation, J. Atmos. Sci., 28 (1971), 587-597.  doi: 10.1175/1520-0469(1971)028<0587:FABWWS>2.0.CO;2.

[24]

J. Pedlosky, The effect of $\beta$ on the chaotic behavior of unstable baroclinic wave, J. Atmos. Sci., 38 (1981), 717-731. 

[25]

W.-X. Qin and G. Chen, On the boundedness of solutions of the Chen system, J. Math. Anal. Appl., 329 (2007), 445-451.  doi: 10.1016/j.jmaa.2006.06.091.

[26]

D. Ruelle and F. Takens, On the nature of turbulence, Commun. Math. Phys., 20 (1971), 167-192.  doi: 10.1007/BF01646553.

[27]

H. Saberi NikS. Effati and J. Saberi-Nadjafi, New ultimate bound sets and exponential finite-time synchronization for the complex Lorenz system, J. Complexity, 31 (2015), 715-730.  doi: 10.1016/j.jco.2015.03.001.

[28]

R. SaravananO. NarayanK. Banerjee and J. K. Bhattacharjee, Chaos in a periodically forced Lorenz system, Phys. Rev. A, 31 (1985), 520-522.  doi: 10.1103/PhysRevA.31.520.

[29]

E. A. Sataev, Non-existence of stable trajectories in non-autonomous perturbations of systems of Lorenz type, Mat. Sb., 196 (2005), 99-134.  doi: 10.1070/SM2005v196n04ABEH000892.

[30]

P. Sooraksa and G. Chen, Chen system as a controlled weather model –physical principle, engineering design and real applications, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 28 (2018), 1830009, 12 pp. doi: 10.1142/S0218127418300094.

[31] C. Sparrow, The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors, New York, Oxford, 1982. 
[32] A. Tsinober, The Essence of Turbulence as a Physical Phenomenon with Emphasis on Issues of Paradigmatic Nature, Springer Nature Switzerland AG, 2019.  doi: 10.1007/978-3-319-99531-1.
[33]

V. Yu. Toronov and V. L. Derbov, Boundedness of attractors in the complex Lorenz model, Phys. Rev. E, 55 (1997), 3689-3692.  doi: 10.1103/PhysRevE.55.3689.

[34]

F. ZhangX. LiaoC. MuG. Zhang and Y.-A. Chen, On global boundedness of the Chen system, Discrete Contin. Dyna. Syst.-B, 22 (2017), 1673-1681.  doi: 10.3934/dcdsb.2017080.

[35]

F. ZhangX. LiaoG. ZhangC. MuM. Xiao and P. Zhou, Dynamical behaviors of a generalized Lorenz system, Discrete Contin. Dyna. Syst.-B, 22 (2017), 3707-3720.  doi: 10.3934/dcdsb.2017184.

[36]

F. Zhang and G. Zhang, Boundedness solutions of the complex Lorenz chaotic system, Appl. Math. Comput., 243 (2014), 12-23.  doi: 10.1016/j.amc.2014.05.102.

[37]

X. Zhang, Dynamics of a class of nonautonomous Lorenz-type systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 26 (2016), 1650208, 12 pp. doi: 10.1142/S0218127416502084.

[38]

X. Zhang, Dynamics of a class of fractional-order nonautonomous Lorenz-type systems, Chaos, 27 (2017), 041104, 7 pp. doi: 10.1063/1.4981909.

[39]

X. Zhang, Boundedness of a class of complex Lorenz systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 31 (2021), 2150101, 22 pp. doi: 10.1142/S0218127421501017.

[40]

Q. ZhaoS. Zhou and X. Li, Synchronization slaved by partial-states in lattices of non-autonomous coupled Lorenz equation, Commun. Nonlinear Sci. Numer. Simul., 13 (2008), 928-938.  doi: 10.1016/j.cnsns.2006.09.001.

Table 1.   
$ \widetilde{Z_i} $ $ U_i $ $ U_i-\tfrac{1}{2}\big(\tfrac{\Re\,B-1}{C}- \widetilde{Z_i}\tfrac{\Re\,B}{AC}\big)^2 $
$ i=0 $ $ -\tfrac{37}{12} $ $ \tfrac{5013121}{352800}\approx14.2095 $
$ i=1 $ $ -3.85753 $ $ 14.6399 $ $ 14.2579 $
$ i=2 $ $ -4.73162 $ $ 15.0607 $ $ 14.5737 $
$ i=3 $ $ -5.71849 $ $ 15.4508 $ $ 14.8301 $
$ i=4 $ $ -6.83271 $ $ 15.7805 $ $ 14.9893 $
$ i=5 $ $ -8.09069 $ $ 16.0086 $ $ 14.9999 $
$ i=6 $ $ -9.51099 $ $ 16.0782 $ $ 14.7925 $
$ i=7 $ $ -11.1146 $ $ 15.9126 $ $ 14.2737 $
$ i=8 $ $ -12.925 $ $ 15.4081 $ $ 13.3189 $
$ i=9 $ $ -14.9691 $ $ 14.426 $ $ 11.7629 $
$ i=10 $ $ -17.277 $ $ 12.7821 $ $ 9.38745 $
$ i=11 $ $ -19.8826 $ $ 10.2326 $ $ 5.90533 $
$ i=12 $ $ -22.8245 $ $ 6.45598 $ $ 0.940016 $
$ i=13 $ $ -26.1459 $ $ 1.03054 $ $ -6.00074 $
$ \widetilde{Z_i} $ $ U_i $ $ U_i-\tfrac{1}{2}\big(\tfrac{\Re\,B-1}{C}- \widetilde{Z_i}\tfrac{\Re\,B}{AC}\big)^2 $
$ i=0 $ $ -\tfrac{37}{12} $ $ \tfrac{5013121}{352800}\approx14.2095 $
$ i=1 $ $ -3.85753 $ $ 14.6399 $ $ 14.2579 $
$ i=2 $ $ -4.73162 $ $ 15.0607 $ $ 14.5737 $
$ i=3 $ $ -5.71849 $ $ 15.4508 $ $ 14.8301 $
$ i=4 $ $ -6.83271 $ $ 15.7805 $ $ 14.9893 $
$ i=5 $ $ -8.09069 $ $ 16.0086 $ $ 14.9999 $
$ i=6 $ $ -9.51099 $ $ 16.0782 $ $ 14.7925 $
$ i=7 $ $ -11.1146 $ $ 15.9126 $ $ 14.2737 $
$ i=8 $ $ -12.925 $ $ 15.4081 $ $ 13.3189 $
$ i=9 $ $ -14.9691 $ $ 14.426 $ $ 11.7629 $
$ i=10 $ $ -17.277 $ $ 12.7821 $ $ 9.38745 $
$ i=11 $ $ -19.8826 $ $ 10.2326 $ $ 5.90533 $
$ i=12 $ $ -22.8245 $ $ 6.45598 $ $ 0.940016 $
$ i=13 $ $ -26.1459 $ $ 1.03054 $ $ -6.00074 $
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