A piece-wise affine contracting map with positive entropy

We construct the simplest chaotic system with a two-point attractor.Comment: 2 page

A parsimonious description and cross-country analysis of COVID-19 epidemic curve

In a given country, the cumulative death toll of the first wave of the COVID-19 epidemic follows a sigmoid curve as a function of time. In most cases, the curve is well described by the Gompertz function, which is characterized by two essential parameters, the initial growth rate and the decay rate as the first epidemic wave subsides. These parameters are determined by socioeconomic factors and the countermeasures to halt the epidemic. The Gompertz model implies that the total death toll depends exponentially, and hence very sensitively, on the ratio between these rates. The remarkably different epidemic curves for the first epidemic wave in Sweden and Norway and many other countries are classified and discussed in this framework, and their usefulness for the planning of mitigation strategies is discussed.Comment: 27 pages, 14 figure

Statistical properties of a filtered Poisson process with additive random noise: Distributions, correlations and moment estimation

Filtered Poisson processes are often used as reference models for intermittent fluc- tuations in physical systems. Such a process is here extended by adding a noise term, either as a purely additive term to the process or as a dynamical term in a stochastic differential equation. The lowest order moments, probability density function, auto-correlation function and power spectral density are derived and used to identify and compare the effects of the two different noise terms. Monte-Carlo studies of synthetic time series are used to investigate the accuracy of model pa- rameter estimation and to identify methods for distinguishing the noise types. It is shown that the probability density function and the three lowest order moments provide accurate estimations of the parameters, but are unable to separate the noise types. The auto-correlation function and the power spectral density also provide methods for estimating the model parameters, as well as being capable of identifying the noise type. The number of times the signal crosses a prescribed threshold level in the positive direction also promises to be able to differentiate the noise type.Comment: 34 pages, 25 figure

Statistical significance of rising and oscillatory trends in global ocean and land temperature in the past 160 years

Submitted manuscript version.Various interpretations of the notion of a trend in the context of global warming are discussed, contrasting the difference between viewing a trend as the deterministic response to an external forcing and viewing it as a slow variation which can be separated from the background spectral continuum of long-range persistent climate noise. The emphasis in this paper is on the latter notion, and a general scheme is presented for testing a multi-parameter trend model against a null hypothesis which models the observed climate record as an autocorrelated noise. The scheme is employed to the instrumental global sea-surface temperature record and the global land temperature record. A trend model comprising a linear plus an oscillatory trend with period of approximately 70 yr, and the statistical significance of the trends, are tested against three different null models: first-order autoregressive process, fractional Gaussian noise, and fractional Brownian motion. The parameters of the null models are estimated from the instrumental record, but are also checked to be consistent with a Northern Hemisphere temperature reconstruction prior to 1750 for which an anthropogenic trend is negligible. The linear trend in the period 1850–2010 AD is significant in all cases, but the oscillatory trend is insignificant for ocean data and barely significant for land data. However, by using the significance of the linear trend to constrain the null hypothesis, the oscillatory trend in the land record appears to be statistically significant. The results suggest that the global land record may be better suited for detection of the global warming signal than the ocean record

Early-warning signals for the onsets of Greenland interstadials and the younger dryas-preboreal transition

Source: Journal of Climatedoi: 10.4236/am.2016.715143 © 2016 American Meteorological Society (AMS). Permission to use figures, tables, and brief excerpts from this work in scientific and educational works is hereby granted provided that the source is acknowledged. Any use of material in this work that is determined to be “fair use” under Section 107 of the U.S. Copyright Act September 2010 Page 2 or that satisfies the conditions specified in Section 108 of the U.S. Copyright Act (17 USC §108, as revised by P.L. 94-553) does not require the AMS’s permission. Republication, systematic reproduction, posting in electronic form, such as on a website or in a searchable database, or other uses of this material, except as exempted by the above statement, requires written permission or a license from the AMS.The climate system approaches a tipping point if the prevailing climate state loses stability, making a transition to a different state possible. A result from the theory of randomly driven dynamical systems is that the reduced stability in the vicinity of a tipping point is accompanied by increasing fluctuation levels and longer correlation times (critical slowing down) and can in principle serve as early-warning signals of an upcoming tipping point. This study demonstrates that the high-frequency band of the δ18O variations in the North Greenland Ice Core Project displays fluctuation levels that increase as one approaches the onset of an interstadial (warm) period. Similar results are found for the locally estimated Hurst exponent for the high-frequency fluctuations, signaling longer correlation times. The observed slowing down is found to be even stronger in the Younger Dryas, suggesting that both the Younger Dryas–Preboreal transition and the onsets of the Greenland interstadials are preceded by decreasing stability of the climate state. It is also verified that the temperature fluctuations during the stadial periods can be approximately modeled as a scale-invariant persistent noise, which can be approximated as an aggregation of processes that respond to perturbations on certain characteristic time scales. The results are consistent with the hypothesis that both the onsets of the Greenland interstadials and the Younger Dryas–Preboreal transition are caused by tipping points in dynamical processes with characteristic time scales on the order of decades and that the variability of other processes on longer time scales masks the early-warning signatures in the δ18O signal

Long-range persistence in global surface temperatures explained by linear multibox energy balance models

Source at: http://doi.org/10.1175/JCLI-D-16-0877.1 The temporal fluctuations in global mean surface temperature are an example of a geophysical quantity that can be described using the notions of long-range persistence and scale invariance/scaling, but this description has suffered from lack of a generally accepted physical explanation. Processes with these statistical signatures can arise from nonlinear effects, for instance, through cascade-like energy transfer in turbulent fluids, but they can also be produced by linear models with scale-invariant impulse–response functions. This paper demonstrates that, on time scales from months to centuries, the scale-invariant impulse–response function of global surface temperature can be explained by simple linear multibox energy balance models. This explanation describes both the scale invariance of the internal variability and the lack of a characteristic time scale of the response to external forcings. With parameters estimated from observational data, the climate response is approximately scaling in these models, even if the response function is not chosen to be scaling a priori. It is also demonstrated that the differences in scaling exponents for temperatures over land and for sea surface temperatures can be reproduced by a version of the multibox energy balance model with two distinct surface boxes