Institut für Physik und Astronomie
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We propose a novel approach based on the fluctuation of similarity to identify regimes of distinct dynamical complexity in short time series. A statistical test is developed to estimate the significance of the identified transitions. Our method is verified by uncovering bifurcation structures in several paradigmatic models, providing more complex transitions compared with traditional Lyapunov exponents. In a real-world situation, we apply this method to identify millennial-scale dynamical transitions in Plio-Pleistocene proxy records of the South Asian summer monsoon system. We infer that many of these transitions are induced by the external forcing of the solar insolation and are also affected by internal forcing on Monsoonal dynamics, i.e., the glaciation cycles of the Northern Hemisphere and the onset of the Walker circulation.
Dynamical organization of connection weights is studied in scale-free networks of chaotic oscillators, where the coupling strength of a node from its neighbors develops adaptively according to the local synchronization property between the node and its neighbors. We find that when complete synchronization is achieved, the coupling strength becomes weighted and correlated with the topology due to a hierarchical transition to synchronization in heterogeneous networks. Importantly, such an adaptive process enhances significantly the synchronizability of the networks, which could have meaningful implications in the manipulation of dynamical networks
We introduce a modified dynamical optimization coupling scheme to enhance the synchronizability in the scale- free networks as well as to keep uniform and converging intensities during the transition to synchronization. Further, the size of networks that can be synchronizable exceeds by several orders of magnitude the size of unweighted networks.
Heterogeneity in the degree (connectivity) distribution has been shown to suppress synchronization in networks of symmetrically coupled oscillators with uniform coupling strength (unweighted coupling). Here we uncover a condition for enhanced synchronization in weighted networks with asymmetric coupling. We show that, in the optimum regime, synchronizability is solely determined by the average degree and does not depend on the system size and the details of the degree distribution. In scale-free networks, where the average degree may increase with heterogeneity, synchronizability is drastically enhanced and may become positively correlated with heterogeneity, while the overall cost involved in the network coupling is significantly reduced as compared to the case of unwcighted coupling
Climatic changes are of major importance in landslide generation in the Argentine Andes. Increased humidity as a potential influential factor was inferred from the temporal clustering of landslide deposits during a period of significantly wetter climate, 30,000 years ago. A change in seasonality was tested by comparing past (inferred from annual-layered lake deposits, 30,000 years old) and modern (present-day observations) precipitation changes. Quantitative analysis of cross recurrence plots were developed to compare the influence of the El Nino/Southern Oscillation (ENSO) on present and past rainfall variations. This analysis has shown the stronger influence of NE trades in the location of landslide deposits in the intra-andean basin and valleys, what caused a higher contrast between summer and winter rainfall and an increasing of precipitation in La Nina years. This is believed to reduce thresholds for landslide generation in the arid to semiarid intra-andean basins and valleys.
We present a modern method used in nonlinear time series analysis to investigate the relation of two oscillating systems with respect to their phases, independently of their amplitudes. We study the difference of the phase dynamics between El Nino/Southern Oscillation (ENSO) and the Indian Monsoon on inter-annual time scales. We identify distinct epochs, especially two intervals of phase coherence, 1886 - 1908 and 1964 - 1980, corroborating earlier findings from a new point of view. A significance test shows that the coherence is very unlikely to be the result of stochastic fluctuations. We also detect so far unknown periods of coupling which are invisible to linear methods. These findings suggest that the decreasing correlation during the last decades might be a typical epoch of the ENSO/ Monsoon system having occurred repeatedly. The high time resolution of the method enables us to present an interpretation of how volcanic radiative forcing could cause the coupling
In this paper we show that two dynamical invariants, the second order Renyi entropy and the correlation dimension, can be estimated from recurrence plots (RPs) with arbitrary embedding dimension and delay. This fact is interesting as these quantities are even invariant if no embedding is used. This is an important advantage of RPs compared to other techniques of nonlinear data analysis. These estimates for the correlation dimension and entropy are robust and, moreover, can be obtained at a low numerical cost. We exemplify our results for the Rossler system, the funnel attractor and the Mackey-Glass system. In the last part of the paper we estimate dynamical invariants for data from some fluid dynamical experiments and confirm previous evidence for low dimensional chaos in this experimental system. (C) 2004 American Institute of Physics
We propose a technique to calculate large-scale dimension densities in both higher-dimensional spatio-temporal systems and low-dimensional systems from only a few data points, where known methods usually have an unsatisfactory scaling behavior. This is mainly due to boundary and finite size effects. With our rather simple method we normalize boundary effects and get a significant correction of the dimension estimate. This straightforward approach is basing on rather general assumptions. So even weak coherent structures obtained from small spatial couplings can be detected with this method, what is impossible by using the Lyapunov-dimension density. We demonstrate the efficiency of our technique for coupled logistic maps, coupled tent maps, the Lorenz-attractor and the Roessler-attractor.
Estimation of parameters and unobserved components for nonlinear systems from noisy time series
(2002)
We study the problem of simultaneous estimation of parameters and unobserved states from noisy data of nonlinear time-continuous systems, including the case of additive stochastic forcing. We propose a solution by adapting the recently developed statistical method of unscented Kalman filtering to this problem. Due to its recursive and derivative-free structure, this method minimizes the cost function in a computationally efficient and robust way. It is found that parameters as well as unobserved components can be estimated with high accuracy, including confidence bands, from heavily noise-corrupted data.
Standard time and frequency parameters of heart rate variability (HRV) describe only linear and periodic behaviour, whereas more complex relationships cannot be recognised. A method that may be capable of assessing more complex properties is the non-linear measure of 'renormalised entropy.' A new concept of the method, RE(AR), has been developed, based on a non-linear renormalisation of autoregressive spectral distributions. To test the hypothesis that renormalised entropy may improve the result of high-risk stratification after myocardial infarction, it is applied to a clinical pilot study (41 subjects) and to prospective data of the St George's Hospital post- infarction database (572 patients). The study shows that the new RE(AR) method is more reproducible and more stable in time than a previously introduced method (p<0.001). Moreover, the results of the study confirm the hypothesis that on average, the survivors have negative values of RE(AR) (-0.11+/-0.18), whereas the non-survivors have positive values (0.03+/-0.22, p<0.01). Further, the study shows that the combination of an HRV triangular index and RE(AR) leads to a better prediction of sudden arrhythmic death than standard measurements of HRV. In summary, the new RE(AR) method is an independent measure in HRV analysis that may be suitable for risk stratification in patients after myocardial infarction.
Interacting human activities underlie the patterns of many social, technological, and economic phenomena. Here we present clear empirical evidence from Short Message correspondence that observed human actions are the result of the interplay of three basic ingredients: Poisson initiation of tasks and decision making for task execution in individual humans as well as interaction among individuals. This interplay leads to new types of interevent time distribution, neither completely Poisson nor power-law, but a bimodal combination of them. We show that the events can be separated into independent bursts which are generated by frequent mutual interactions in short times following random initiations of communications in longer times by the individuals. We introduce a minimal model of two interacting priority queues incorporating the three basic ingredients which fits well the distributions using the parameters extracted from the empirical data. The model can also embrace a range of realistic social interacting systems such as e-mail and letter communications when taking the time scale of processing into account. Our findings provide insight into various human activities both at the individual and network level. Our analysis and modeling of bimodal activity in human communication from the viewpoint of the interplay between processes of different time scales is likely to shed light on bimodal phenomena in other complex systems, such as interevent times in earthquakes, rainfall, forest fire, and economic systems, etc.
Experimental evidence of anomalous phase synchronization in two diffusively coupled Chua oscillators
(2006)
We study the transition to phase synchronization in two diffusively coupled, nonidentical Chua oscillators. In the experiments, depending on the used parameterization, we observe several distinct routes to phase synchronization, including states of either in-phase, out-of-phase, or antiphase synchronization, which may be intersected by an intermediate desynchronization regime with large fluctuations of the frequency difference. Furthermore, we report the first experimental evidence of an anomalous transition to phase synchronization, which is characterized by an initial enlargement of the natural frequency difference with coupling strength. This results in a maximal frequency disorder at intermediate coupling levels, whereas usual phase synchronization via monotonic decrease in frequency difference sets in only for larger coupling values. All experimental results are supported by numerical simulations of two coupled Chua models
We study prebifurcation fluctuation amplification in nonlinear oscillators subject to bifurcations of spontaneous symmetry breaking which are manifest in the doubling of stable equilibrium states. Our theoretical estimates of both the linear growth and the nonlinear saturation of the fluctuations are in good agreement with our results from numerical simulations. We show that in the saturation mode, the fluctuation variance is proportional to the standard deviation of the external noise, whereas in the linear mode, the fluctuation variance is proportional to the noise variance. It is demonstrated that the phenomenon of prebifurcation noise amplification is more pronounced in the case of a slow transition through the bifurcation point. The amplification of fluctuations in this case makes it easier to form a symmetric probability of the final equilibrium states. In contrast, for a fast transition through the bifurcation point, the effect of amplification is much less pronounced. Under backward and forward passages through the bifurcation point, a loop of noise-dependent hysteresis emerges here. We find that for a fast transition of the nonlinear oscillator through the bifurcation point, the probability symmetry of the final equilibrium states is destroyed
We give evidence of frequency entrainment of dominant peaks in the chaotic spectra of two coupled chaotic nonautonomous oscillators. At variance with the autonomous case, the phenomenon is here characterized by the vanishing of a previously positive Lyapunov exponent in the spectrum, which takes place for a broad range of the coupling strength parameter. Such a state is studied also for the case of chaotic oscillators with ill-defined phases due to the absence of a unique center of rotation. Different phase synchronization indicators are used to circumvent this difficulty