Showing papers by "Peter D. Ditlevsen published in 2020"

Crossover and peaks in the Pleistocene climate spectrum; understanding from simple ice age models

Abstract: The power spectrum provides a compact representation of the scale dependence of the variability in time series. At multi-millennial time scales the spectrum of the Pleistocene climate is composed of a set of narrow band spectral modes attributed to the regularly varying changes in insolation from the astronomical change in Earth’s orbit and rotation superimposed on a continuous background generally attributed to stochastic variations. Quantitative analyses of paleoclimatic records indicate that the continuous part comprises a dominant part of the variance. It exhibits a power-law dependency typical of stochastic, self-similar processes, but with a scale break at the frequency of glacial-interglacial cycles. Here we discuss possible origins of this scale break, the connection between the continuous background and the narrow bands, and the apparently modest spectral power above the continuum at these scales. We demonstrate that the observed scale break around 100 ka can have a variety of different origins and does not imply an internal time scale of correlation as implied by the simplest linear stochastic model.

Multiscale measures of phase-space trajectories.

Abstract: Characterizing the multiscale nature of fluctuations from nonlinear and nonstationary time series is one of the most intensively studied contemporary problems in nonlinear sciences. In this work, we address this problem by combining two established concepts—empirical mode decomposition (EMD) and generalized fractal dimensions—into a unified analysis framework. Specifically, we demonstrate that the intrinsic mode functions derived by EMD can be used as a source of local (in terms of scales) information about the properties of the phase-space trajectory of the system under study, allowing us to derive multiscale measures when looking at the behavior of the generalized fractal dimensions at different scales. This formalism is applied to three well-known low-dimensional deterministic dynamical systems (the Henon map, the Lorenz ’63 system, and the standard map), three realizations of fractional Brownian motion with different Hurst exponents, and two somewhat higher-dimensional deterministic dynamical systems (the Lorenz ’96 model and the on–off intermittency model). These examples allow us to assess the performance of our formalism with respect to practically relevant aspects like additive noise, different initial conditions, the length of the time series under study, low- vs high-dimensional dynamics, and bursting effects. Finally, by taking advantage of two real-world systems whose multiscale features have been widely investigated (a marine stack record providing a proxy of the global ice volume variability of the past 5 × 10 6 years and the SYM-H geomagnetic index), we also illustrate the applicability of this formalism to real-world time series.

Refined central limit theorem and infinite density tail of the Lorentz gas from Levy walk

Abstract: We consider point particle that collides with a periodic array of hard-core elastic scatterers where the length of the free flights is unbounded (the infinite-horizon Lorentz gas, LG). The Bleher central limit theorem (CLT) states that the distribution of the particle displacement divided by $\sqrt{t\ln t}$ is Gaussian in the limit of infinite time $t$. However it was stressed recently that the slow convergence makes this result unobservable. Using a Levy walk model (LW) of the LG, it was proposed that the use of a rescaled Lambert function instead of $\sqrt{t\ln t}$ provides a fast convergent, observable CLT, which was confirmed by the LG simulations. We demonstrate here that this result can simplified to a mixed CLT where the scaling factor combines normal and anomalous diffusions. For narrow infinite corridors the particle for long time obeys the usual normal diffusion, which explains the previous numerical observations. In the opposite limit of small scatterers the Bleher CLT gives a good guiding. In the intermediate cases the mixed CLT applies. The Gaussian peak determines moments of order smaller than two. In contrast, the CLT gives only half the coordinate dispersion. The missing half of the dispersion and also moments of order higher than two are described by the distribution's tail (the infinite density) which we derive here. The tail is supported along the infinite corridors and formed by anomalously long flights whose duration is comparable with the whole time of observation. The moments' calculation from the tail is confirmed by direct calculation of the fourth moment from the statistics of the backward recurrence time defined as time that elapsed since the last collision. This completes the solution of the LW model allowing full comparison with the LG.

Bifurcation of critical sets and relaxation oscillations in singular fast-slow systems

Abstract: Fast-slow dynamical systems have subsystems that evolve on vastly different timescales, and bifurcations in such systems can arise due to changes in any or all subsystems. We classify bifurcations of the critical set (the equilibria of the fast subsystem) and associated fast dynamics, parametrized by the slow variables. Using a distinguished parameter approach we are able to classify bifurcations for one fast and one slow variable. Some of these bifurcations are associated with the critical set losing manifold structure. We also conjecture a list of generic bifurcations of the critical set for one fast and two slow variables. We further consider how the bifurcations of the critical set can be associated with generic bifurcations of attracting relaxation oscillations under an appropriate singular notion of equivalence.

Multiscale measures of phase-space trajectories

Abstract: Characterizing the multiscale nature of fluctuations from nonlinear and nonstationary time series is one of the most intensively studied contemporary problems in nonlinear sciences. In this work, we address this problem by combining two established concepts—empirical mode decomposition (EMD) and generalized fractal dimensions—into a unified analysis framework. Specifically, we demonstrate that the intrinsic mode functions derived by EMD can be used as a source of local (in terms of scales) information about the properties of the phase-space trajectory of the system under study, allowing us to derive multiscale measures when looking at the behavior of the generalized fractal dimensions at different scales. This formalism is applied to three well-known low-dimensional deterministic dynamical systems (the Henon map, the Lorenz ’63 system, and the standard map), three realizations of fractional Brownian motion with different Hurst exponents, and two somewhat higher-dimensional deterministic dynamical systems (the Lorenz ’96 model and the on–off intermittency model). These examples allow us to assess the performance of our formalism with respect to practically relevant aspects like additive noise, different initial conditions, the length of the time series under study, low- vs high-dimensional dynamics, and bursting effects. Finally, by taking advantage of two real-world systems whose multiscale features have been widely investigated (a marine stack record providing a proxy of the global ice volume variability of the past 5 × 10 6 years and the SYM-H geomagnetic index), we also illustrate the applicability of this formalism to real-world time series.

Predicting past tipping points: The Dansgaard-Oeschger events of the last glacial period

Abstract:

The Dansgaard-Oeschger (DO) events of the last glacial period provide a unique example of large-scale climate change on centennial time scales. Despite significant progress in modeling DO-like transitions with realistic climate models, it is still unknown what ultimately drives these changes. It is an outstanding problem whether they are driven by a self-sustained oscillation of the earth system, or by stochastic perturbations in terms of freshwater discharges into the North Atlantic or extremes in atmospheric dynamics.

This work addresses the question of whether DO events fall into the realm of tipping points in the mathematical sense, either driven by an underlying bifurcation, noise or a rate-dependent instability, or whether they are a true and possibly chaotic oscillation. To do this, different ice core proxy data and empirical predictability can be used as a discriminator.

The complex temporal pattern of DO events has been investigated previously to suggest that the transitions in between cold (stadial) and warm (interstadial) phases are purely noise-induced and thus unpredictable. In contrast, evidence is presented that trends in proxy records of Greenland ice cores within the stadial and interstadial phases pre-determine the impending abrupt transitions and allow their prediction. As a result, they cannot be purely noise-induced.

The observed proxy trends manifest consistent reorganizations of the climate system at specific time scales, and can give some hints on the physical processes being involved. Nevertheless, the complex temporal pattern, i.e., what sets the highly variable and largely uncorrelated time scales of individual DO excursions remains to be explained.