New results of the ultimate bound on the trajectories of the family of the Lorenz systems

Fuchen Zhang; Chunlai Mu; Shouming Zhou; Pan Zheng

doi:10.3934/dcdsb.2015.20.1261

Article Contents

2015, Volume 20, Issue 4: 1261-1276. Doi: 10.3934/dcdsb.2015.20.1261

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New results of the ultimate bound on the trajectories of the family of the Lorenz systems

1.
College of Mathematics and Statistics, Chongqing Technology and Business, University, Chongqing 400067
2.
College of Mathematics and Statistics, Chongqing University, Chongqing 401331
3.
College of Mathematics Science, Chongqing Normal University, Chongqing 400047
4.
College of Mathematics and Physics, Chongqing University of Posts, and Telecommunications, Chongqing 400065

Received: October 2013

Revised: August 2014

Published: June 2015

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Abstract

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.

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Mathematics Subject Classification: Primary: 37B25; Secondary: 45M10, 11D41, 26A18, 37A60.

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