An Extension of the Lyapunov Analysis for the Predictability Problem

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Abstract
The predictability problem for systems with different characteristic timescales is investigated. It is shown that even in simple chaotic dynamical systems, the leading Lyapunov exponent is not sufficient to estimate the predictability time. This fact is due to the saturation of the error on the fast components of the system, which therefore do not contribute to the exponential growth of the uncertainty at large error levels. It is proposed to adopt a generalization of the Lyapunov exponent that is based on the natural concept of error growing time at fixed error size. The predictability time defined in terms of the finite size Lyapunov exponent displays a strong dependence on the error magnitude, as already recognized by other authors. The method is first illustrated on a simple numerical model obtained by coupling two Lorenz systems with different timescales. As a more realistic example, the analysis is then applied to a “toy” model of the atmospheric circulation recently introduced by Lorenz. Abstract The predictability problem for systems with different characteristic timescales is investigated. It is shown that even in simple chaotic dynamical systems, the leading Lyapunov exponent is not sufficient to estimate the predictability time. This fact is due to the saturation of the error on the fast components of the system, which therefore do not contribute to the exponential growth of the uncertainty at large error levels. It is proposed to adopt a generalization of the Lyapunov exponent that is based on the natural concept of error growing time at fixed error size. The predictability time defined in terms of the finite size Lyapunov exponent displays a strong dependence on the error magnitude, as already recognized by other authors. The method is first illustrated on a simple numerical model obtained by coupling two Lorenz systems with different timescales. As a more realistic example, the analysis is then applied to a “toy” model of the atmospheric circulation recently introduced by Lorenz.
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