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October  2014, 34(10): 3901-3920. doi: 10.3934/dcds.2014.34.3901

Asymptotic behaviour of a non-autonomous Lorenz-84 system

María Anguiano 1, and  Tomás Caraballo 2,

1. 

Dpto. Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, Campus Reina Mercedes, Apdo. de Correos 1160, 41080-Sevilla
2. 

Dpto. Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, Campus Reina Mercedes, Apdo. de Correos 1160, 41080 Sevilla

Received  March 2013 Revised  April 2013 Published  April 2014

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.
Keywords: pullback attractor, Lorenz system, non-autonomous equation, Hausdorff dimensionn., uniform attractor.
Mathematics Subject Classification: 37B55, 35B41, 37L3.
Citation: María Anguiano, Tomás Caraballo. Asymptotic behaviour of a non-autonomous Lorenz-84 system. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 3901-3920. doi: 10.3934/dcds.2014.34.3901
References:
[1]

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

[2]

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

[3]

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.

[4]

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.

[5]

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.

[6]

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.

[7]

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.

[8]

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

[9]

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.

[10]

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

[11]

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

[12]

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.

[13]

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

[14]

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

[15]

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.

[16]

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.

[17]

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.

[18]

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.

[19]

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.

[20]

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

[21]

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

[22]

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.

[23]

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.

[24]

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

[25]

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.

[26]

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.

[27]

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.

[28]

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.

[29]

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.

[30]

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.

[31]

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.

[32]

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

show all references

References:
[1]

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

[2]

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

[3]

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.

[4]

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.

[5]

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.

[6]

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.

[7]

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.

[8]

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

[9]

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.

[10]

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

[11]

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

[12]

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.

[13]

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

[14]

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

[15]

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.

[16]

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.

[17]

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.

[18]

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.

[19]

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.

[20]

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

[21]

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

[22]

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.

[23]

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.

[24]

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

[25]

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.

[26]

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.

[27]

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.

[28]

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.

[29]

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.

[30]

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.

[31]

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.

[32]

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

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