The response of Earth’s climate system to orbital forcing has been highly state dependent over the past 66 million years.

The knowledge of transitions between regular, laminar or chaotic behaviors is essential to understand the underlying mechanisms behind complex systems. While several linear approaches are often insufficient to describe such processes, there are several nonlinear methods that, however, require rather long time observations. To overcome these difficulties, we propose measures of complexity based on vertical structures in recurrence plots and apply them to the logistic map as well as to heart-rate-variability data. For the logistic map these measures enable us not only to detect transitions between chaotic and periodic states, but also to identify laminar states, i.e., chaos-chaos transitions. The traditional recurrence quantification analysis fails to detect the latter transitions. Applying our measures to the heart-rate-variability data, we are able to detect and quantify the laminar phases before a life-threatening cardiac arrhythmia occurs thereby facilitating a prediction of such an event. Our findings could be of importance for the therapy of malignant cardiac arrhythmias.

Linear-stability measures for assessing the state of a dynamical system are inherently local, and thus insufficient to quantify stability against substantial perturbations. The volume of a state’s basin of attraction offers a powerful alternative—and points towards a plausible explanation for regularity in real-world networks. The human brain1,2, power grids3, arrays of coupled lasers4 and the Amazon rainforest5,6 are all characterized by multistability7. The likelihood that these systems will remain in the most desirable of their many stable states depends on their stability against significant perturbations, particularly in a state space populated by undesirable states. Here we claim that the traditional linearization-based approach to stability is too local to adequately assess how stable a state is. Instead, we quantify it in terms of basin stability, a new measure related to the volume of the basin of attraction. Basin stability is non-local, nonlinear and easily applicable, even to high-dimensional systems. It provides a long-sought-after explanation for the surprisingly regular topologies8,9,10 of neural networks and power grids, which have eluded theoretical description based solely on linear stability11,12,13. We anticipate that basin stability will provide a powerful tool for complex systems studies, including the assessment of multistable climatic tipping elements14.

Recurrence networks-a novel paradigm for nonlinear time series analysis View the table of contents for this issue, or go to the journal homepage for more 2010 New J. Phys. 12 033025