Evidence Of Low-dimensional Chaos Research Articles

Nonlinear dynamic analysis of Dα signals for type I edge localized modes characterization on JET with a carbon wall

In this paper, the dynamic characteristics of type-I ELM time-series from the JET tokamak, the world’s largest magnetic confinement plasma physics experiment, have been investigated. The dynamic analysis has been focused on the detection of nonlinear structure in Dα radiation time series. Firstly, the method of surrogate data has been applied to evaluate the statistical significance of the null hypothesis of static nonlinear distortion of an underlying Gaussian linear process. Several nonlinear statistics have been evaluated, such us the time delayed mutual information, the correlation dimension and the maximal Lyapunov exponent. The obtained results allow us to reject the null hypothesis, giving evidence of underlying nonlinear dynamics. Moreover, no evidence of low-dimensional chaos has been found; indeed, the analysed time series are better characterized by the power law sensitivity to initial conditions which can suggest a motion at the ‘edge of chaos’, at the border between chaotic and regular non-chaotic dynamics. This uncertainty makes it necessary to further investigate about the nature of the nonlinear dynamics. For this purpose, a second surrogate test to distinguish chaotic orbits from pseudo-periodic orbits has been applied. In this case, we cannot reject the null hypothesis which means that the ELM time series is possibly pseudo-periodic. In order to reproduce pseudo-periodic dynamical properties, a periodic state-of-the-art model, proposed to reproduce the ELM cycle, has been corrupted by a dynamical noise, obtaining time series qualitatively in agreement with experimental time series.

Reliability of unstable periodic orbit based control strategies in biological systems.

Presence of recurrent and statistically significant unstable periodic orbits (UPOs) in time series obtained from biological systems is now routinely used as evidence for low dimensional chaos. Extracting accurate dynamical information from the detected UPO trajectories is vital for successful control strategies that either aim to stabilize the system near the fixed point or steer the system away from the periodic orbits. A hybrid UPO detection method from return maps that combines topological recurrence criterion, matrix fit algorithm, and stringent criterion for fixed point location gives accurate and statistically significant UPOs even in the presence of significant noise. Geometry of the return map, frequency of UPOs visiting the same trajectory, length of the data set, strength of the noise, and degree of nonstationarity affect the efficacy of the proposed method. Results suggest that establishing determinism from unambiguous UPO detection is often possible in short data sets with significant noise, but derived dynamical properties are rarely accurate and adequate for controlling the dynamics around these UPOs. A repeat chaos control experiment on epileptic hippocampal slices through more stringent control strategy and adaptive UPO tracking is reinterpreted in this context through simulation of similar control experiments on an analogous but stochastic computer model of epileptic brain slices. Reproduction of equivalent results suggests that far more stringent criteria are needed for linking apparent success of control in such experiments with possible determinism in the underlying dynamics.

Evidence for low dimensional chaos in sunspot cycles

Sunspot cycles are widely used for investigating solar activity. In 1953 Bracewell argued that it is sometimes desirable to introduce the inversion of the magnetic field polarity, and that can be done with a sign change at the beginning of each cycle. It will be shown in this paper that, for topological reasons, this so-called Bracewell index is inappropriate and that the symmetry must be introduced in a more rigorous way by a coordinate transformation. The resulting symmetric dynamics is then favourably compared with a symmetrized phase portrait reconstructed from the z -variable of the Rossler system. Such a link with this latter variable – which is known to be a poor observable of the underlying dynamics – could explain the general difficulty encountered in finding evidence of low-dimensional dynamics in sunspot data.

Estimation of dynamical invariants without embedding by recurrence plots.

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.

Geometrical characteristics of magnetospheric energetic ion time series: evidence for low dimensional chaos

Abstract. In the first part of the paper we study the geometrical characteristics of the magnetospheric ions’ time series in the reconstructed phase space by using the SVD extended chaotic analysis, and we test the strong null hypothesis supposing that the ions’ time series is caused by a linear stochastic process perturbed by a static nonlinear distortion. The SVD reconstructed spectrum of the ions’ signal reveals a strong component of high dimensional, external coloured noise, as well as an internal low dimensional nonlinear deterministic component. Also, the stochastic Lorenz system produced by coloured noise perturbation of the deterministic Lorenz system was used as an archetype model in comparison with the dynamics of the magnetrospheric ions.Key words. Magnetospheric physics (energetic particles) – Radio science (nonlinear phenomena)

Dynamical characteristics of magnetospheric energetic ion time series: evidence for low dimensional chaos

Abstract. This paper is a companion to the first work (Pavlos et al., 2003), which contains significant results concerning the dynamical characteristics of the magnetospheric energetic ions’ time series. The low dimensional and nonlinear deterministic characteristics of the same time series were described in Pavlos et al. (2003). In this second work we present significant results concerning the Lyapunov spectrum, the mutual information and prediction models. The dynamical characteristics of the magnetospheric ions’ signals are compared with corresponding characteristics obtained for the stochastic Lorenz system when a coloured noise perturbation is present. In addition, the null hypothesis is tested for the dynamical characteristics of the magnetospheric ions’ signal by using nonlinear surrogate data. The results of the above comparisons provide significant evidence for the existence of low dimensional chaotic dynamics underlying the energetic ions’ time series.Key words. Magnetospheric physics (energetic particles) – Radio sciences (nonlinear phenomena)

Dynamics and Correlates of Microscopic Changes in Affect

How are momentary microscopic assessments of behavioral dynamics related to traditional macroscopic, static measures? The present study used the computerized “mouse paradigm” to assess near-instantaneous changes in affect over a short time. Dynamic indices were computed from approximately 1400 measurements from the last 2.5 minutes of a 3-minute assessment. Subjects monitored and reported emotional states by moving a cursor along a dimension from “Sadder” to “Happier.” They also completed self-report measures: Scales 2 (Depression) and 9 (Mania) of the MMPI-2, the Positive and Negative Affect Scale, and the Wisconsin scales of Physical Anhedonia and Hypomanic traits. The instantaneous data yielded a summary measure of mean affect, as well as indices of variability. Complexity indices were derived and examined. Mean affect ratings correlated in expected directions with the paper and pencil measures. There was evidence for low-dimensional chaos in short-term affect dynamics. As hypothesized, greater complexity was associated with Pleasant Affect and Hypomania but was negatively correlated with Anhedonia.

Chaos and stochasticity in volcanic eruptions the case of Mount Etna and Vesuvius

The series of historical eruptions of Mount Etna and Vesuvius volcanoes are analyzed to verify the presence of low-dimensional chaos in the mechanism driving eruptive activity. A recently developed optimal methodology which is efficient on relatively small sets of data is used. The results indicate that there is no evidence of low-dimensional chaos in the sequences considered, and the mechanism appears better described by a classical stationary stochastic process.

Evidence of low dimensional chaos in renal blood flow control in genetic and experimental hypertension

We applied a surrogate data technique to test for nonlinear structure in spontaneous fluctuations of hydrostatic pressure in renal tubules of hypertensive rats. Tubular pressure oscillates at 0.03–0.05 Hz in animals with normal blood pressure, but the fluctuations become irregular with chronic hypertension. Using time series from rats with hypertension we produced surrogate data sets to test whether they represent linearly correlated noise or ‘static’ nonlinear transforms of a linear stochastic process. The correlation dimension and the forecasting error were used as discriminating statistics to compare surrogate with experimental data. The results show that the original experimental time series can be distinguished from both linearly and static nonlinearly correlated noise, indicating that the nonlinear behavior is due to the intrinsic dynamics of the system. Together with other evidence this strongly suggests that a low dimensional chaotic attractor governs renal hemodynamics in hypertension. This appears to be the first demonstration of a transition to chaotic dynamics in an integrated physiological control system occurring in association with a pathological condition.

Evidence of low dimensional chaos in renal blood flow control in genetic and experimental hypertension

Abstract We applied a surrogate data technique to test for nonlinear structure in spontaneous fluctuations of hydrostatic pressure in renal tubules of hypertensive rats. Tubular pressure oscillates at 0.03–0.05 Hz in animals with normal blood pressure, but the fluctuations become irregular with chronic hypertension. Using time series from rats with hypertension we produced surrogate data sets to test whether they represent linearly correlated noise or ‘static’ nonlinear transforms of a linear stochastic process. The correlation dimension and the forecasting error were used as discriminating statistics to compare surrogate with experimental data. The results show that the original experimental time series can be distinguished from both linearly and static nonlinearly correlated noise, indicating that the nonlinear behavior is due to the intrinsic dynamics of the system. Together with other evidence this strongly suggests that a low dimensional chaotic attractor governs renal hemodynamics in hypertension. This appears to be the first demonstration of a transition to chaotic dynamics in an integrated physiological control system occurring in association with a pathological condition.

On the evidence for low-dimensional chaos in an epileptic electroencephalogram

A variant of the method of surrogate data is applied to a single time series from an electroencephalogram (EEG) recording of a patient undergoing an epileptic seizure. The time series is a nearly periodic pattern of spike-and-wave complexes. The surrogate data sets are generated by shuffling the individual spike-and-wave cycles, and correspond to a null hypothesis that there is no deterministic structure in the cycle-to-cycle variability of the orginal data. Using estimates of autocorrelation, correlation dimension, and Lyapunov exponent as discriminating statistics, the evidence for dynamical correlations between successive spike-and-wave patterns is evaluated both formally and informally.

On the evidence for low-dimensional chaos in an epileptic electroencephalogram

A variant of the method of surrogate data is applied to a single time series from an electroencephalogram (EEG) recording of a patient undergoing an epileptic seizure. The time series is a nearly periodic pattern of spike-and-wave complexes. The surrogate data sets are generated by shuffling the individual spike-and-wave cycles, and correspond to a null hypothesis that there is no deterministic structure in the cycle-to-cycle variability of the original data. Using estimates of autocorrelation, correlation dimension, and Lyapunov exponent as discriminating statistics, the evidence for dynamical correlations between successive spike-and-wave patterns is evaluated both formally and informally.

Lack of evidence for low-dimensional chaos in heart rate variability.

The term chaos is used to describe erratic or apparently random time-dependent behavior in deterministic systems. It has been suggested that the variability observed in the normal heart rate may be due to chaos, but this question has not been settled. Heart rate variability was assessed by recordings of consecutive RR intervals in ten healthy subjects using ambulatory ECG. All recordings were performed with the subjects at rest in the supine position. To test for the presence of nonlinearities and/or chaotic dynamics, ten surrogate time series were constructed from each experimental dataset. The surrogate data were tailored to have the same linear dynamics and the same amplitude distribution as the original data. Experimental and surrogate data were then compared using various nonlinear measures. Power spectral analysis of the RR intervals showed a 1/f pattern. The correlation dimension differed only slightly between the experimental and the surrogate data, indicating that linear correlations, and not a "strange" attractor, are the major determinants of the calculated correlation dimension. A test for nonlinear predictability showed coherent nonlinear dynamic structure in the experimental data, but the prediction error as a function of the prediction length increased at a slower rate than characteristic of a low-dimensional chaotic system. There is no evidence for low-dimensional chaos in the time series of RR intervals from healthy human subjects. However, nonlinear determinism is present in the data, and various mechanisms that could generate such determinism are discussed.

Luminosity variations of 3C 345: Is there any evidence of low-dimensional chaos ?

view Abstract Citations (11) References (40) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Luminosity Variations of 3C 345: Is There Any Evidence of Low-dimensional Chaos? Provenzale, A. ; Vio, R. ; Cristiani, S. Abstract We analyze the light curve of the optically violent variable quasar 3C 345, with the goal of determining the possible presence of low- dimensional chaos in the system dynamics. The results of the phase-space reconstruction and of the correlation integral analysis provide a convergent, noninteger estimate of the correlation dimension, naively suggesting the presence of deterministic chaos. However, we show that this result is generated by the long memory and by the power-law shape of the power spectrum of the signal, with no relationship to any underlying chaotic behavior. In fact, the light curve of 3C 345 is consistent with the output of a nonlinear stochastic process, and it is characterized by a well-defined intermittent nature. Publication: The Astrophysical Journal Pub Date: June 1994 DOI: 10.1086/174267 Bibcode: 1994ApJ...428..591P Keywords: Light Curve; Luminous Intensity; Nonlinearity; Quasars; Stochastic Processes; Variations; Correlation; Embedding; Integrals; Numerical Analysis; Procedures; Time Dependence; Astrophysics; METHODS: NUMERICAL; GALAXIES: QUASARS: INDIVIDUAL ALPHANUMERIC: 3C 345 full text sources ADS | data products SIMBAD (1) NED (1)

Discriminating randomness from chaos with application to a weather time series

A new method is presented which permits one to discriminate low-dimensional chaos from randomness. The method consists in fitting autoregressive processes to the data and forecasting future values of the system on the basis of the model selected. We distinguish between 2 possible forecasting techniques of a dynamical system given by experimental series of observations. The “global autoregressive approximation” views the observations as a realization of a stochastic process, whereas the “local autoregressive approximation” views the observations as the realizations of a truly deterministic process. A proper comparison between the predictive skills of the 2 techniques allows us to gain insight into distinguishing low-dimensional chaos from randomness. The procedure has been applied to a daily temperature time series recorded in Trieste (Italy) over the past 40 years (1950–1989). The analysis gives no evidence for low-dimensional chaos, the dynamics being compatible with a limit cycle blurred by red noise. DOI: 10.1034/j.1600-0870.1994.t01-2-00005.x

Finding Chaos in Noisy Systems

SUMMARYIn the past 20 years there has been much interest in the physical and biological sciences in non‐linear dynamical systems that appear to have random, unpredictable behaviour. One important parameter of a dynamical system is the dominant Lyapunov exponent (LE). When the behaviour of the system is compared for two similar initial conditions, this exponent is related to the rate at which the subsequent trajectories diverge. A bounded system with a positive LE is one operational definition of chaotic behaviour. Most methods for determining the LE have assumed thousands of observations generated from carefully controlled physical experiments. Less attention has been given to estimating the LE for biological and economic systems that are subjected to random perturbations and observed over a limited amount of time. Using nonparametric regression techniques (neural networks and thin plate splines) it is possible to estimate the LE consistently. The properties of these methods have been studied with simulated data and are applied to a biological time series: marten fur returns for the Hudson Bay Company (1820–1900). On the basis of a nonparametric analysis there is little evidence for low dimensional chaos in these data. Although these methods appear to work well for systems perturbed by small amounts of noise, finding chaos in a system with a significant stochastic component may be difficult.

Investigation of the nonlinear behavior of a partially ionized, turbulent plasma in a magnetic field

In this paper, a number of the methods of nonlinear dynamics are applied to the study of electrostatic turbulence in a magnetized, steady-state, partially ionized plasma. Electrostatic potential fluctuations were obtained by using a capacitative probe. These signals were captured, digitized, and recorded with a LeCroy transient recorder system interfaced to an IBM-AT personal computer. A commercially available software program was used to calculate power spectra, to reconstruct and plot phase portraits, take Poincaré sections, compute correlation dimensions and Lyapunov exponents, and to perform other manipulations of the time series of electrostatic potential fluctuations obtained from the plasma. Evidence of low-dimensional chaos was sought, and trends were investigated which related the state of the turbulence to such plasma parameters as the anode voltage (rms electrostatic potential), background gas pressure (collisionality), and magnetic induction. These variables were found to have a significant effect on the nonlinear dynamics of the plasma.