Breeding and predictability in chaotic model

Physical Review E, 2010

Bred vectors are a type of finite perturbation used in prediction studies of atmospheric models that exhibit spatially extended chaos. We study the structure, spatial correlations, and the growthrates of logarithmic bred vectors (which are constructed by using a given norm). We find that, after a suitable transformation, logarithmic bred vectors are roughly piecewise copies of the leading Lyapunov vector. This fact allows us to deduce a scaling law for the bred vector growth rate as a function of their amplitude. In addition, we relate growth rates with the spectrum of Lyapunov exponents corresponding to the most expanding directions. We illustrate our results with simulations of the Lorenz '96 model.

Nonlinear Processes in Geophysics, 2008

ABSTRACT Due to the chaotic nature of atmospheric dynamics, numerical weather prediction systems are sensitive to errors in the initial conditions. To estimate the forecast uncertainty, forecast centres produce ensemble forecasts based on perturbed initial conditions. How to optimally perturb the initial conditions remains an open question and different methods are in use. One is the singular vector (SV) method, adapted by ECMWF, and another is the breeding vector (BV) method (previously used by NCEP). In this study we compare the two methods with a modified version of breeding vectors in a low-order dynamical system (Lorenz-63). We calculate the Empirical Orthogonal Functions (EOF) of the subspace spanned by the breeding vectors to obtain an orthogonal set of initial perturbations for the model. We will also use Normal Mode perturbations. Evaluating the results, we focus on the fastest growth of a perturbation. The results show a large improvement for the BV-EOF perturbations compared to the non-orthogonalised BV. The BV-EOF technique also shows a larger perturbation growth than the SVs of this system, except for short time-scales. The highest growth rate is found for the second BV-EOF for the long-time scale. The differences between orthogonal and non-orthogonal breeding vectors are also investigated using the ECMWF IFS-model. These results confirm the results from the Loernz-63 model regarding the dependency on orthogonalisation.

Physical Review E, 2005

Spatial configuration of initial errors strongly affects predictability of space-time chaotic systems. The predictability of numerical models can be adjusted by using prepared ensembles of initial conditions. We present a natural way of preparing ensembles based in using finite-amplitude perturbations with varying correlation. This allows one to take into account the underlying dynamics to generate initial perturbations with spatial correlations varying from fully correlated ͑bred vectors͒ to random fluctuations.

Proceedings of the 6th International Conference on Nonlinear Science and Complexity, 2016

Nonlinear Processes in Geophysics, 2008

Nonlinear Processes in Geophysics, 2013

1994

It is frequently asserted that in a chaotic system two initially close points will separate at an exponential rate governed by the largest global Lyapunov exponent. Local Lyapunov exponents, however, are more directly relevant to predictability. The difference between the local and global Lyapunov exponents, the large variations of local exponents over an attractor, and the saturation of error growth near the size of the attractor-all result in nonexponential scalings in errors at both short and long prediction times, sometimes even obscuring evidence of exponential growth. Failure to observe exponential error scaling cannot rule out deterministic chaos as an explanation. We demonstrate a simple model that quantitatively predicts observed error scaling from the local Lyapunov exponents, for both short and surprisingly long times. We comment on the relevance to atmospheric predictability as studied in the meteorological literature.

Discrete Dynamics in Nature and Society, 2001

It is commonly found in the fixed-step numerical integration of nonlinear differential equations that the size of the integration step is opposite related to the numerical stability of the scheme and to the speed of computation. We present a procedure that establishes a criterion to select the largest possible step size before the onset of chaotic numerical instabilities, based upon the observation that computational chaos does not occur in a smooth, continuous way, but rather abruptly, as detected by examining the largest Lyapunov exponent as a function of the step size. For completeness, examination of the bifurcation diagrams with the step reveals the complexity imposed by the algorithmic discretization, showing the robustness of a scheme to numerical instabilities, illustrated here for explicit and implicit Euler schemes. An example of numerical suppression of chaos is also provided.

2014

The growth of small errors in weather prediction is exponential on average. As an error becomes larger, its growth slows down and then stops with the magnitude of the error saturating at about the average distance between two states chosen randomly. This paper studies the error growth in a low-dimensional atmospheric model before, during and after the initial exponential divergence occurs. We test cubic, quartic and logarithmic hypotheses by ensemble prediction method. Furthermore, the quadratic hypothesis suggested by Lorenz in 1969 is compared with the ensemble prediction method. The study shows that a small error growth is best modeled by the quadratic hypothesis. After the error exceeds about a half of the average value of variables, logarithmic approximation becomes superior. It is also shown that the time length of the exponential growth in the model data is a function of the size of small initial error and the largest Lyapunov exponent. We conclude that the size of the error at the least upper bound (supremum) of time length is equal to 1 and it is invariant to these variables. Predictability, as a time interval, where the model error is growing, is for small initial error, the sum of the least upper bound of time interval of exponential growth and predictability for the size of initial error equal to 1.

Journal of The Atmospheric Sciences, 2009

The transient evolution of prediction errors in the short to intermediate time regime is considered under the combined effect of initial condition and model errors. Some generic features are brought out and connected with intrinsic properties. Under the assumption of small uncorrelated initial errors and of small parameter errors, the conditions of existence of a time at which the mean quadratic error reaches a minimum and of a crossover time at which the contribution of initial condition errors matches that of model errors are determined. The results are illustrated and tested on representative low-order models of atmospheric dynamics exhibiting bistability, saddle-point behavior, and chaotic behavior. * Current affiliation: Instituto Dom Luiz,

2022

2007

We describe a control method based on optimization techniques to control of spatiotemporal chaos in a globally coupled map lattice (CML) system. We have developed a method for updating a CML model emulating complex spatial dynamics in an epileptic brain that exhibits characteristic spatiotemporal changes seen during transitions into a seizure susceptible state. Our updating algorithm uses metaheuristic techniques for obtaining feedback control parameters for controlling spatiotemporal chaos (local and global Lyapunov exponents). This methodology can be used in systems with hidden variables, i.e. where not all variables can be observed, such as the human brain, to reconstruct evolution maps and complex spatial patterns. Results from numerical simulations show that this algorithm is robust and effective in achieving controllability of the lattice model. We discuss the computational aspects of this learning methodology and its potential application in epileptic seizure control.

2012

The status of the work and the recent results of the application of the Assimilation in the Unstable Subspace are reported with special emphasis to the analysis of the stability induced by the assimilation of observations.