# 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 & Continuous Dynamical Systems - B, 2017, 22 (10) : 3707-3720. doi: 10.3934/dcdsb.2017184
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