2015, Maluck, J., Donner, R.V.

2017, Mitra, C., Kittel, T., Choudhary, A., Kurths, J., Donner, R.V.

Maintaining the synchronous motion of dynamical systems interacting on complex networks is often critical to their functionality. However, real-world networked dynamical systems operating synchronously are prone to random perturbations driving the system to arbitrary states within the corresponding basin of attraction, thereby leading to epochs of desynchronized dynamics with a priori unknown durations. Thus, it is highly relevant to have an estimate of the duration of such transient phases before the system returns to synchrony, following a random perturbation to the dynamical state of any particular node of the network. We address this issue here by proposing the framework of single-node recovery time (SNRT) which provides an estimate of the relative time scales underlying the transient dynamics of the nodes of a network during its restoration to synchrony. We utilize this in differentiating the particularly slow nodes of the network from the relatively fast nodes, thus identifying the critical nodes which when perturbed lead to significantly enlarged recovery time of the system before resuming synchronized operation. Further, we reveal explicit relationships between the SNRT values of a network, and its global relaxation time when starting all the nodes from random initial conditions. Earlier work on relaxation time generally focused on investigating its dependence on macroscopic topological properties of the respective network. However, we employ the proposed concept for deducing microscopic relationships between topological features of nodes and their respective SNRT values. The framework of SNRT is further extended to a measure of resilience of the different nodes of a networked dynamical system. We demonstrate the potential of SNRT in networks of Rössler oscillators on paradigmatic topologies and a model of the power grid of the United Kingdom with second-order Kuramoto-type nodal dynamics illustrating the conceivable practical applicability of the proposed concept.

2014, Zou, Y., Donner, R.V., Marwan, N., Small, M., Kurths, J.

Solar activity is characterized by complex dynamics superimposed onto an almost periodic, approximately 11-year cycle. One of its main features is the presence of a marked, time-varying hemispheric asymmetry, the deeper reasons for which have not yet been completely uncovered. Traditionally, this asymmetry has been studied by considering amplitude and phase differences. Here, we use visibility graphs, a novel tool of nonlinear time series analysis, to obtain complementary information on hemispheric asymmetries in dynamical properties. Our analysis provides deep insights into the potential and limitations of this method, revealing a complex interplay between factors relating to statistical and dynamical properties, i.e., effects due to the probability distribution and the regularity of observed fluctuations. We demonstrate that temporal changes in the hemispheric predominance of the graph properties lag those directly associated with the total hemispheric sunspot areas. Our findings open a new dynamical perspective on studying the north-south sunspot asymmetry, which is to be further explored in future work.

2015, Mitra, C., Kurths, J., Donner, R.V.

2010, Donner, R.V., Zou, Y., Donges, J.F., Marwan, N., Kurths, J.

This paper presents a new approach for analysing the structural properties of time series from complex systems. Starting from the concept of recurrences in phase space, the recurrence matrix of a time series is interpreted as the adjacency matrix of an associated complex network, which links different points in time if the considered states are closely neighboured in phase space. In comparison with similar network-based techniques the new approach has important conceptual advantages, and can be considered as a unifying framework for transforming time series into complex networks that also includes other existing methods as special cases. It has been demonstrated here that there are fundamental relationships between many topological properties of recurrence networks and different nontrivial statistical properties of the phase space density of the underlying dynamical system. Hence, this novel interpretation of the recurrence matrix yields new quantitative characteristics (such as average path length, clustering coefficient, or centrality measures of the recurrence network) related to the dynamical complexity of a time series, most of which are not yet provided by other existing methods of nonlinear time series analysis. © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft.

2011, Donges, J.F., Donner, R.V., Rehfeld, K., Marwan, N., Trauth, M.H., Kurths, J.

The analysis of palaeoclimate time series is usually affected by severe methodological problems, resulting primarily from non-equidistant sampling and uncertain age models. As an alternative to existing methods of time series analysis, in this paper we argue that the statistical properties of recurrence networks - a recently developed approach - are promising candidates for characterising the system's nonlinear dynamics and quantifying structural changes in its reconstructed phase space as time evolves. In a first order approximation, the results of recurrence network analysis are invariant to changes in the age model and are not directly affected by non-equidistant sampling of the data. Specifically, we investigate the behaviour of recurrence network measures for both paradigmatic model systems with non-stationary parameters and four marine records of long-term palaeoclimate variations. We show that the obtained results are qualitatively robust under changes of the relevant parameters of our method, including detrending, size of the running window used for analysis, and embedding delay. We demonstrate that recurrence network analysis is able to detect relevant regime shifts in synthetic data as well as in problematic geoscientific time series. This suggests its application as a general exploratory tool of time series analysis complementing existing methods.

2013, Donner, R.V., Balasis, G.

The dynamical behaviour of the magnetosphere is known to be a sensitive indicator for the response of the system to solar wind coupling. Since the solar activity commonly displays very interesting non-stationary and multi-scale dynamics, the magnetospheric response also exhibits a high degree of dynamical complexity associated with fundamentally different characteristics during periods of quiescence and magnetic storms. The resulting temporal complexity profile has been explored using several approaches from applied statistics, dynamical systems theory and statistical mechanics. Here, we propose an alternative way of looking at time-varying dynamical complexity of nonlinear geophysical time series utilising subtle but significant changes in the linear autocorrelation structure of the recorded data. Our approach is demonstrated to sensitively trace the dynamic signatures associated with intense magnetic storms, and to display reasonable skills in distinguishing between quiescence and storm periods. The potentials and methodological limitations of this new viewpoint are discussed in some detail.

2015, Donges, J.F., Donner, R.V., Marwan, N., Breitenbach, S.F.M., Rehfeld, K., Kurths, J.

The Asian monsoon system is an important tipping element in Earth's climate with a large impact on human societies in the past and present. In light of the potentially severe impacts of present and future anthropogenic climate change on Asian hydrology, it is vital to understand the forcing mechanisms of past climatic regime shifts in the Asian monsoon domain. Here we use novel recurrence network analysis techniques for detecting episodes with pronounced non-linear changes in Holocene Asian monsoon dynamics recorded in speleothems from caves distributed throughout the major branches of the Asian monsoon system. A newly developed multi-proxy methodology explicitly considers dating uncertainties with the COPRA (COnstructing Proxy Records from Age models) approach and allows for detection of continental-scale regime shifts in the complexity of monsoon dynamics. Several epochs are characterised by non-linear regime shifts in Asian monsoon variability, including the periods around 8.5–7.9, 5.7–5.0, 4.1–3.7, and 3.0–2.4 ka BP. The timing of these regime shifts is consistent with known episodes of Holocene rapid climate change (RCC) and high-latitude Bond events. Additionally, we observe a previously rarely reported non-linear regime shift around 7.3 ka BP, a timing that matches the typical 1.0–1.5 ky return intervals of Bond events. A detailed review of previously suggested links between Holocene climatic changes in the Asian monsoon domain and the archaeological record indicates that, in addition to previously considered longer-term changes in mean monsoon intensity and other climatic parameters, regime shifts in monsoon complexity might have played an important role as drivers of migration, pronounced cultural changes, and the collapse of ancient human societies.

2012, Donner, R.V., Ehrcke, R., Barbosa, S.M., Wagner, J., Donges, J.F., Kurths, J.

The study of long-term trends in tide gauge data is important for understanding the present and future risk of changes in sea-level variability for coastal zones, particularly with respect to the ongoing debate on climate change impacts. Traditionally, most corresponding analyses have exclusively focused on trends in mean sea-level. However, such studies are not able to provide sufficient information about changes in the full probability distribution (especially in the more extreme quantiles). As an alternative, in this paper we apply quantile regression (QR) for studying changes in arbitrary quantiles of sea-level variability. For this purpose, we chose two different QR approaches and discuss the advantages and disadvantages of different settings. In particular, traditional linear QR poses very restrictive assumptions that are often not met in reality. For monthly data from 47 tide gauges from along the Baltic Sea coast, the spatial patterns of quantile trends obtained in linear and nonparametric (spline-based) frameworks display marked differences, which need to be understood in order to fully assess the impact of future changes in sea-level variability on coastal areas. In general, QR demonstrates that the general variability of Baltic sea-level has increased over the last decades. Linear quantile trends estimated for sliding windows in time reveal a wide-spread acceleration of trends in the median, but only localised changes in the rates of changes in the lower and upper quantiles.

2016, Barbosa, S.M., Donner, R.V.