The present work applies singular spectrum analysis (SSA) to the study of macroeconomic fluctuations in three European countries: Italy, The Netherlands, and the United Kingdom. This advanced spectral method provides valuable spatial and frequency information for multivariate data sets and goes far beyond the classical forms of time domain analysis. In particular, SSA enables us to identify dominant cycles that characterize the deterministic behavior of each time series separately, as well as their shared behavior. We demonstrate its usefulness by analyzing several fundamental indicators of the three countries' real aggregate economy in a univariate, as well as a multivariate setting. Since business cycles are international phenomena, which show common characteristics across countries, our aim is to uncover supranational behavior within the set of representative European economies selected herein. Finally, the analysis is extended to include several indicators from the U.S. economy, in order to examine its influence on the European economies under study and their interrelationships.

This paper addresses the effect of interannual variability in jet stream orientation on weather systems over the North Atlantic basin (NAB). The observational analysis relies on 65 yr of NCEP–NCAR reanalysis (1948–2012). The total daily kinetic energy of the geostrophic wind (GTKE) is taken as a measure of storm activity over the North Atlantic. The NAB is partitioned into four rectangular regions, and the winter average of GTKE is calculated for each quadrant. The spatial GTKE average over all four quadrants shows striking year-to-year variability and is strongly correlated with the North Atlantic Oscillation (NAO).The GTKE strength in the northeast quadrant is closely related to the diffluence angle of the jet stream in the northwest quadrant. To gain insight into the relationship between the diffluence angle and its downstream impact, a quasigeostrophic baroclinic model is used. The results show that an initially zonal jet persists at its initial latitude over 30 days or longer, while a tilted jet propagates meridionally according to the Rossby wave group velocity, unless kept stationary by external forcing.A Gulf Stream–like narrow sea surface temperature (SST) front provides the requisite forcing for an analytical steady-state solution to this problem. This SST front influences the atmospheric jet in the northwest quadrant: it both strengthens the jet and tilts it northward at higher levels, while its effect is opposite at lower levels. Reanalysis data confirm these effects, which are consistent with thermal wind balance. The results suggest that the interannual variability found in the GTKE may be caused by intrinsic variability of the thermal Gulf Stream front.

The recent development of dense and continuously operating Global Navigation Satellite System (GNSS) networks worldwide has led to a significant increase in geodetic data sets that sometimes capture transient-deformation signals. It is challenging, however, to extract such transients of geophysical origin from the background noise inherent to GNSS time series and, even more so, to separate them from other signals, such as seasonal redistributions of geophysical fluid mass loads. In addition, because of the very large number of continuously recording GNSS stations now available, it has become impossible to systematically inspect each time series and visually compare them at all neighboring sites. Here we show that Multichannel Singular Spectrum Analysis (M-SSA), a method derived from the analysis of dynamical systems, can be used to extract transient deformations, seasonal oscillations, and background noise present in GNSS time series. M-SSA is a multivariate, nonparametric, statistical method that simultaneously exploits the spatial and temporal correlations of geophysical fields. The method allows for the extraction of common modes of variability, such as trends with nonconstant slopes and oscillations shared across time series, without a priori hypotheses about their spatiotemporal structure or their noise characteristics. We illustrate this method using synthetic examples and show applications to actual GPS data from Alaska to detect seasonal signals and microdeformation at the Akutan active volcano. The geophysically coherent spatiotemporal patterns of uplift and subsidence thus detected are compared to the results of an idealized model of such processes in the presence of a magma chamber source.

A suite of empirical model experiments under the empirical model reduction framework are conducted to advance the understanding of ENSO diversity, nonlinearity, seasonality, and the memory effect in the simulation and prediction of tropical Pacific sea surface temperature (SST) anomalies. The model training and evaluation are carried out using 4000-yr preindustrial control simulation data from the coupled model GFDL CM2.1. The results show that multivariate models with tropical Pacific subsurface information and multilevel models with SST history information both improve the prediction skill dramatically. These two types of models represent the ENSO memory effect based on either the recharge oscillator or the time-delayed oscillator viewpoint. Multilevel SST models are a bit more efficient, requiring fewer model coefficients. Nonlinearity is found necessary to reproduce the ENSO diversity feature for extreme events. The nonlinear models reconstruct the skewed probability density function of SST anomalies and improve the prediction of the skewed amplitude, though the role of nonlinearity may be slightly overestimated given the strong nonlinear ENSO in GFDL CM2.1. The models with periodic terms reproduce the SST seasonal phase locking but do not improve the prediction appreciably. The models with multiple ingredients capture several ENSO characteristics simultaneously and exhibit overall better prediction skill for more diverse target patterns. In particular, they alleviate the spring/autumn prediction barrier and reduce the tendency for predicted values to lag the target month value.

A low-order quasigeostrophic double-gyre ocean model is subjected to an aperiodic forcing that mimics time dependence dominated by interdecadal variability. This model is used as a prototype of an unstable and nonlinear dynamical system with time-dependent forcing to explore basic features of climate change in the presence of natural variability. The study relies on the theoretical framework of nonautonomous dynamical systems and of their pullback attractors (PBAs), that is, of the time-dependent invariant sets attracting all trajectories initialized in the remote past. The existence of a global PBA is rigorously demonstrated for this weakly dissipative nonlinear model. Ensemble simulations are carried out and the convergence to PBAs is assessed by computing the probability density function (PDF) of localization of the trajectories. A sensitivity analysis with respect to forcing amplitude shows that the PBAs experience large modifications if the underlying autonomous system is dominated by small-amplitude limit cycles, while less dramatic changes occur in a regime characterized by large-amplitude relaxation oscillations. The dependence of the attracting sets on the choice of the ensemble of initial states is then analyzed. Two types of basins of attraction coexist for certain parameter ranges; they contain chaotic and nonchaotic trajectories, respectively. The statistics of the former does not depend on the initial states whereas the trajectories in the latter converge to small portions of the global PBA. This complex scenario requires separate PDFs for chaotic and nonchaotic trajectories. General implications for climate predictability are finally discussed.

This article revisits the approximation problem of systems of nonlinear delay differential equations (DDEs) by a set of ordinary differential equations (ODEs). We work in Hilbert spaces endowed with a natural inner product including a point mass, and introduce polynomials orthogonal with respect to such an inner product that live in the domain of the linear operator associated with the underlying DDE. These polynomials are then used to design a general Galerkin scheme for which we derive rigorous convergence results and show that it can be numerically implemented via simple analytic formulas. The scheme so obtained is applied to three nonlinear DDEs, two autonomous and one forced: (i) a simple DDE with distributed delays whose solutions recall Brownian motion; (ii) a DDE with a discrete delay that exhibits bimodal and chaotic dynamics; and (iii) a periodically forced DDE with two discrete delays arising in climate dynamics. In all three cases, the Galerkin scheme introduced in this article provides a good approximation by low-dimensional ODE systems of the DDE's strange attractor, as well as of the statistical features that characterize its nonlinear dynamics.

Evaluating the effects of climate variation on ecosystems is of paramount importance for our ability to forecast and mitigate the consequences of global change. However, the ways in which complex food webs respond to climate variations remain poorly understood. Here, we use long-term time series to investigate the effects of temperature variation on the intraguild-predation (IGP) system of Windermere (UK), a lake where pike (Esox lucius, top predator) feed on small-sized perch (Perca fluviatilis) but compete with large-sized perch for the same food sources. Spectral analyses of time series reveal that pike recruitment dynamics are temperature controlled. In 1976, expansion of a size-truncating perch pathogen into the lake severely impacted large perch and favoured pike as the IGP-dominant species. This pathogen-induced regime shift to a pike-dominated IGP apparently triggered a temperature-controlled trophic cascade passing through pike down to dissolved nutrients. In simple food chains, warming is predicted to strengthen top–down control by accelerating metabolic rates in ectothermic consumers, while pathogens of top consumers are predicted to dampen this top–down control. In contrast, the local IGP structure in Windermere made warming and pathogens synergistic in their top–down effects on ecosystem functioning. More generally, our results point to top predators as major mediators of community response to global change, and show that size-selective agents (e.g. pathogens, fishers or hunters) may change the topological architecture of food webs and alter whole ecosystem sensitivity to climate variation.

Abstract The Rosenzweig–MacArthur model is a set of ordinary differential equations (ODEs) that provides an aggregate description of the dynamics of a predator–prey system. When including an Allee effect on the prey, this model exhibits bistability and contains a pitchfork bifurcation, a Hopf bifurcation and a heteroclinic bifurcation. We develop an agent-based model (ABM) on a two-dimensional, square lattice that encompasses the key assumptions of the aggregate model. Although the two modelling approaches – \ODE\ and \ABM\ – differ, both models exhibit similar bifurcation patterns. The \ABM\ model's behaviour is richer and it is analysed using advanced statistical methods. In particular, singular spectrum analysis is used to robustly locate the transition between apparently random, small-amplitude fluctuations around a fixed point and stable, large-amplitude oscillations. Critical slowing down of model trajectories anticipates the heteroclinic bifurcation. Systematic comparison between the \ABM\ and the \ODE\ models’ behaviour helps one understand the predator–prey system better; it provides guidance in model exploration and allows one to draw more robust conclusions on the nature of predator–prey interactions.

This paper has two interrelated foci: (i) obtaining stable and efficient data-driven closure models by using a multivariate time series of partial observations from a large-dimensional system; and (ii) comparing these closure models with the optimal closures predicted by the Mori–Zwanzig (MZ) formalism of statistical physics. Multilayer stochastic models (MSMs) are introduced as both a generalization and a time-continuous limit of existing multilevel, regression-based approaches to closure in a data-driven setting; these approaches include empirical model reduction (EMR), as well as more recent multi-layer modeling. It is shown that the multilayer structure of MSMs can provide a natural Markov approximation to the generalized Langevin equation (GLE) of the MZ formalism. A simple correlation-based stopping criterion for an EMR–MSM model is derived to assess how well it approximates the GLE solution. Sufficient conditions are derived on the structure of the nonlinear cross-interactions between the constitutive layers of a given MSM to guarantee the existence of a global random attractor. This existence ensures that no blow-up can occur for a broad class of MSM applications, a class that includes non-polynomial predictors and nonlinearities that do not necessarily preserve quadratic energy invariants. The EMR–MSM methodology is first applied to a conceptual, nonlinear, stochastic climate model of coupled slow and fast variables, in which only slow variables are observed. It is shown that the resulting closure model with energy-conserving nonlinearities efficiently captures the main statistical features of the slow variables, even when there is no formal scale separation and the fast variables are quite energetic. Second, an MSM is shown to successfully reproduce the statistics of a partially observed, generalized Lotka–Volterra model of population dynamics in its chaotic regime. The challenges here include the rarity of strange attractors in the model’s parameter space and the existence of multiple attractor basins with fractal boundaries. The positivity constraint on the solutions’ components replaces here the quadratic-energy–preserving constraint of fluid-flow problems and it successfully prevents blow-up.

Singular spectrum analysis (SSA) along with its multivariate extension (M-SSA) provides an efficient way to identify weak oscillatory behavior in high-dimensional data. To prevent the misinterpretation of stochastic fluctuations in short time series as oscillations, Monte Carlo (MC)–type hypothesis tests provide objective criteria for the statistical significance of the oscillatory behavior. Procrustes target rotation is introduced here as a key method for refining previously available MC tests. The proposed modification helps reduce the risk of type-I errors, and it is shown to improve the test’s discriminating power. The reliability of the proposed methodology is examined in an idealized setting for a cluster of harmonic oscillators immersed in red noise. Furthermore, the common method of data compression into a few leading principal components, prior to M-SSA, is reexamined, and its possibly negative effects are discussed. Finally, the generalized Procrustes test is applied to the analysis of interannual variability in the North Atlantic’s sea surface temperature and sea level pressure fields. The results of this analysis provide further evidence for shared mechanisms of variability between the Gulf Stream and the North Atlantic Oscillation in the interannual frequency band.

We apply multivariate singular spectrum analysis to the study of U.S. business cycle dynamics. This method provides a robust way to identify and reconstruct oscillations, whether intermittent or modulated. We show such oscillations to be associated with comovements across the entire economy. The problem of spurious cycles generated by the use of detrending filters is addressed and we present a Monte Carlo test to extract significant oscillations. The behavior of the U.S. economy is shown to change significantly from one phase of the business cycle to another: the recession phase is dominated by a five-year mode, while the expansion phase exhibits more complex dynamics, with higher-frequency modes coming into play. We show that the variations so identified cannot be generated by random shocks alone, as assumed in ‘real’ business-cycle models, and that endogenous, deterministically generated variability has to be involved.