Predictability: a way to characterize complexity

Boffetta, G.; Cencini, Massimo; Falcioni, M.; Vulpiani, Angelo

doi:10.1016/s0370-1573(01)00025-4

Cited by 335 publications

(242 citation statements)

References 247 publications

(477 reference statements)

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“…Despite these simplifications the Lorenz-96 model has emerged as an important and often used model system for the testing of new ideas in the atmospheric sciences [24][25][26] and in the general study of spatiotemporal chaos. 27 In our work, we explore the Lorenz-96 model as a numerically accessible model with phenomenological relevance to fluid systems.…”

Section: The Lorenz-96 Modelmentioning

confidence: 99%

Extensive chaos in the Lorenz-96 model

Karimi

Paul

2010

Chaos: An Interdisciplinary Journal of Nonlinear Science

147

123

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We explore the high-dimensional chaotic dynamics of the Lorenz-96 model by computing the variation of the fractal dimension with system parameters. The Lorenz-96 model is a continuous in time and discrete in space model first proposed by Lorenz to study fundamental issues regarding the forecasting of spatially extended chaotic systems such as the atmosphere. First, we explore the spatiotemporal chaos limit by increasing the system size while holding the magnitude of the external forcing constant. Second, we explore the strong driving limit by increasing the external forcing while holding the system size fixed. As the system size is increased for small values of the forcing we find dynamical states that alternate between periodic and chaotic dynamics. The windows of chaos are extensive, on average, with relative deviations from extensivity on the order of 20%. For intermediate values of the forcing we find chaotic dynamics for all system sizes past a critical value. The fractal dimension exhibits a maximum deviation from extensivity on the order of 5% for small changes in system size and the deviation from extensivity decreases nonmonotonically with increasing system size. The length scale describing the deviations from extensivity is consistent with the natural chaotic length scale in support of the suggestion that deviations from extensivity are due to the addition of chaotic degrees of freedom as the system size is increased. We find that each wavelength of the deviation from extensive chaos contains on the order of two chaotic degrees of freedom. As the forcing is increased, at constant system size, the dimension density grows monotonically and saturates at a value less than unity. We use this to quantify the decreasing size of chaotic degrees of freedom with increased forcing which we compare with spatial features of the patterns. © 2010 American Institute of Physics. ͓doi:10.1063/1.3496397͔We explore fundamental questions regarding the composition and description of spatiotemporal chaos in highdimensional systems. The Lorenz-96 model is used for its phenomenological relevance to fluid convection and atmospheric dynamics. We compute the variation of the fractal dimension over a wide range of system parameters, for very long-times, and over many initial conditions. We explore two limits: the "spatiotemporal chaos" limit where the system size is increased while holding the external forcing constant and the "strong driving" limit where the external forcing is increased for a system of fixed size. The variations in the fractal dimension with system parameters are used to provide estimates for characteristic length and time scales describing the chaotic dynamics. These findings are directly compared with experimentally accessible diagnostics of the pattern dynamics to provide new physical insights into the underlying features of high-dimensional chaos.

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Section: The Lorenz-96 Modelmentioning

confidence: 99%

Extensive chaos in the Lorenz-96 model

Karimi

Paul

2010

Chaos: An Interdisciplinary Journal of Nonlinear Science

147

123

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“…We are currently studying the bifurcation diagram of this 1-D spatio-temporal chaotic system using a more dynamical approach (e.g., Boffeta et al (2002)). …”

Section: Modelingmentioning

confidence: 99%

The magnetosphere as a complex system

Valdivia

Rogan

Muñoz

et al. 2005

Advances in Space Research

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The magnetosphere is a complex system, with multi-scale spatio-temporal behavior. Self-organization is a possible solution to two seemingly contradicting observations: (a) the repeatable and coherent substorm phenomena, and, (b) the underlying selfsimilar turbulent behavior in the plasma sheet. Such states, are seen to emerge naturally in a plasma physics model with sporadic dissipation, through spatio-temporal chaos.

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“…However, there is a number of reasons to examine Weibull statistics in large ocean and atmospheric models. First, extremum statistics are often observed to arise in multi-dimensional systems, exhibiting correlation over a broad range of scales, leading to emergent phenomenology, such as self-similarity and in some case fractional dimension (Boffetta et al, 2002). Second, Weibull statistics seems to be a good mathematical tool for the parametrical estimate of PE distributions in small forecast ensembles and from limited observation samples.…”

Section: Discussionmentioning

confidence: 99%

On stochastic stability of regional ocean models to finite-amplitude perturbations of initial conditions

Ivanov

Chu

2007

Dynamics of Atmospheres and Oceans

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We consider error propagation near an unstable equilibrium state (classified as an unstable focus) for spatially uncorrelated and correlated finite-amplitude initial perturbations using short-(up to several weeks) and intermediate (up to 2 months) range forecast ensembles produced by a barotropic regional ocean model. An ensemble of initial perturbations is generated by the Latin Hypercube design strategy, and its optimal size is estimated through the Kullback-Liebler distance (the relative entropy). Although the ocean model is simple, the prediction error (PE) demonstrates non-trivial behavior similar to that existing in 3D ocean circulation models. In particular, in the limit of zero horizontal viscosity, the PE at first decays with time for all scales due to dissipation caused by non-linear bottom friction, and then grows faster than (quasi)-exponentially. Statistics of a prediction time scale (the irreversible predictability time (IPT)) quickly depart from Gaussian (the linear predictability regime) and becomes Weibullian (the non-linear predictability regime) as amplitude of initial perturbations grows. A transition from linear to non-linear predictability is clearly detected by the specific behavior of IPT variance. A new analytical formula for the model predictability horizon is introduced and applied to estimate the limit of predictability for the ocean model.

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Predictability: a way to characterize complexity

Cited by 335 publications

References 247 publications

Extensive chaos in the Lorenz-96 model

Extensive chaos in the Lorenz-96 model

The magnetosphere as a complex system

On stochastic stability of regional ocean models to finite-amplitude perturbations of initial conditions

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