Advanced Search
Article Contents
Article Contents

Asymptotic behaviour of a non-autonomous Lorenz-84 system

Abstract Related Papers Cited by
  • The so called Lorenz-84 model has been used in climatological studies, for example by coupling it with a low-dimensional model for ocean dynamics. The behaviour of this model has been studied extensively since its introduction by Lorenz in 1984. In this paper we study the asymptotic behaviour of a non-autonomous Lorenz-84 version with several types of non-autonomous features. We prove the existence of pullback and uniform attractors for the process associated to this model. In particular we consider that the non-autonomous forcing terms are more general than almost periodic. Finally, we estimate the Hausdorff dimension of the pullback attractor. We illustrate some examples of pullback attractors by numerical simulations.
    Mathematics Subject Classification: 37B55, 35B41, 37L30.


    \begin{equation} \\ \end{equation}
  • [1]

    V. A. Boichenko and G. A. Leonov, The Hausdorff dimension of attractors of the Lorenz system, Differentsial'nye Uravneniya, 25 (1989), 1999-2000.


    V. A. Boichenko, G. A. Leonov and V. Reitmann, Dimension Theory for Ordinary Differential Equations, Vieweg-Teubner, Wiesbaden, 2005.


    H. Broer, C. Simó and R. Vitolo, Bifurcations and strange attractors in the Lorenz-84 climate model with seasonal forcing, Nonlinearity, 15 (2002), 1205-1267.doi: 10.1088/0951-7715/15/4/312.


    T. Caraballo, J. A. Langa and J. C. Robinson, Attractors for differential equations with variable delays, Journal of Mathematical Analysis and Applications, 260 (2001), 421-438.doi: 10.1006/jmaa.2000.7464.


    T. Caraballo, G. Lukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Analysis, 64 (2006), 484-498.doi: 10.1016/j.na.2005.03.111.


    T. Caraballo, G. Lukaszewicz and J. Real, Pullback attractors for non-autonomous 2D Navier-Stokes equations in unbounded domains, C. R. Math. Acad. Sci. Paris, 342 (2006), 263-268.doi: 10.1016/j.crma.2005.12.015.


    A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, 182, Springer, New York, 2013.doi: 10.1007/978-1-4614-4581-4.


    V. V. Chepyzhov and M. I. Vishik, Attractors of nonautonomous dynamical systems and their dimension, J. Math. Pures Appl., 73 (1994), 279-333.


    V. V. Chepyzhov and M. I. Vishik, Attractors of periodic processes and estimates of their dimension, Mathematical Notes, 57 (1995), 127-140.doi: 10.1007/BF02309145.


    V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Colloquium Publications, 49, 2002.doi: 10.1070/RM2013v068n02ABEH004832.


    H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dyn. Differential Equations, 9 (1997), 341-397.doi: 10.1007/BF02219225.


    A. Haraux, Attractors of asymptotically compact processes and applications to nonlinear partial differential equations, Comm. Partial Differential Equations, 13 (1988), 1383-1414.doi: 10.1080/03605308808820580.


    W. Hurewicz and H. Wallman, Dimension Theory, Princeton University Press, 1941.


    P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, 176, American Mathematical Society, Province, RI, 2011.


    P. E. Kloeden and H. M. Rodrigues, Dynamics of a class of ODEs more general than almost periodic, Nonlinear Analysis, 74 (2011), 2695-2719.doi: 10.1016/j.na.2010.12.025.


    P. E. Kloeden and B. Schmalfuss, Nonautonomous systems, cocycle attractors and variable time-step discretization, Numer. Algorithms, 14 (1997), 141-152.doi: 10.1023/A:1019156812251.


    P. E. Kloeden and B. Schmalfuss, Asymptotic behaviour of non-autonomous difference inclusions, Systems Control Lett., 33 (1998), 275-280.doi: 10.1016/S0167-6911(97)00107-2.


    P. E. Kloeden and D. J. Stonier, Cocycle attractors in nonautonomously perturbed differential equations, Dynamics of Continuous, Discrete and Impulsive Systems, 4 (1998), 211-226.


    G. A. Leonov and V. A. Boichenko, Lyapunov's direct method in the estimation of the Hausdorff dimension of attractors, Acta Appl. Math., 26 (1992), 1-60.doi: 10.1007/BF00046607.


    G. A. Leonov, Formulas for the Lyapunov dimension of Hénon and Lorenz attractors, Algebra Anal., 13 (2001), 155-170.


    G. A. Leonov, Strange Attractors and Classical Stability Theory, St. Petersburg Univ. Press, St. Petersburg, 2008.


    G. A. Leonov, V. Reitmann and A. S. Slepukhin, Upper estimates for the Hausdorff dimension of negatively invariant sets of local cocycles, Doklady Mathematics, 84 (2011), 551-554.doi: 10.1134/S1064562411050103.


    E. N. Lorenz, Deterministic nonperiodic flow, Journal of the atmospheric sciences, 20 (1963), 130-141.doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.


    E. N. Lorenz, Irregularity: A fundamental property of the atmosphere, Tellus, 36A (1984), 98-110.doi: 10.1111/j.1600-0870.1984.tb00230.x.


    E. N. Lorenz, Can chaos and intransitivity lead to interannual variability?, Tellus, 42A (1990), 378-389.doi: 10.1034/j.1600-0870.1990.t01-2-00005.x.


    C. Masoller, A. C. Sicardi Schifino and L. Romanelli, Regular and chaotic behavior in the new Lorenz system, Physics Letters A, 167 (1992), 185-190.doi: 10.1016/0375-9601(92)90226-C.


    C. Masoller, A. C. Schifino and L. Romanelli, Characterization of strange attractors of Lorenz model of general circulation of the atmosphere, Chaos, Solitons & Fractals, 6 (1995), 357-366.doi: 10.1016/0960-0779(95)80041-E.


    B. Schmalfuss, Attractors for the non-autonomous dynamical systems, in Proceedings of Equadiff 99 (eds. B. Fiedler, K. Gröger and J. Sprekels), Berlin, Singapore World Scientific, Singapore, 2000, 684-689.


    A. Shil'nikov, G. Nicolis and C. Nicolis, Bifurcation and predictability analysis of a low-order atmospheric circulation model, Int. J. Bifur. Chaos, 5 (1995), 1701-1711.doi: 10.1142/S0218127495001253.


    A. Sicardi and C. Masoller, Analytical study of the codimension two bifurcation of the new Lorenz system, Instabilities and Nonequilibrium Structures, V (1996), 345-348.doi: 10.1007/978-94-009-0239-8_32.


    H. L. Swinney and J. P. Gollub, Characterization of hydrodynamic strange attractors, Physica D, 18 (1986), 448-454.doi: 10.1016/0167-2789(86)90213-7.


    R. Temam, Infinite-Dimensional Dynamical System in Mechanics and Physics, Springer-Verlag, New York, 1988.doi: 10.1007/978-1-4684-0313-8.

  • 加载中

Article Metrics

HTML views() PDF downloads(76) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint