# American Institute of Mathematical Sciences

June  2015, 20(4): 1261-1276. doi: 10.3934/dcdsb.2015.20.1261

## 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, China 2 College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China 3 College of Mathematics Science, Chongqing Normal University, Chongqing 400047, China 4 College of Mathematics and Physics, Chongqing University of Posts, and Telecommunications, Chongqing 400065, China

Received  October 2013 Revised  August 2014 Published  February 2015

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.
Citation: Fuchen Zhang, Chunlai Mu, Shouming Zhou, Pan Zheng. New results of the ultimate bound on the trajectories of the family of the Lorenz systems. Discrete and Continuous Dynamical Systems - B, 2015, 20 (4) : 1261-1276. doi: 10.3934/dcdsb.2015.20.1261
##### References:
 [1] V. O. Bragin, V. I. Vagaitsev, N. V. Kuznetsov and G. A. 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] T. Gheorghe and C. Dana, Heteroclinic orbits in the T and the Lu systems, Chaos Solitons Fractals, 42 (2009), 20-23. doi: 10.1016/j.chaos.2008.10.024. [3] T. Gheorghe and O. Dumitru, Analysis of a 3D chaotic system, Chaos Solitons Fractals, 36 (2008), 1315-1319. doi: 10.1016/j.chaos.2006.07.052. [4] B. Jiang, X. J. Han and Q. S. Bi, Hopf bifurcation analysis in the T system, Nonlinear Anal., Real World Appl., 11 (2010), 522-527. doi: 10.1016/j.nonrwa.2009.01.007. [5] G. A. Leonov, Lyapunov dimension formulas for Henon and Lorenz attractors, St Petersburg Math. J., 13 (2001), 155-170. [6] G. A. Leonov, Bound for attractors and the existence of homoclinic orbit in the Lorenz system, J. Appl. Math. Mech., 65 (2001), 19-32. doi: 10.1016/S0021-8928(01)00004-1. [7] G. A. Leonov, Localization of the attractors of the non-autonomous Lienard equation by the method of discontinuous comparison systems, J. Appl. Maths Mechs, 60 (1996), 329-332. doi: 10.1016/0021-8928(96)00042-1. [8] G. A. Leonov, A. I. Bunin and N. Koksch, Attractor localization of the Lorenz system, Z.Angew. Math. Mech., 67 (1987), 649-656. doi: 10.1002/zamm.19870671215. [9] X. F. Li, Y. D. Chu, J. G. Zhang and Y. X. Chang, Nonlinear dynamics and circuit implementation for a new Lorenz-like attractor, Chaos Solitons Fractals, 41 (2009), 2360-2370. doi: 10.1016/j.chaos.2008.09.011. [10] X. X. Liao, Y. L. Fu and S. L. 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, Sci. China Ser.F Inform. Sci., 48 (2005), 304-321. doi: 10.1360/04yf0087. [11] G. A. Leonov and N. V. Kuznetsov, Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 23 (2013), 1330002, 69pp. doi: 10.1142/S0218127413300024. [12] G. A. Leonov, N. V. Kuznetsov and V. I. Vagaitsev, Hidden attractor in smooth Chua systems, Physica D, 241 (2012), 1482-1486. doi: 10.1016/j.physd.2012.05.016. [13] G. A. Leonov, N. V. Kuznetsov and V. I. Vagaitsev, Localization of hidden Chua's attractors, Phys. Lett. A, 375 (2011), 2230-2233. doi: 10.1016/j.physleta.2011.04.037. [14] L. Liu, C. X. Liu and Y. B. Zhang, Experimental confirmation of a modified Lorenz system, Chinese Physics Letters, 24 (2007), 2756-2758. [15] G. A. Leonov, D. V. Ponomarenko and V. B. Smirnova, Frequency-Domain Methods for Nonlinear Analysis, Theory and applications. World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, 9. World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/9789812798695. [16] G. A. Leonov, D. V. Ponomarenko and V. B. Smirnova, Local instability and Localization of attractors. From stochastic generator to Chua's systems, Acta Appl. Math., 40 (1995), 179-243. doi: 10.1007/BF00992721. [17] Y. J. Liu and Q. G. Yang, Dynamics of a new Lorenz-like chaotic system, Nonlinear Anal., Real World Appl., 11 (2010), 2563-2572. doi: 10.1016/j.nonrwa.2009.09.001. [18] A. Y. 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. [19] A. Y. Pogromsky and H. Nijmeijer, On estimates of the Hausdorff dimension of invariant compact sets, Nonlinearity, 13 (2000), 927-945. doi: 10.1088/0951-7715/13/3/324. [20] P. Yu and X. X. Liao, Globally attractive and positive invariant set of the Lorenz system, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 16 (2006), 757-764. doi: 10.1142/S0218127406015143. [21] Q. G. Yang and Y. J. Liu, A hyperchaotic system from a chaotic system with one saddle and two stable node-foci, J. Math. Appl., 360 (2009), 293-306. doi: 10.1016/j.jmaa.2009.06.051. [22] F. C. Zhang, Y. L. Shu and H. L. Yang, Bounds for a new chaotic system and its application in chaos synchronization, Commun. Nonlin. Sci. Numer. Simulat., 16 (2011), 1501-1508. doi: 10.1016/j.cnsns.2010.05.032.

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##### References:
 [1] V. O. Bragin, V. I. Vagaitsev, N. V. Kuznetsov and G. A. 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] T. Gheorghe and C. Dana, Heteroclinic orbits in the T and the Lu systems, Chaos Solitons Fractals, 42 (2009), 20-23. doi: 10.1016/j.chaos.2008.10.024. [3] T. Gheorghe and O. Dumitru, Analysis of a 3D chaotic system, Chaos Solitons Fractals, 36 (2008), 1315-1319. doi: 10.1016/j.chaos.2006.07.052. [4] B. Jiang, X. J. Han and Q. S. Bi, Hopf bifurcation analysis in the T system, Nonlinear Anal., Real World Appl., 11 (2010), 522-527. doi: 10.1016/j.nonrwa.2009.01.007. [5] G. A. Leonov, Lyapunov dimension formulas for Henon and Lorenz attractors, St Petersburg Math. J., 13 (2001), 155-170. [6] G. A. Leonov, Bound for attractors and the existence of homoclinic orbit in the Lorenz system, J. Appl. Math. Mech., 65 (2001), 19-32. doi: 10.1016/S0021-8928(01)00004-1. [7] G. A. Leonov, Localization of the attractors of the non-autonomous Lienard equation by the method of discontinuous comparison systems, J. Appl. Maths Mechs, 60 (1996), 329-332. doi: 10.1016/0021-8928(96)00042-1. [8] G. A. Leonov, A. I. Bunin and N. Koksch, Attractor localization of the Lorenz system, Z.Angew. Math. Mech., 67 (1987), 649-656. doi: 10.1002/zamm.19870671215. [9] X. F. Li, Y. D. Chu, J. G. Zhang and Y. X. Chang, Nonlinear dynamics and circuit implementation for a new Lorenz-like attractor, Chaos Solitons Fractals, 41 (2009), 2360-2370. doi: 10.1016/j.chaos.2008.09.011. [10] X. X. Liao, Y. L. Fu and S. L. 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, Sci. China Ser.F Inform. Sci., 48 (2005), 304-321. doi: 10.1360/04yf0087. [11] G. A. Leonov and N. V. Kuznetsov, Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 23 (2013), 1330002, 69pp. doi: 10.1142/S0218127413300024. [12] G. A. Leonov, N. V. Kuznetsov and V. I. Vagaitsev, Hidden attractor in smooth Chua systems, Physica D, 241 (2012), 1482-1486. doi: 10.1016/j.physd.2012.05.016. [13] G. A. Leonov, N. V. Kuznetsov and V. I. Vagaitsev, Localization of hidden Chua's attractors, Phys. Lett. A, 375 (2011), 2230-2233. doi: 10.1016/j.physleta.2011.04.037. [14] L. Liu, C. X. Liu and Y. B. Zhang, Experimental confirmation of a modified Lorenz system, Chinese Physics Letters, 24 (2007), 2756-2758. [15] G. A. Leonov, D. V. Ponomarenko and V. B. Smirnova, Frequency-Domain Methods for Nonlinear Analysis, Theory and applications. World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, 9. World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/9789812798695. [16] G. A. Leonov, D. V. Ponomarenko and V. B. Smirnova, Local instability and Localization of attractors. From stochastic generator to Chua's systems, Acta Appl. Math., 40 (1995), 179-243. doi: 10.1007/BF00992721. [17] Y. J. Liu and Q. G. Yang, Dynamics of a new Lorenz-like chaotic system, Nonlinear Anal., Real World Appl., 11 (2010), 2563-2572. doi: 10.1016/j.nonrwa.2009.09.001. [18] A. Y. 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. [19] A. Y. Pogromsky and H. Nijmeijer, On estimates of the Hausdorff dimension of invariant compact sets, Nonlinearity, 13 (2000), 927-945. doi: 10.1088/0951-7715/13/3/324. [20] P. Yu and X. X. Liao, Globally attractive and positive invariant set of the Lorenz system, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 16 (2006), 757-764. doi: 10.1142/S0218127406015143. [21] Q. G. Yang and Y. J. Liu, A hyperchaotic system from a chaotic system with one saddle and two stable node-foci, J. Math. Appl., 360 (2009), 293-306. doi: 10.1016/j.jmaa.2009.06.051. [22] F. C. Zhang, Y. L. Shu and H. L. Yang, Bounds for a new chaotic system and its application in chaos synchronization, Commun. Nonlin. Sci. Numer. Simulat., 16 (2011), 1501-1508. doi: 10.1016/j.cnsns.2010.05.032.
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