Skip Navigation Links
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu


Exact Lyapunov Dimension of Attractors and Convergence for the Lorenz System

Journal of Applied Nonlinear Dynamics 7(3) (2018) 319--327 | DOI:10.5890/JAND.2018.09.009

G. A. Leonov

Department of Applied Cybernetics, Saint-Petersburg State University, Universitetsky pr. 28, Saint-Petersburg, Russia, 198504

Download Full Text PDF

 

Abstract

Lyapunov dimension formula for Lorenz system is obtained. The methods of attractors localization and Lyapunov functions are developed.

Acknowledgments

This work was supported by Russian Science Foundation project 14-21-00041.

References

  1. [1]  Lorenz, E. (1963), Deterministic nonperiodic flow, Journal of Atmos. Sci., 20, 130-141.
  2. [2]  Smith, R. (1986), Some application of Hausdorff dimension inequalities for ordinary differential equation, Proc. Soc. Edinb., 104, 3045-3050.
  3. [3]  Doering, C. and Gibbon, J. (1995), On the shape and dimension of the Lorenz attractor, Dynamics and Stability of Systems, 10(3), 255-268.
  4. [4]  Eden, A. (1990), Local estimates for the Hausdorff dimension of an attractor, J. Math. Anal. Appl., 150(1), 100-119.
  5. [5]  Eden, A., Foias, C. and Temam, R. (1991), Local and global Lyapunov exponents, J. Dyn. Differ. Equ., 3(1), 133-177.
  6. [6]  Leonov, G.A. (2002), Lyapunov dimension formulas for Henon and Lorenz attractors, St. Petersburg Math. J., 13, 453-464.
  7. [7]  Leonov, G.A. (2008), Strange Attractors and Classical Stability Theory, St. Petersburg University Press: St. Petersburg.
  8. [8]  Leonov, G.A. (2012), Lyapunov functions in the attractors dimension theory, Appl. Math. and Mech., 76, 129-141.
  9. [9]  Leonov, G.A. (2013), Formulas for the Lyapunov dimension of attractors of the generalized Lorenz system, Doklady mathematics, 450(1), 13-18.
  10. [10]  Leonov, G.A., Kuznetsov, N.V., Korzhemanova, N.A., and Kusakin, D.V. (2016), Lyapunov dimension formula for the global attractor of the Lorenz system, Communications in Nonlinear Science and Numerical Simulation, 41, 84-103.
  11. [11]  Leonov, G.A., Pogromsky, A.Y., and Starkov, D.V. (2011), Dimension formula for the Lorenz attractor, Phys. Lett. A, 375(8), 1179-1182.
  12. [12]  Leonov, G.A. (2016), Lyapunov dimension formulas for Lorenz-Like Systems, International Journal of Bifurcations and Chaos, 26, 1650240 (7pages).
  13. [13]  Ladyzhenskaya, O.A. (1987), Determination of Minimal Global Attractors for the Navier-Stokes Equations and other Particl Differential Equations, Russian Mathematical Surveys, 42, 25-60.
  14. [14]  Kaplan, J. and Yorke, J. (1979) Chaotic behavior of multidimensional difference equations, Functional Differential Equations and Approximations of Fixed Points, Springer, Berlin (eds. H. Peitgen and H. Walter), 204-227.
  15. [15]  Kuznetsov, N.V., Alexeeva, T.A., and Leonov, G.A. (2016), Invariance of Lyapunov exponents and Lyapunov dimension for regular and irregular linearizations, Nonlinear Dynamics, 85(1), 195-201.
  16. [16]  Leonov, G.A., Bunin, A.I., and Kokch, N. (1987), Attraktor lokalisiering des Lorenz-systems, Zeitschrift angewandte Mathemat and Mechanik, 67, 649-656.
  17. [17]  Boichenko, V.A., Leonov, G.A., and Reitmann, V. (2005), Dimension Theory for Ordinary Differential Equations, Teubner: Stuttgart
  18. [18]  Leonov, G.A. (2014), Fishing principle for homoclinic and heteroclinic trajectories, Nonlinear Dynamics, 78(3), 2751-2758.