# American Institute of Mathematical Sciences

December  2017, 22(10): 3707-3720. doi: 10.3934/dcdsb.2017184

## Dynamical behaviors of a generalized Lorenz family

 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 Automation, Nanjing University of Posts and Telecommunications, Nanjing 210003, China 6 Key Laboratory of Network Control and Intelligent Instrument of Ministry of Education, Chongqing University of Posts and Telecommunications, Chongqing 400065, China

Received  September 2016 Revised  June 2017 Published  July 2017

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 $0 ≤ α < \frac{1}{{29}}.$ The ultimate bound set and globally exponential attractive set of this generalized Lorenz system are still unknown for $\alpha \notin \left[ {0, \frac{1}{{29}}} \right).$ 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 $\frac{1}{{29}} ≤ α < \frac{{14}}{{173}}.$ 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 $0 ≤ α < \frac{1}{{29}}.$

Citation: Fuchen Zhang, Xiaofeng Liao, Guangyun Zhang, Chunlai Mu, Min Xiao, Ping Zhou. Dynamical behaviors of a generalized Lorenz family. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3707-3720. doi: 10.3934/dcdsb.2017184
##### References:
 [1] V. Bragin, V. Vagaitsev, N. Kuznetsov and G. Leonov, Algorithms for finding hidden oscillations 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. [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. [4] T. Huang, G. Chen and J. Kurths, Synchronization of chaotic systems with time-varying coupling delays, Discrete Continuous Dyn. Syst. Ser. B., 16 (2011), 1071-1082.  doi: 10.3934/dcdsb.2011.16.1071. [5] N. Kuznetsov, T. 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. [6] E. Lorenz, Deterministic non-periods flows, J. Atmos. Sci., 20 (1963), 130-141. [7] 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. [8] 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. [9] G. 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. [10] G. Leonov, The Tricomi problem for the Shimizu-Morioka dynamical system, Dokl. Math., 86 (2012), 850-853.  doi: 10.1134/S1064562412060324. [11] 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. [12] G. Leonov, A. Bunin and N. Koksch, Attractor localization of the Lorenz system, Z. Angew. Math. Mech., 67 (1987), 649-656.  doi: 10.1002/zamm.19870671215. [13] Lü J. and G. Chen, A new chaotic attractor coined, Int. J. Bifurc. Chaos Appl. Sci. Eng., 12 (2002), 659-661.  doi: 10.1142/S0218127402004620. [14] Lü J., G. Chen, D. Cheng and S. Celikovsky, Bridge the gap between the Lorenz system and the Chen system, Int. J. Bifurc. Chaos Appl. Sci. Eng., 12 (2002), 2917-2926.  doi: 10.1142/S021812740200631X. [15] X. Liao, Y. Fu and S. Xie, On the new results of global attractive set and positive invariant set of the Lorenz chaotic system and the applications to chaos control and synchronization, Science in China Series F: Information Sciences, 48 (2005), 304-321.  doi: 10.1360/04yf0087. [16] X. Liao, Y. Fu, S. Xi 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. [17] G. Leonov and N. 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. [18] G. Leonov and N. Kuznetsov, Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits, Int. J. Bifurc. Chaos Appl. Sci. Eng. , 23 (2013), 1330002, 69pp. doi: 10.1142/S0218127413300024. [19] G. Leonov, N. Kuznetsov, M. Kiseleva, E. Solovyeva and A. Zaretskiy, Hidden oscillations in mathematical model of drilling system actuated by induction motor with a wound rotor, Nonlinear Dyn., 77 (2014), 277-288.  doi: 10.1007/s11071-014-1292-6. [20] G. Leonov, N. 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. [21] G. Leonov, N. Kuznetsov and V. Vagaitsev, Hidden attractor in smooth Chua systems, Phys. D., 241 (2012), 1482-1486.  doi: 10.1016/j.physd.2012.05.016. [22] D. Li, J. Lu, X. 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. [23] X. Liao, P. Yu, S. 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. [24] A. Pogromsky, G. Santoboni and H. Nijmeijer, An ultimate bound on the trajectories of the Lorenz system and its applications, Nonlinearity, 16 (2003), 1597-1605.  doi: 10.1088/0951-7715/16/5/303. [25] P. Yu and X. Liao, Globally attractive and positive invariant set of the Lorenz system, Int. J. Bifurc. Chaos Appl. Sci. Eng., 16 (2006), 757-764.  doi: 10.1142/S0218127406015143. [26] P. Yu, X. Liao, S. Xie and Y. Fu, A constructive proof on the existence of globally exponentially attractive set and positive invariant set of general Lorenz family, Commun. Nonlinear Sci. Numer. Simul., 14 (2009), 2886-2896.  doi: 10.1016/j.cnsns.2008.10.008. [27] F. Zhang, C. Mu and X. Li, On the boundedness of some solutions of the Lü system, Int. J. Bifurc. Chaos Appl. Sci. Eng. , 22 (2012), 1250015, 5pp. doi: 10.1142/S0218127412500150. [28] F. Zhang, C. Mu, P. Zheng, D. 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. [29] 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. [30] 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.

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##### References:
 [1] V. Bragin, V. Vagaitsev, N. Kuznetsov and G. Leonov, Algorithms for finding hidden oscillations 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. [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. [4] T. Huang, G. Chen and J. Kurths, Synchronization of chaotic systems with time-varying coupling delays, Discrete Continuous Dyn. Syst. Ser. B., 16 (2011), 1071-1082.  doi: 10.3934/dcdsb.2011.16.1071. [5] N. Kuznetsov, T. 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. [6] E. Lorenz, Deterministic non-periods flows, J. Atmos. Sci., 20 (1963), 130-141. [7] 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. [8] 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. [9] G. 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. [10] G. Leonov, The Tricomi problem for the Shimizu-Morioka dynamical system, Dokl. Math., 86 (2012), 850-853.  doi: 10.1134/S1064562412060324. [11] 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. [12] G. Leonov, A. Bunin and N. Koksch, Attractor localization of the Lorenz system, Z. Angew. Math. Mech., 67 (1987), 649-656.  doi: 10.1002/zamm.19870671215. [13] Lü J. and G. Chen, A new chaotic attractor coined, Int. J. Bifurc. Chaos Appl. Sci. Eng., 12 (2002), 659-661.  doi: 10.1142/S0218127402004620. [14] Lü J., G. Chen, D. Cheng and S. Celikovsky, Bridge the gap between the Lorenz system and the Chen system, Int. J. Bifurc. Chaos Appl. Sci. Eng., 12 (2002), 2917-2926.  doi: 10.1142/S021812740200631X. [15] X. Liao, Y. Fu and S. Xie, On the new results of global attractive set and positive invariant set of the Lorenz chaotic system and the applications to chaos control and synchronization, Science in China Series F: Information Sciences, 48 (2005), 304-321.  doi: 10.1360/04yf0087. [16] X. Liao, Y. Fu, S. Xi 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. [17] G. Leonov and N. 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. [18] G. Leonov and N. Kuznetsov, Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits, Int. J. Bifurc. Chaos Appl. Sci. Eng. , 23 (2013), 1330002, 69pp. doi: 10.1142/S0218127413300024. [19] G. Leonov, N. Kuznetsov, M. Kiseleva, E. Solovyeva and A. Zaretskiy, Hidden oscillations in mathematical model of drilling system actuated by induction motor with a wound rotor, Nonlinear Dyn., 77 (2014), 277-288.  doi: 10.1007/s11071-014-1292-6. [20] G. Leonov, N. 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. [21] G. Leonov, N. Kuznetsov and V. Vagaitsev, Hidden attractor in smooth Chua systems, Phys. D., 241 (2012), 1482-1486.  doi: 10.1016/j.physd.2012.05.016. [22] D. Li, J. Lu, X. 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. [23] X. Liao, P. Yu, S. 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. [24] A. Pogromsky, G. Santoboni and H. Nijmeijer, An ultimate bound on the trajectories of the Lorenz system and its applications, Nonlinearity, 16 (2003), 1597-1605.  doi: 10.1088/0951-7715/16/5/303. [25] P. Yu and X. Liao, Globally attractive and positive invariant set of the Lorenz system, Int. J. Bifurc. Chaos Appl. Sci. Eng., 16 (2006), 757-764.  doi: 10.1142/S0218127406015143. [26] P. Yu, X. Liao, S. Xie and Y. Fu, A constructive proof on the existence of globally exponentially attractive set and positive invariant set of general Lorenz family, Commun. Nonlinear Sci. Numer. Simul., 14 (2009), 2886-2896.  doi: 10.1016/j.cnsns.2008.10.008. [27] F. Zhang, C. Mu and X. Li, On the boundedness of some solutions of the Lü system, Int. J. Bifurc. Chaos Appl. Sci. Eng. , 22 (2012), 1250015, 5pp. doi: 10.1142/S0218127412500150. [28] F. Zhang, C. Mu, P. Zheng, D. 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. [29] 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. [30] 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.
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