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We propose a RAndom Interacting Network (RAIN) model to study the interactions between a pair of complex networks. The model involves two major steps: (i) the selection of a pair of nodes, one from each network, based on intra-network node-based characteristics, and (ii) the placement of a link between selected nodes based on the similarity of their relative importance in their respective networks. Node selection is based on a selection fitness function and node linkage is based on a linkage probability defined on the linkage scores of nodes. The model allows us to relate within-network characteristics to between-network structure. We apply the model to the interaction between the USA and Schengen airline transportation networks (ATNs). Our results indicate that two mechanisms: degree-based preferential node selection and degree-assortative link placement are necessary to replicate the observed inter-network degree distributions as well as the observed inter-network assortativity. The RAIN model offers the possibility to test multiple hypotheses regarding the mechanisms underlying network interactions. It can also incorporate complex interaction topologies. Furthermore, the framework of the RAIN model is general and can be potentially adapted to various real-world complex systems.

We show a scenario of a two-frequeney torus breakdown, in which a global bifurcation occurs due to the collision of a quasi-periodic torus T-2 with saddle points, creating a heteroclinic saddle connection. We analyze the geometry of this torus-saddle collision by showing the local dynamics and the invariant manifolds (global dynamics) of the saddle points. Moreover, we present detailed evidences of a heteroclinic saddle-focus orbit responsible for the type- if intermittency induced by this global bifurcation. We also characterize this transition to chaos by measuring the Lyapunov exponents and the scaling laws.

Identifying abrupt transitions is a key question in various disciplines. Existing transition detection methods, however, do not rigorously account for time series uncertainties, often neglecting them altogether or assuming them to be independent and qualitatively similar. Here, we introduce a novel approach suited to handle uncertainties by representing the time series as a time-ordered sequence of probability density functions. We show how to detect abrupt transitions in such a sequence using the community structure of networks representing probabilities of recurrence. Using our approach, we detect transitions in global stock indices related to well-known periods of politico-economic volatility. We further uncover transitions in the El Nino-Southern Oscillation which coincide with periods of phase locking with the Pacific Decadal Oscillation. Finally, we provide for the first time an 'uncertainty-aware' framework which validates the hypothesis that ice-rafting events in the North Atlantic during the Holocene were synchronous with a weakened Asian summer monsoon.

A method for the multivariate analysis of statistical phase synchronization phenomena in empirical data is presented. A first statistical approach is complemented by a stochastic dynamic model, to result in a data analysis algorithm which can in a specific sense be shown to be a generic multivariate statistical phase synchronization analysis. The method is applied to EEG data from a psychological experiment, obtaining results which indicate the relevance of this method in the context of cognitive science as well as in other fields.

A method for the multivariate analysis of statistical phase synchronization phenomena in empirical data is presented. A first statistical approach is complemented by a stochastic dynamic model, to result in a data analysis algorithm which can in a specific sense be shown to be a generic multivariate statistical phase synchronization analysis. The method is applied to EEG data from a psychological experiment, obtaining results which indicate the relevance of this method in the context of cognitive science as well as in other fields

The dynamical structure of genetic networks determines the occurrence of various biological mechanisms, such as cellular differentiation. However, the question of how cellular diversity evolves in relation to the inherent stochasticity and intercellular communication remains still to be understood. Here, we define a concept of stochastic bifurcations suitable to investigate the dynamical structure of genetic networks, and show that under stochastic influence, the expression of given proteins of interest is defined via the probability distribution of the phase variable, representing one of the genes constituting the system. Moreover, we show that under changing stochastic conditions, the probabilities of expressing certain concentration values are different, leading to different functionality of the cells, and thus to differentiation of the cells in the various types.

Analysis of blood pressure dynamics in male and female rats using the continuous wavelet transform
(2009)

We study gender-related particularities in cardiovascular responses to stress and nitric oxide (NO) deficiency in rats using HR, mean arterial pressure (MAP) and a proposed wavelet-based approach. Blood pressure dynamics is analyzed: (1) under control conditions, (2) during immobilization stress and recovery and (3) during nitric oxide blockade by N-G-nitro-L-arginine-methyl ester (L-NAME). We show that cardiovascular sensitivity to stress and NO deficiency depends upon gender. Actually, in females the chronotropic effect of stress is more pronounced, while the pressor effect is weakened compared with males. We conclude that females demonstrate more favorable patterns of cardiovascular responses to stress and more effective NO control of cardiovascular activity than males.

Using quantities of symbolic dynamics, such as mutual information, Shannon information and algorithmic complexity, we have searched for interrelations of spikes emitted simultaneously at different frequencies during the impulsive phase of a flare event. As the spikes are related to the flare energy release and are interpreted as emissions originating at different sites having different magnetic field strengths, any relation in frequency is interpretated as a relation in space. This approach is appropriate to characterize such spatio-temporal patterns, whereas the popular estimate of fractal dimensions can be applied to low-dimensional systems only. Depending on the energy release and emission processes, two types of fragmentation are possible: a scenario of global organization (spikes are emitted in a succession of similar events by the same system) or a scenario of local organization (many systems triggered by an initial event). Mutual information which is a generalization of correlation indicates a relation in frequency beyond the bandwidth of individual spikes. The scans in the spectrograms with large mutual information also show a low level of Shannon information and algorithmic complexity, indicating that the simultaneous appearance of spikes at other frequencies is not a completely stochastic phenomenon (white noise). It may be caused by a nonlinear deterministic system or by a Markov process. By means of mutual information we find a memory over frequency intervals up to 60 MHz. Shannon information and algorithmic complexity concern the mbox{whole} frequency region, i.e. the global source region. A global organization is also apparent in quasi-periodic changes of the Shannon information and algorithmic complexity in the range of 2 - 8 seconds. The finding is compatible with a scenario of local organization in which the information of one event spreads spatially and triggers further events at different places. The region is not an ensemble of independently flashing sources, each representing a system that cascades in energy after an initial trigger. On the contrary, there is a causal connection between the sources at any time. The analysis of the four spike events suggests that the structure in frequency is not stochastic but a process in which spikes at nearby locations are simultaneously triggered by a common exciter.

Analysis of spatial and temporal extreme monsoonal rainfall over South Asia using complex networks
(2012)

We present a detailed analysis of summer monsoon rainfall over the Indian peninsular using nonlinear spatial correlations. This analysis is carried out employing the tools of complex networks and a measure of nonlinear correlation for point processes such as rainfall, called event synchronization. This study provides valuable insights into the spatial organization, scales, and structure of the 90th and 94th percentile rainfall events during the Indian summer monsoon (June-September). We furthermore analyse the influence of different critical synoptic atmospheric systems and the impact of the steep Himalayan topography on rainfall patterns. The presented method not only helps us in visualising the structure of the extreme-event rainfall fields, but also identifies the water vapor pathways and decadal-scale moisture sinks over the region. Furthermore a simple scheme based on complex networks is presented to decipher the spatial intricacies and temporal evolution of monsoonal rainfall patterns over the last 6 decades.

We look for structural properties in the light curve of the dwarf nova SS Cyg by means of techniques from nonlinear dynamics. Applying the popular Grassberger-Procaccia procedure, Cannizzo and Goddings (1988) showed that there is no evidence for a low-dimensional attractor underlying this record. Because there are some hints for order in the light curve, we search for other signatures of deterministic systems. Therefore, we use other methods recently developed in this theory, such as local linear prediction and recurrence maps. Our main findings are: i] the prediction error grows exponentially during outburst phases, but via a power law in the quiescent states, ii] there are some rather regular patterns in this light curve which sometimes recur, but the recurrence is not regular. This leads to the following conclusions: i] The outburst dynamics shows a higher degree of order than the quiescent one. There are some hints for deterministic chaos in the outburst behavior. ii] The light curve is a complex mixture of deterministic and stochastic structures. The analysis presented in this paper shows that methods of nonlinear dynamics can be an efficient tool for the study of complex processes, even if there is no evidence for a low-dimensional attractor.

Attractor-repeller collision and eyelet intermittency at the transition to phase synchronization
(1997)

The chaotically driven circle map is considered as the simplest model ofphase synchronization of a chaotic continuous-time oscillator by external periodic force. The phase dynamics is analyzed via phase-locking regions of the periodic cycles embedded in the strange attractor. It is shown that full synchronization, where all the periodic cycles are phase locked, disappears via the attractor-repeller collision. Beyond the transition an intermittent regime with exponentially rare phase slips, resulting from the trajectory's hits on an eyelet, is observed.

We present results of physical experiments where we measure the autocorrelation function (ACF) and the spectral linewidth of the basic frequency of a spiral chaotic attractor in a generator with inertial nonlinearity both without and in the presence of external noise. It is shown that the ACF of spiral attractors decays according to an exponential law with a decrement which is defined by the phase diffusion coefficient. It is also established that the evolution of the instantaneous phase can be approximated by a Wiener random process

We present an automatic control method for phase locking of regular and chaotic non-identical oscillations, when all subsystems interact via feedback. This method is based on the well known principle of feedback control which takes place in nature and is successfully used in engineering. In contrast to unidirectional and bidirectional coupling, the approach presented here supposes the existence of a special controller, which allows to change the parameters of the controlled systems. First we discuss general principles of automatic phase synchronization (PS) for arbitrary coupled systems with a controller whose input is given by a special quadratic form of coordinates of the individual systems and its output is a result of the application of a linear differential operator. We demonstrate the effectiveness of our approach for controlled PS on several examples: (i) two coupled regular oscillators, (ii) coupled regular and chaotic oscillators, (iii) two coupled chaotic R"ossler oscillators, (iv) two coupled foodweb models, (v) coupled chaotic R"ossler and Lorenz oscillators, (vi) ensembles of locally coupled regular oscillators, (vii) ensembles of locally coupled chaotic oscillators, and (viii) ensembles of globally coupled chaotic oscillators.

In this paper, we analytically study a star motif of Stuart-Landau oscillators, derive the bifurcation diagram and discuss the different forms of synchronization arising in such a system. Despite the parameter mismatch between the central node and the peripheral ones, an analytical approach independent of the number of units in the system has been proposed. The approach allows to calculate the separatrices between the regions with distinct dynamical behavior and to determine the nature of the different transitions to synchronization appearing in the system. The theoretical analysis is supported by numerical results.

The EEG is one of the most commonly used tools in brain research. Though of high relevance in research, the data obtained is very noisy and nonstationary. In the present article we investigate the applicability of a nonlinear data analysis method, the recurrence quantification analysis (RQA), to Such data. The method solely rests on the natural property of recurrence which is a phenomenon inherent to complex systems, such as the brain. We show that this method is indeed suitable for the analysis of EEG data and that it might improve contemporary EEG analysis.

Complex dynamical systems with many degrees of freedom may exhibit a wealth of collective phenomena related to high-dimensional chaos. This paper focuses on a lattice of coupled logistic maps to investigate the relationship between the loss of chaos synchronization and the onset of shadowing breakdown via unstable dimension variability in complex systems. In the neighborhood of the critical transition to strongly non-hyperbolic behavior, the system undergoes on-off intermittency with respect to the synchronization manifold. This has been confirmed by numerical diagnostics of synchronization and non-hyperbolic behavior, the latter using the statistical properties of finite-time Lyapunov exponents. (c) 2005 Elsevier B.V. All rights reserved

In one of the data mining techniques, change-point detection is of importance in evaluating time series measured in real world. For decades this technique has been developed as a nonlinear dynamics. We apply the method for detecting the change points, Singular Spectrum Transformation (SST), to the climate time series. To know where the structures of climate data sets change can reveal a climate background. In this paper we discuss the structures of precipitation data in Kenya and Wrangel Island (Arctic land) by using the SST.

The application of chaos theory has become popular to understand the nature of various features of solar activity because most of them are far from regular. The usual approach, however, that is basing on finding low- dimensional structures of the underlying processes seems to be successful only in a few exceptional cases, such as in rather coherent phenomena as coronal pulsations. It is important to note that most phenomena in solar radio emission are more complex. We present two kinds of techniques from nonlinear dynamics which can be useful to analyse such phenomena: i] Fragmentation processes observed in solar spike events are studied by means of symbolic dynamics methods. Different measures of complexity calculated from such observations reveal that there is some order in this fragmentation. ii] Bursts are a typical transient phenomenon. To study energization processes causing impulsive microwave bursts, the wavelet analysis is applied. It exhibits structural differences of the pre- and post-impulsive phase in cases where the power spectra of both are not distinct.

Chaotic channel
(2005)

This work combines the theory of chaotic synchronization with the theory of information in order to introduce the chaotic channel, an active medium formed by connected chaotic systems. This subset of a large chaotic net represents the path along which information flows. We show that the possible amount of information exchange between the transmitter, where information enters the net, and the receiver, the destination of the information, is proportional to the level of synchronization between these two special subsystems

We investigate the characteristics of time-delay systems in the presence of Gaussian noise. We show that the delay time embedded in the time series of time-delay system with constant delay cannot be estimated in the presence noise for appropriate values of noise intensity thereby forbidding any possibility of phase space reconstruction. We also demonstrate the existence of complete synchronization between two independent identical time-delay systems driven by a common noise without explicitly establishing any external coupling between them.

The dynamics of noisy bistable systems is analyzed by means of Lyapunov exponents and measures of complexity. We consider both the classical Kramers problem with additive white noise and the case when the barrier fluctuates due to additional external colored noise. In case of additive noise we calculate the Lyapunov exponents and all measures of complexity analytically as functions of the noise intensity resp. the mean escape time. For the problem of fluctuating barrier the usual description of the dynamics with the mean escape time is not sufficient. The application of the concept of measures of complexity allows to describe the structures of motion in more detail. Most complexity measures sign the value of correlation time at which the phenomenon of resonant activation occurs with an extremum.

We study the dynamics of the excitable Fitz Hugh-Nagumo system under external noisy driving. Noise activates the system producing a sequence of pulses. The coherence of these noise-induced oscillations is shown to be maximal for a certain noise amplitude. This new effect of coherence resonance is explained by different noise dependencies of the activation and the excursion times. A simple one-dimensional model based on the Langevin dynamics is proposed for the quantitative description of this phenomenon.

Chen et al. [Phys. Rev. E 61, 2559 (2000)] recently proposed an extension of the concept of phase for discrete chaotic systems. Using the newly introduced definition of phase they studied the dynamics of coupled map lattices and compared these dynamics with phase synchronization of coupled continuous-time chaotic systems. In this paper we illustrate by two simple counterexamples that the angle variable introduced by Chen et al. fails to satisfy the basic requirements to the proper phase. Furthermore, we argue that an extension of the notion of phase synchronization to generic discrete maps is doubtful.

In the recent article "Stochastic analysis of recurrence plots with applications to the detection of deterministic signals" (Physica D 237 (2008) 619-629), Rohde et al. stated that the performance of RQA in order to detect deterministic signals would be below traditional and well-known detectors. However, we have concerns about such a general statement. Based on our own studies we cannot confirm their conclusions. Our findings suggest that the measures of complexity provided by RQA are useful detectors outperforming well-known traditional detectors, in particular for the detection of signals of complex systems, with phase differences or signals modified due to the measurement process.

Higher variability in rainfall and river discharge could be of major importance in landslide generation in the north-western Argentine Andes. Annual layered (varved) deposits of a landslide dammed lake in the Santa Maria Basin (26°S, 66°W) with an age of 30,000 14C years provide an archive of precipitation variability during this time. The comparison of these data with present-day rainfall observations tests the hypothesis that increased rainfall variability played a major role in landslide generation. A potential cause of such variability is the El Niño/ Southern Oscillation (ENSO). The causal link between ENSO and local rainfall is quantified by using a new method of nonlinear data analysis, the quantitative analysis of cross recurrence plots (CRP). This method seeks similarities in the dynamics of two different processes, such as an ocean-atmosphere oscillation and local rainfall. Our analysis reveals significant similarities in the statistics of both modern and palaeo-precipitation data. The similarities in the data suggest that an ENSO-like influence on local rainfall was present at around 30,000 14C years ago. Increased rainfall, which was inferred from a lake balance modeling in a previous study, together with ENSO-like cyclicities could help to explain the clustering of landslides at around 30,000 14C years ago.

Competitive random sequential adsorption of point and fixed-sized particles: analytical results
(2001)

Complex networks in climate dynamics : comparing linear and nonlinear network construction methods
(2009)

Complex network theory provides a powerful framework to statistically investigate the topology of local and non- local statistical interrelationships, i.e. teleconnections, in the climate system. Climate networks constructed from the same global climatological data set using the linear Pearson correlation coefficient or the nonlinear mutual information as a measure of dynamical similarity between regions, are compared systematically on local, mesoscopic and global topological scales. A high degree of similarity is observed on the local and mesoscopic topological scales for surface air temperature fields taken from AOGCM and reanalysis data sets. We find larger differences on the global scale, particularly in the betweenness centrality field. The global scale view on climate networks obtained using mutual information offers promising new perspectives for detecting network structures based on nonlinear physical processes in the climate system.

We investigate the relationship between precipitation and runoff data from a small forested catchment in the Harz mountains (Germany). For this purpose, we develop a conceptual model including memory effects to predict the runoff signal using the precipitation data as input. An enhanced variant of the model also includes air temperature as input variable. We show in terms of correlation functions that this model describes main dynamical properties of the runoff, especially the delay between rain event and runoff response as the annual persistence in the runoff data.

Despite many previous Studies on the association between hyperthyroidism and the hyperadrenergic state, controversies still exist. Detrended fluctuation analysis (DFA) is a well recognized method in the nonlinear analysis of heart rate variability (HRV), and it has physiological significance related to the autonomic nervous system. In particular, an increased short-term scaling exponent alpha 1 calculated from DFA is associated with both increased sympathetic activity and decreased vagal activity. No study has investigated the DFA of HRV in hyperthyroidism. This study was designed to assess the sympathovagal balance in hyperthyroidism. We performed the DFA along with the linear analysis of HRV in 36 hyperthyroid Graves' disease patients (32 females and 4 males; age 30 +/- 1 years, means +/- SE) and 36 normal controls matched by sex, age and body mass index. Compared with the normal controls, the hyperthyroid patients revealed a significant increase (P < 0.001) in alpha 1 (hyperthyroid 1.28 +/- 0.04 versus control 0.91 +/- 0.02), long-term scaling exponent alpha 2 (1.05 +/- 0.02 versus 0.90 +/- 0.01), overall scaling exponent alpha (1.11 +/- 0.02 versus 0.89 +/- 0.01), low frequency power in normalized units (LF%) and the ratio of low frequency power to high frequency power (LF/HF); and a significant decrease (P < 0.001) in the standard deviation of the R-R intervals (SDNN) and high frequency power (HF). In conclusion, hyperthyroidism is characterized by concurrent sympathetic activation and vagal withdrawal. This sympathovagal imbalance state in hyperthyroidism helps to explain the higher prevalence of atrial fibrillation and exercise intolerance among hyperthyroid patients.

In the recent past, recurrence quantification analysis (RQA) has gained an increasing interest in various research areas. The complexity measures the RQA provides have been useful in describing and analysing a broad range of data. It is known to be rather robust to noise and nonstationarities. Yet, one key question in empirical research concerns the confidence bounds of measured data. In the present Letter we suggest a method for estimating the confidence bounds of recurrence-based complexity measures. We study the applicability of the suggested method with model and real- life data.

Biochemical and genetic regulatory systems that involve low concentrations of molecules are inherently noisy. This intrinsic stochasticity, has received considerable interest recently, leading to new insights about the sources and consequences of noise in complex systems of genetic regulation. However, most prior work was devoted to the reduction of fluctuation and the robustness of cellular function with respect to intrinsic noise. Here, we focus on several scenarios in which the inherent molecular fluctuations are not merely a nuisance, but act constructively and bring about qualitative changes in the dynamics of the system. It will be demonstrated that in many typical situations biochemical and genetic regulatory systems may utilize intrinsic noise to their advantage. (C) 2002 Elsevier Ireland Ltd. All rights reserved

Control of noise-induced oscillations of a pendulum with a rondomly vibrating suspension axis
(1997)

We present a study of ocean convection parameterization based on a novel approach which includes both eddy diffusion and advection and consists of a two-dimensional lattice of bistable maps. This approach retains important features of usual grid models and allows to assess the relative roles of diffusion and advection in the spreading of convective cells. For large diffusion our model exhibits a phase transition from convective patterns to a homogeneous state over the entire lattice. In hysteresis experiments we find staircase behavior depending on stability thresholds of local convection patterns. This nonphysical behavior is suspected to induce spurious abrupt changes in the spreading of convection in ocean models. The final steady state of convective cells depends not only on the magnitude of the advective velocity but also on its direction, implying a possible bias in the development of convective patterns. Such bias points to the need for an appropriate choice of grid geometry in ocean modeling