Christopher Essex

The climate attractor over short timescales

Recent work has highlighted the possibilities of using certain ideas from the theory of dynamical systems for the study of global climate. These ideas include the geometrical notion of correlation or scaling dimension, first used to analyse attractors arising in mathematical and laboratory systems1–4 and later applied in a geophysical context5,6. The conclusion of the original geophysical work5 has been criticized in the light of a reanalysis7 which questions the existence of a low-dimensional climate attractor. Here results of a similar analysis conducted on daily meteorological observations from 1946 to 1982, are announced. This new work overcomes limitations of previous analyses, and supports the existence of such an attractor.

Numerical methods can suppress chaos

Abstract Numerical methods for the solution of ordinary differential equations are one of the main tools used in the theoretical investigation of nonlinear continuous dynamical systems. These replace the continuous dynamical system under study by a discrete dynamical system that is then usually simulated on a digital computer. It is well known that such discrete dynamical systems may be chaotic even when the underlying continuous dynamical system is not chaotic. We here show that some numerical methods may produce discrete dynamical systems that are not chaotic, even when the underlying continuous dynamical system is thought to be chaotic. We find in this case that the transition to chaos from false stability mimics the transition to chaos that has been previously observed as parameters were changed in the Rossler system.

Fractional Diffusion and Entropy Production

The entropy production rate for fractional diffusion processes is calculated and shows an apparently counter-intuitive increase with the transition from dissipative diffusion behaviour to reversible wave propagation. This is deduced directly from invariant and non-invariant factors of the (probability) density function, arising from a group method applied to the fractional differential equation which exists between the pure wave and diffusion equations. However, the counter-intuitive increase of the entropy production rate within the transition turns out to be a consequence of increasing quickness of processes as the wave case is approached. When this aspect is removed the entropy shows the expected decrease with the approach to the reversible wave limit.

Tsallis Relative Entropy and Anomalous Diffusion

In this paper we utilize the Tsallis relative entropy, a generalization of the Kullback–Leibler entropy in the frame work of non-extensive thermodynamics to analyze the properties of anomalous diffusion processes. Anomalous (super-) diffusive behavior can be described by fractional diffusion equations, where the second order space derivative is extended to fractional order α ∈ (1, 2). They represent a bridging regime, where for α = 2 one obtains the diffusion equation and for α = 1 the (half) wave equation is given. These fractional diffusion equations are solved by so-called stable distributions, which exhibit heavy tails and skewness. In contrast to the Shannon or Tsallis entropy of these distributions, the Kullback and Tsallis relative entropy, relative to the pure diffusion case, induce a natural ordering of the stable distributions consistent with the ordering implied by the pure diffusion and wave limits.

Tsallis and Rényi entropies in fractional diffusion and entropy production

The entropy production rate for fractional diffusion processes using Shannon entropy was calculated previously, which showed an apparently counter intuitive increase with the transition from dissipative diffusion behaviour to reversible wave propagation. Renyi and Tsallis entropies, which have an additional parameter q generalizing the Shannon case (q=1), are shown here to have similar counter intuitive behaviours. However, the issue can be successfully treated in exactly the same manner as with Shannon entropy for q being not too large (i.e., generalizations near the Shannon case), whereas for larger q the Renyi and Tsallis entropies behave in a different way.

Estimation of chaotic parameter regimes via generalized competitive mode approach

Abstract A procedure, we call it generalized competitive mode (GCM), is proposed to estimate the parameter regimes of chaos in nonlinear systems by implementing a mathematical version of mode competition. The idea is that for a system to be chaotic there must exist at least two GCMs in the system. The Lorenz system and a thin plate in flow-induced vibrations system are analyzed to find chaotic regimes by this procedure.

The similarity group and anomalous diffusion equations

A number of distinct differential equations, known as generalized diffusion equations, have been proposed to describe the phenomenon of anomalous diffusion on fractal objects. Although all are constructed to correctly reproduce the basic subdiffusive property of this phenomenon, using similarity methods it becomes very clear that this is far from sufficient to confirm their validity. The similarity group that they all have in common is the natural basis for making comparisons between these otherwise different equations, and a practical basis for comparisons between the very different modelling assumptions that their solutions each represent. Similarity induces a natural space in which to compare these solutions both with one another and with data from numerical experiments on fractals. It also reduces the differential equations to (extra-) ordinary ones, which are presented here for the first time. It becomes clear here from this approach that the proposed equations cannot agree even qualitatively with either each other or the data, suggesting that a new approach is needed.

Comments on ‘Deterministic chaos: the science and the fiction' by D. Ruelle

The results of several works concerning estimates of correlation dimension have recently been criticized on the basis of a new upper bound on such estimates (i. e. 2 log10 N, where N is the number of data). It is shown here that this bound is not valid for correlation dimension estimates in general, and furthermore it is too weak to be useful for régimes where it does hold. This casts the indicated criticism into question.

Correlation Dimension and Data Sample Size

A general method for estimating the amount of data sufficient to reliably determine the correlation dimension from a time series is presented. The results of this method are discussed in terms of previous attempts to calculate this quantity from climatological data.

COMPETITIVE MODES AND THEIR APPLICATION

We investigate nonlinear dynamical systems from the mode competition point of view, and propose the necessary conditions for a system to be chaotic. We conjecture that a chaotic system has at least two competitive modes (CMs). For a general nonlinear dynamical system, we give a simple, dynamically motivated definition of mode suitable for this concept. Since for most chaotic systems it is difficult to obtain the form of a CM, we focus on the competition between the corresponding modulated frequency components of the CMs. Some direct applications result from the explicit form of the frequency functions. One application is to estimate parameter regimes which may lead to chaos. It is shown that chaos may be found by analyzing the frequency function of the CMs without applying a numerical integration scheme. Another application is to create new chaotic systems using custom-designed CMs. Several new chaotic systems are reported.