DCDS-B
Dynamical behaviors of a generalized Lorenz family
Fuchen Zhang Xiaofeng Liao Guangyun Zhang Chunlai Mu Min Xiao Ping Zhou
Discrete & Continuous Dynamical Systems - B 2017, 22(10): 3707-3720 doi: 10.3934/dcdsb.2017184

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 $0 ≤ α < \frac{1}{{29}}.$

keywords: Generalized Lorenz system Lyapunov-like functions ultimate boundedness global attractive sets
DCDS-B
New results of the ultimate bound on the trajectories of the family of the Lorenz systems
Fuchen Zhang Chunlai Mu Shouming Zhou Pan Zheng
Discrete & Continuous Dynamical Systems - B 2015, 20(4): 1261-1276 doi: 10.3934/dcdsb.2015.20.1261
In this paper, the global exponential attractive sets of a class of continuous-time dynamical systems defined by $\dot x = f\left( x \right),{\kern 1pt} {\kern 1pt} {\kern 1pt} x \in {R^3},$ are studied. The elements of main diagonal of matrix $A$ are both negative numbers and zero, where matrix $A$ is the Jacobian matrix $\frac{{df}}{{dx}}$ of a continuous-time dynamical system defined by $\dot x = f\left( x \right),{\kern 1pt} {\kern 1pt} {\kern 1pt} x \in {R^3},$ evaluated at the origin ${x_0} = \left( {0,0,0} \right).$ The former equations [1-6] that we are searching for a global bounded region have a common characteristic: The elements of main diagonal of matrix $A$ are all negative, where matrix $A$ is the Jacobian matrix $\frac{{df}}{{dx}}$ of a continuous-time dynamical system defined by $\dot x = f\left( x \right),{\kern 1pt} {\kern 1pt} {\kern 1pt} x \in {R^n},$ evaluated at the origin ${x_0} = {\left( {0,0, \cdots ,0} \right)_{1 \times n}}.$ For the reason that the elements of main diagonal of matrix $A$ are both negative numbers and zero for this class of dynamical systems , the method for constructing the Lyapunov functions that applied to the former dynamical systems does not work for this class of dynamical systems. We overcome this difficulty by adding a cross term $xy$ to the Lyapunov functions of this class of dynamical systems and get a perfect result through many integral inequalities and the generalized Lyapunov functions.
keywords: global attractive set Lyapunov stability generalized Lyapunov functions. Dynamical systems
CPAA
Blow-up for a non-local diffusion equation with exponential reaction term and Neumann boundary condition
Shouming Zhou Chunlai Mu Yongsheng Mi Fuchen Zhang
Communications on Pure & Applied Analysis 2013, 12(6): 2935-2946 doi: 10.3934/cpaa.2013.12.2935
This paper deals with the blow-up for a non-local diffusion equation with exponential reaction term and Neumann boundary condition. The local existence and uniqueness of the solution are obtained. Furthermore, we prove that the solution of the equation blows up in finite time. Under appropriate hypotheses, we give the estimates of the blow-up rate, and obtain that the blow-up set is a single point $x=0$ for radially symmetric solution with a single maximum at the origin. Finally, some numerical experiments are performed, which illustrate our results.
keywords: Nonlocal diffusion equation blow-up single point blow-up. blow-up rates
DCDS-B
On global boundedness of the Chen system
Fuchen Zhang Xiaofeng Liao Chunlai Mu Guangyun Zhang Yi-An Chen
Discrete & Continuous Dynamical Systems - B 2017, 22(4): 1673-1681 doi: 10.3934/dcdsb.2017080

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.

keywords: Chen system Lyapunov stability attractor global exponential attractive sets

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