Delay differential equations evolve in an infinite-dimensional phase space. In this paper, we consider the effect of external fluctuations ͑noise͒ on delay differential equations involving one variable, thus leading to univariate stochastic delay differential equations ͑SDDE's͒. For small delays, a univariate nondelayed stochastic differential equation approximating such a SDDE is presented. Another approximation, complementary to the first, is also obtained using an average of the SDDE's drift term over the delayed dynamical variable, which defines a conditional average drift. This second approximation is characterized by the fact that the diffusion term is identical to that of the original SDDE. For small delays, our approach yields a steady-state probability density and a conditional average drift which are in close agreement with numerical simulations of the original SDDE. We illustrate this scheme with the delayed linear Langevin equation and a stochastic version of the delayed logistic equation. The technique can be used with any type of noise, and is easily generalized to multiple delays. ͓S1063-651X͑99͒08304-X͔

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A Fokker-Planck formulation of systems described by stochastic delay differential equations has been recently proposed. A separation of time scales approximation allowing this Fokker-Planck equation to be simplified in the case of multistable systems is hereby introduced, and applied to a system consisting of a particle coupled to a delayed quartic potential. In that approximation, population numbers in each well obey a phenomenological rate law. The corresponding transition rate is expressed in terms of the noise variance and the steady-state probability density. The same type of expression is also obtained for the mean first passage time from a given point to another one. The steady-state probability density appearing in these formulas is determined both from simulations and from a small delay expansion. The results support the validity of the separation of time scales approximation. However, the results obtained using a numerically determined steady-state probability are more accurate than those obtained using the small delay expansion, thereby stressing the high sensitivity of the transition rate and mean first passage time to the shape of the steady-state probability density. Simulation results also indicate that the transition rate and the mean first passage time both follow Arrhenius' law when the noise variance is small, even if the delay is large. Finally, deterministic unbounded solutions are found to coexist with the bounded ones. In the presence of noise, the transition rate from bounded to unbounded solutions increases with the delay.

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A one-dimensional reactive transport model including mass, momentum and volume conservation for the solid, aqueous, and gaseous phases is developed to explore the fate of free methane gas in marine sediments. The model assumes steady-state compaction for the solid phase in addition to decoupled gas and aqueous phase transport, instigated by processes such as buoyancy, externally impressed flows and compaction. Chemical species distributions are governed by gas advection, dissolved advection and diffusion as well as by reaction processes, which include organoclastic sulfate reduction, methanogenesis and anaerobic oxidation of methane (AOM). The model is applied to Eckernfo ¨rde Bay, a shallow-water environment where acoustic profiles confirm a widespread occurrence of year-round biogenic free methane gas within the muddy regions of the sediment, and where subsurface methanogenesis, overlaid by a zone of AOM has been reported. The model results reveal that, under steady-state conditions, upward gas migration is an effective methane transport mechanism from oversaturated to undersaturated intervals of the sediment. Furthermore, sensitivity tests show that when methanogenesis rates increase, the gas flux to the AOM zone becomes progressively more important and may reach values comparable to those of the aqueous methane diffusive flux. Nevertheless, the model also proves that the gas transport rates always remain smaller than the removal rates by combined gaseous methane dissolution and oxidation. Consequently, for the range of environmental conditions investigated here, the AOM zone acts as an efficient subsurface barrier for both aqueous and gaseous methane, preventing methane escape from the sediments to the water column.

Chemical oscillating patterns are ubiquitous in geochemical systems. Although many such patterns result from systematic variations in the external environmental conditions, it is recognized that some patterns are due to intrinsic self-organized processes in a non-equilibrium nonlinear system with positive feedback. In rocks and minerals, periodic precipitation (Liesegang bands) and oscillatory zoning constitute good examples of patterns that can be explained using concepts from nonlinear dynamics. Generally, as the system parameters exceed some threshold values, the steady (time-independent) state characterizing the system loses its stability. The system then evolves towards other time-dependent solutions (‘attractors’) that may have an oscillatory behaviour or a complex chaotic one. In this review, we describe many of these pattern types taken from a variety of geological environments: eruptive, sedimentary, hydrothermal or metamorphic. One particular example (periodic precipitation of pyrite bands in an evolving sapropel sediment) is presented here for the first time. This will help in convincing the reader that the tools of nonlinear dynamics may be useful to understand the history of our planet.

We show that several time series analysis methods that are often used for detecting self-affine fractal scaling and determining Hurst exponents in data sets may lead to spurious results when applied to short discretized data series. We show that irregularities in the series, such as jumps or spikes (as are often found in geophysical data) may lead to spurious scaling and consequently to an incorrect determination of the Hurst exponent. We also illustrate the statistical error in measuring Hurst exponent in series where self-affine fractal scaling does exist. Users should be aware of these caveats when interpreting the results of short time series analysis. r

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