Determining the spectrum of the nonlinear local Lyapunov exponents in a multidimensional chaotic system

Barana, G., and I. Tsuda, 1993: A new method for computing Lyapunov exponents. Phys. Lett. A, 175, 421–427, doi: 10.1016/0375-9601(93)90994-B.

Björck, A., 1967: Solving linear least squares problems by Gram-Schmidt orthogonalization. BIT Numerical Mathematics, 7, 1–21, doi: 10.1007/BF01934122.

Björck, A., 1994: Numerics of Gram-Schmidt orthogonalization. Linear Algebra and its Applications, 197–198, 297–316, doi: 10.1016/0024-3795(94)90493-6.

Brown, R., P. Bryant, and H. D. I. Abarbanel, 1991: Computing the Lyapunov spectrum of a dynamical system from an observed time series. Phys. Rev. A, 43, 2787–2806, doi: 10.1103/Phys-RevA.43.2787.

Dalcher, A., and E. Kalnay, 1987: Error growth and predictability in operational ECMWF forecasts. Tellus A, 39, 474–491, doi: 10.3402/tellusa.v39i5.11774.

Ding, R. Q., and J. P. Li, 2007: Nonlinear finite-time Lyapunov exponent and predictability. Phys. Lett. A, 364, 396–400, doi: 10.1016/j.physleta.2006.11.094.

Ding, R. Q., J. P. Li, and K. J. Ha, 2008: Trends and interdecadal changes of weather predictability during 1950s–1990s. J. Geophys. Res., 113, D24112, doi: 10.1029/2008JD010404.

Ding, R. Q., J. P. Li, and K. H. Seo, 2010: Predictability of the Madden-Julian oscillation estimated using observational data. Mon. Wea. Rev., 138, 1004–1013, doi: 10.1175/2009MWR 3082.1.

Ding, R. Q., J. P. Li, and K. H. Seo, 2011: Estimate of the predictability of boreal summer and winter intraseasonal oscillations from observations. Mon. Wea. Rev., 139, 2421–2438, doi: 10.1175/2011MWR3571.1.

Ding, R. Q., P. J. Li, F. Zheng, J. Feng, and D. Q. Liu, 2016: Estimating the limit of decadal-scale climate predictability using observational data. Climate Dyn., 46, 1563–1580, doi: 10.1007/s00382-015-2662-6.

Duan, W. S., and M. Mu, 2009: Conditional nonlinear optimal perturbation: Applications to stability, sensitivity, and predictability. Science in China Series D: Earth Sciences, 52, 883–906, doi: 10.1007/s11430-009-0090-3.

Eckmann, J. P., and D. Ruelle, 1985: Ergodic theory of chaos and strange attractors. Reviews of Modern Physics, 57, 617–656, doi: 10.1103/RevModPhys.57.617.

Feng, G. L., and W. J. Dong, 2003: Evaluation of the applicability of a retrospective scheme based on comparison with several difference schemes. Chinese Physics, 12, 1076–1086, doi: 10.1088/1009-1963/12/10/307.

Fraedrich, K., 1987: Estimating weather and climate predictability on attractors. J. Atmos. Sci., 44, 722–728, doi: 10.1175/1520- 0469(1987)044〈0722:EWACPO〉2.0.CO;2.

Fraedrich, K., 1988: El Ni˜no/southern oscillation predictability. Mon. Wea. Rev., 116, 1001–1012, doi: 10.1175/1520-0493 (1988)116〈1001:ENOP〉2.0.CO;2.

Houtekamer, P. L., 1991: Variation of the predictability in a loworder spectral model of the atmospheric circulation. Tellus A, 43, 177–190, doi: 10.3402/tellusa.v43i3.11925.

Kalnay, E., and Z. Toth, 1995: The breeding method. Vol. I, Proceedings of the ECMWF seminar on predictability, 4–8 September 1995, ECMWF, Reading, UK, 69–82.

Kolmogorov, A. N., 1941: The local structure of turbulence in incompressible viscous fluid for very large Reynolds’ numbers. Doklady Akademiia Nauk SSSR, 30, 301–305.

Lacarra, J. F., and O. Talagrand, 1988: Short-range evolution of small perturbations in a barotropic model. Tellus A, 40, 81–95, doi: 10.1111/j.1600-0870.1988.tb00408.x.

Li, J. P., and R. Q. Ding, 2008: Temporal-spatial distributions of predictability limit of short-term climate. Chinese Journal of Atmospheric Sciences, 32, 975–986. (in Chinese)

Li, J. P., and R. Q. Ding, 2011: Temporal-spatial distribution of atmospheric predictability limit by local dynamical analogs. Mon. Wea. Rev., 139, 3265–3283, doi: 10.1175/MWR-D-10-05020.1.

Li, J. P., and R. Q. Ding, 2013: Temporal–spatial distribution of the predictability limit of monthly sea surface temperature in the global oceans. Int. J. Climatol., 33, 1936–1947, doi: 10.1002/joc.3562.

Li, Y. X., W. K. S. Tang, and G. R. Chen, 2005: Hyperchaos evolved from the generalized Lorenz equation. International Journal of Circuit Theory and Applications, 33, 235–251, doi: 10.1002/cta.318.

Lorenz, E. N., 1963: Deterministic nonperiodic flow. J. Atmos. Sci., 20, 130–141, doi: 10.1175/1520-0469(1963)020〈0130: DNF〉2.0.CO;2.

Lorenz, E. N., 1996: Predictability: A problem partly solved. Proc. Seminar on predictability. Vol.1, No.1. Reading, United Kingdom, ECMWF, 1–18.

Mu, M., and W. S. Duan, 2003: A new approach to studying ENSO predictability: Conditional nonlinear optimal perturbation. Chinese Science Bulletin, 48, 1045–1047, doi: 10.1007/BF 03184224.

Nese, J. M., 1989: Quantifying local predictability in phase space. Physica D: Nonlinear Phenomena, 35, 237–250, doi: 10.1016/0167-2789(89)90105-X.

Norwood, A., E. Kalnay, K. Ide, S. C. Yang, and C. Wolfe, 2013: Lyapunov, singular and bred vectors in a multi-scale system: An empirical exploration of vectors related to instabilities. Journal of Physics A: Mathematical and Theoretical, 46, 254021, doi: 10.1088/1751-8113/46/25/254021.

Toth, Z., and E. Kalnay, 1993: Ensemble forecasting at NMC: The generation of perturbations. Bull. Amer. Meteor. Soc., 74, 2317–2330, doi: 10.1175/1520-0477(1993)074〈2317: EFANTG〉2.0.CO;2.

Toth, Z., and E. Kalnay, 1997: Ensemble forecasting at NCEP and the breeding method. Mon. Wea. Rev., 125, 3297–3319, doi: 10.1175/1520-0493(1997)125〈3297:EFANAT〉2.0.CO;2.

Trevisan, A., and R. Legnani, 1995: Transient error growth and local predictability: A study in the Lorenz system. Tellus A, 47, 103–117, doi: 10.1034/j.1600-0870.1995.00006.x.

Wang, F. Q., and C. X. Liu, 2006: Synchronization of hyperchaotic Lorenz system based on passive control. Chinese Physics, 15, 1971, doi: 10.1088/1009-1963/15/9/012.

Wolf, A., J. B. Swift, H. L. Swinney, and J. A. Vastano, 1985: Determining Lyapunov exponents from a time series. Physica D: Nonlinear Phenomena, 16, 285–317, doi: 10.1016/0167-2789(85)90011-9.

Yoden, S., and M. Nomura, 1993: Finite-time Lyapunov stability analysis and its application to atmospheric predictability. J. Atmos. Sci., 50, 1531–1543, doi: 10.1175/1520-0469(1993) 050〈1531:FTLSAA〉2.0.CO;2.

Ziehmann, C., L. A. Smith, and J. Kurths, 2000: Localized Lyapunov exponents and the prediction of predictability. Physics Letters A, 271, 237–251, doi: 10.1016/S0375-9601(00)00336-4.