Quantifying the predictability limits of chaotic systems and their forecast models has attracted much interest among scientists. The attractor radius (AR) and the global attractor radius (GAR), as intrinsic properties of a chaotic system, were introduced in the most recent work (Li et al. 2018). It has been shown that both the AR and GAR provide more accurate, objective metrics to access the global and local predictability limits of forecast models compared with the traditional error saturation or the asymptotic value. In this work, we consider the AR and GAR of fractional Lorenz systems, introduced in Grigorenko and Grigorenko [Phys. Rev. Lett. 91, 034101 (2003)] using the Caputo fractional derivatives and their application to the quantification of the predictability limits. A striking finding is that a fractional Lorenz system with smaller , which is a sum of the orders of all involved equal derivatives, has smaller attractor radius and shorter predictability limits. In addition, we present a new numerical algorithm for the fractional Lorenz system, which is the generalized version of the standard fourth-order Runge–Kutta scheme.
REFERENCES
1.
Amblard
, F.
, Maggs
, A.
, Yurke
, B.
, Pargellis
, A.
, and Leibler
, S.
, “Subdiffusion and anomalous local viscoelasticity in actin networks
,” Phys. Rev. Lett.
77
, 4470
–4473
(1996
). 2.
Bagley
, R. L.
and Calico
, R. A.
, “Fractional order state equations for the control of viscoelastically damped structures
,” J. Guid.
14
, 304
–311
(1991
). 3.
Bauer
, P.
, Thorpe
, A.
, and Brunet
, G.
, “The quiet revolution of numerical weather prediction
,” Nature
525
, 47
–55
(2015
). 4.
Benson
, D. A.
, Wheatcraft
, S. W.
, and Meerschaert
, M. M.
, “Application of a fractional advection-dispersion equation
,” Water Resour. Res.
36
, 1403
–1412
, https://doi.org/10.1029/2000WR900031 (2000a
). 5.
Benson
, D. A.
, Wheatcraft
, S. W.
, and Meerschaert
, M. M.
, “The fractional-order governing equation of Lévy motion
,” Water Resour. Res.
36
, 1413
–1423
, https://doi.org/10.1029/2000WR900032 (2000b
). 6.
Bouchaud
, J.-P.
and Georges
, A.
, “Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications
,” Phys. Rep.
195
, 127
–293
(1990
). 7.
Buizza
, R.
, “Horizontal resolution impact on short- and long-range forecast error
,” Q. J. R. Meteorol. Soc.
136
, 1020
–1035
(2010
). 8.
Charney
, J. G.
, “The feasibility of a global observation and analysis experiment
,” Bull. Amer. Meteor. Soc.
47
, 200
–230
(1966
).9.
Cuomo
, K. M.
and Oppenheim
, A. V.
, “Circuit implementation of synchronized chaos with applications to communications
,” Phys. Rev. Lett.
71
, 65
(1993
). 10.
Dalcher
, A.
and Kalnay
, E.
, “Error growth and predictability in operational ECMWF forecasts
,” Tellus A
39
, 474
–491
(1987
). 11.
Dentz
, M.
, Cortis
, A.
, Scher
, H.
, and Berkowitz
, B.
, “Time behavior of solute transport in heterogeneous media: Transition from anomalous to normal transport
,” Adv. Water Resour.
27
, 155
–173
(2004
). 12.
Diethelm
, K.
, Ford
, N. J.
, and Freed
, A. D.
, “A predictor-corrector approach for the numerical solution of fractional differential equations
,” Nonlinear Dyn.
29
, 3
–22
(2002
). 13.
Ding
, R.
and Li
, J.
, “Nonlinear finite-time Lyapunov exponent and predictability
,” Phys. Lett. A
364
, 396
–400
(2007
). 14.
Ding
, R.
and Li
, J.
, “Study on the regularity of predictability limit of chaotic systems with different initial errors
,” Acta. Phys. Sin.
57
, 7494
–7499
(2008
).15.
Ding
, R.
and Li
, J.
, “Relationships between the limit of predictability and initial error in the uncoupled and coupled Lorenz models
,” Adv. Atmos. Sci.
29
, 1078
–1088
(2012
). 16.
Dmowska
, R.
and Saltzman
, B.
, Advances in Geophysics: Long-range Persistence in Geophysical Time Series
(Academic Press
, 1999
), Vol. 40.17.
Doan
, T. S.
and Kloeden
, P. E.
, “Attractors of Caputo fractional differential equations with triangular vector fields
,” Fract. Calc. Appl. Anal.
25
, 720
–734
(2022
). 18.
Duan
, W.
and Zhao
, P.
, “Revealing the most disturbing tendency error of Zebiak–Cane model associated with El Niño predictions by nonlinear forcing singular vector approach
,” Clim. Dyn.
44
, 2351
–2367
(2015
). 19.
Eichner
, J. F.
, Koscielny-Bunde
, E.
, Bunde
, A.
, Havlin
, S.
, and Schellnhuber
, H. -J.
, “Power-law persistence and trends in the atmosphere: A detailed study of long temperature records
,” Phys. Rev. E
68
, 046133
(2003
). 20.
Goychuk
, I.
, Heinsalu
, E.
, Patriarca
, M.
, Schmid
, G.
, and Hänggi
, P.
, “Current and universal scaling in anomalous transport
,” Phys. Rev. E
73
, 020101
(2006
). 21.
Grigorenko
, I.
and Grigorenko
, E.
, “Chaotic dynamics of the fractional Lorenz system
,” Phys. Rev. Lett.
91
, 034101
(2003
). 22.
Ichise
, M.
, Nagayanagi
, Y.
, and Kojima
, T.
, “An analog simulation of non-integer order transfer functions for analysis of electrode processes
,” J. Electroanal. Chem.
33
, 253
–265
(1971
). 23.
Kalnay
, E.
, Atmospheric Modeling, Data Assimilation and Predictability
(Cambridge University Press
, 2003
).24.
Klages
, R.
, Radons
, G.
, and Sokolov
, I. M.
, Anomalous Transport: Foundations and Applications
(Wiley-VCH
, 2008
).25.
Koeller
, R.
, “Applications of fractional calculus to the theory of viscoelasticity
,” J. Appl. Mech.
51
, 299
–307
(1984
). 26.
Koscielny-Bunde
, E.
, Bunde
, A.
, Havlin
, S.
, Roman
, H. E.
, Goldreich
, Y.
, and Schellnhuber
, H.-J.
, “Indication of a universal persistence law governing atmospheric variability
,” Phys. Rev. Lett.
81
, 729
(1998
). 27.
Leith
, C.
, “Theoretical skill of Monte Carlo forecasts
,” Mon. Weather Rev.
102
, 409
–418
(1974
). 28.
Li
, J.
and Chou
, J.
, “The property of solutions for the equations of large-scale atmosphere with the non-stationary external forcings
,” China Sci. Bull.
41
, 587
–590
(1996
).29.
Li
, J.
and Chou
, J.
, “Existence of the atmosphere attractor
,” Sci. China, Ser. D: Earth Sci.
40
, 215
–220
(1997a
). 30.
Li
, J.
and Chou
, J.
, “Further study on the properties of operators of atmospheric equations and the existence of attractor
,” Acta. Meteorol. Sin.
11
, 216
–223
(1997b
).31.
Li
, J.
and Wang
, S.
, “Some mathematical and numerical issues in geophysical fluid dynamics and climate dynamics
,” Commun. Comput. Phys.
3
, 759
–793
(2008
).32.
Li
, J.
and Ding
, R.
, “Temporal–spatial distribution of atmospheric predictability limit by local dynamical analogs
,” Mon. Weather Rev.
139
, 3265
–3283
(2011
). 33.
Li
, J.
and Ding
, R.
, “Temporal–spatial distribution of the predictability limit of monthly sea surface temperature in the global oceans
,” Int. J. Climatol.
33
, 1936
–1947
(2013
). 34.
Li
, J.
, Feng
, J.
, and Ding
, R.
, “Attractor radius and global attractor radius and their application to the quantification of predictability limits
,” Clim. Dyn.
51
, 2359
–2374
(2018
). 35.
Li
, Y.
and Wang
, Y.
, “The existence and asymptotic behavior of solutions to fractional stochastic evolution equations with infinite delay
,” J. Differ. Equ.
266
, 3514
–3558
(2019
). 36.
Lorenz
, E. N.
, “Deterministic nonperiodic flow
,” J. Atmos. Sci.
20
, 130
–141
(1963
). 37.
Lorenz
, E. N.
, “A study of the predictability of a 28-variable atmospheric model
,” Tellus
17
, 321
–333
(1965
). 38.
Mainardi
, F.
, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models
(World Scientific Publishing Company
, 2010
).39.
Mandelbrot
, B.
, “Some noises with i/f spectrum, a bridge between direct current and white noise
,” IEEE Trans. Inform. Theory
13
, 289
–298
(1967
). 40.
Marks
, R.
and Hall
, M.
, “Differintegral interpolation from a bandlimited signal’s samples
,” IEEE Trans. Acoust. Speech Signal Process.
29
, 872
–877
(1981
). 41.
Meerschaert
, M. M.
and Scalas
, E.
, “Coupled continuous time random walks in finance
,” Phys. A
370
, 114
–118
(2006
). 42.
Mu
, M.
, Duan
, W.
, and Tang
, Y.
, “The predictability of atmospheric and oceanic motions: Retrospect and prospects
,” Sci. China Earth Sci.
60
, 2001
–2012
(2017
). 43.
Mu
, M.
, Duan
, W.
, and Wang
, B.
, “Conditional nonlinear optimal perturbation and its applications
,” Nonlinear Process. Geophys.
10
, 493
–501
(2003
). 44.
Mu
, M.
, Duan
, W.
, Wang
, Q.
, and Zhang
, R.
, “An extension of conditional nonlinear optimal perturbation approach and its applications
,” Nonlinear Process. Geophys.
17
, 211
(2010
). 45.
Ott
, E.
, “Strange attractors and chaotic motions of dynamical systems
,” Rev. Mod. Phys.
53
, 655
(1981
). 46.
Podlubny
, I.
, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications
(Academic Press
, 1999
).47.
Ruelle
, D.
and Takens
, F.
, “On the nature of turbulence
,” Commun. Math. Phys.
12
, 1
–44
(1971
).48.
Schumer
, R.
, Benson
, D. A.
, Meerschaert
, M. M.
, and Baeumer
, B.
, “Multiscaling fractional advection-dispersion equations and their solutions
,” Water Resour. Res.
39
, 1022
(2003
).49.
Simmons
, A. J.
and Hollingsworth
, A.
, “Some aspects of the improvement in skill of numerical weather prediction
,” Q. J. R. Meteorol. Soc.
128
, 647
–677
(2002
). 50.
Smagorinsky
, J.
, “Problems and promises of deterministic extended range forecasting
,” Bull Am. Meteorol. Soc.
50
, 286
–312
(1969
). 51.
Sun
, H.
, Abdelwahab
, A.
, and Onaral
, B.
, “Linear approximation of transfer function with a pole of fractional power
,” IEEE Trans. Automat. Contr.
29
, 441
–444
(1984
). 52.
Syroka
, J.
and Toumi
, R.
, “Scaling and persistence in observed and modeled surface temperature
,” Geophys. Res. Lett.
28
, 3255
–3258
, https://doi.org/10.1029/\break 2000GL012273 (2001
). 53.
Vannitsem
, S.
and Lucarini
, V.
, “Statistical and dynamical properties of covariant Lyapunov vectors in a coupled atmosphere-ocean model—Multiscale effects, geometric degeneracy, and error dynamics
,” J. Phys. A
49
, 224001
(2016
). 54.
Vyushin
, D. I.
and Kushner
, P. J.
, “Power-law and long-memory characteristics of the atmospheric general circulation
,” J. Clim.
22
, 2890
–2904
(2009
). 55.
Vyushin
, D. I.
, Kushner
, P. J.
, and Mayer
, J.
, “On the origins of temporal power-law behavior in the global atmospheric circulation
,” Geophys. Res. Lett.
36
, L14706
(2009
). 56.
Wang
, Y.
, Xu
, J.
, and Kloeden
, P. E.
, “Asymptotic behavior of stochastic lattice systems with a Caputo fractional time derivative
,” Nonlinear Anal. Theory Methods Appl.
135
, 205
–222
(2016
). 57.
Yuan
, N.
, Fu
, Z.
, and Liu
, S.
, “Long-term memory in climate variability: A new look based on fractional integral techniques
,” J. Geophys. Res. Atmos.
118
, 12962
–12969
, https://doi.org/10.1002/2013JD020776 (2013
). 58.
Zaslavsky
, G. M.
, “Chaos, fractional kinetics, and anomalous transport
,” Phys. Rep.
371
, 461
–580
(2002
). © 2023 Author(s). Published under an exclusive license by AIP Publishing.
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