Search results for "Lyapunov exponent"
showing 10 items of 55 documents
Neural net classification of REM sleep based on spectral measures as compared to nonlinear measures
2001
In various studies the implementation of nonlinear and nonconventional measures has significantly improved EEG (electroencephalogram) analyses as compared to using conventional parameters alone. A neural network algorithm well approved in our laboratory for the automatic recognition of rapid eye movement (REM) sleep was investigated in this regard. Originally based on a broad range of spectral power inputs, we additionally supplied the nonlinear measures of the largest Lyapunov exponent and correlation dimension as well as the nonconventional stochastic measures of spectral entropy and entropy of amplitudes. No improvement in the detection of REM sleep could be achieved by the inclusion of …
Nonlinear analysis of continuous ECG during sleep II. Dynamical measures
2000
The hypothesis that cardiac rhythms are associated with chaotic dynamics implicating a healthy flexibility has motivated the investigation of continuous ECG with methods of nonlinear system theory. Sleep is known to be associated with modulations of the sympathetic and parasympathetic control of cardiac dynamics. Thus, the differentiation of ECG signals recorded during different sleep stages can serve to determine the usefulness of nonlinear measures in discriminating ECG states in general. For this purpose the following six nonlinear measures were implemented: correlation dimension D2, Lyapunov exponent L1. Kolmogorov entropy K2, as well as three measures derived from the analysis of unsta…
Surrogate data analysis of sleep electroencephalograms reveals evidence for nonlinearity
1996
We tested the hypothesis of whether sleep electroencephalographic (EEG) signals of different time windows (164 s, 82 s, 41 s and 20.5 s) are in accordance with linear stochastic models. For this purpose we analyzed the all-night sleep electroencephalogram of a healthy subject and corresponding Gaussian-rescaled phase randomized surrogates with a battery of five non-linear measures. The following nonlinear measures were implemented: largest Lyapunov exponent L1, correlation dimension D2, and the Green-Savit measures delta 2, delta 4 and delta 6. The hypothesis of linear stochastic data was rejected with high statistical significance. L1 and D2 yielded the most pronounced effects, while the G…
Alterations of Continuous MEG Measures during Mental Activities
2000
In a pilot study, we investigated the topography of 11 continuous MEG measures for the eyes-opened and eyes-closed condition together with three simple mental tasks (mental arithmetic, visual imagery, word generation). One-minute recordings for each condition from 16 right-handed subjects were analyzed. The electrophysiological measures consisted of 6 spectral band measures together with spectral edge frequency and spectral entropy, plus the time-domain-based entropy of amplitudes (ENA) and the nonlinear measures correlation dimension D2 and Lyapunov exponent L1. In summary, our results indicate a pronounced task-dependent difference between the anterior and the posterior region, but no lat…
Deterministic chaos and the first positive Lyapunov exponent: a nonlinear analysis of the human electroencephalogram during sleep
1993
Under selected conditions, nonlinear dynamical systems, which can be described by deterministic models, are able to generate so-called deterministic chaos. In this case the dynamics show a sensitive dependence on initial conditions, which means that different states of a system, being arbitrarily close initially, will become macroscopically separated for sufficiently long times. In this sense, the unpredictability of the EEG might be a basic phenomenon of its chaotic character. Recent investigations of the dimensionality of EEG attractors in phase space have led to the assumption that the EEG can be regarded as a deterministic process which should not be mistaken for simple noise. The calcu…
The calculation of the first positive Lyapunov exponent in sleep EEG data
1993
To help determine if the EEG is quasiperiodic or chaotic we performed a new analysis by calculating the first positive Lyapunov exponent L1 from sleep EEG data. Lyapunov exponents measure the mean exponential expansion or contraction of a flow in phase space. L1 is zero for periodic as well as quasiperiodic processes, but positive in case of chaotic processes expressing the sensitive dependence on initial conditions. We calculated L1 for sleep EEG segments of 15 healthy male subjects corresponding to sleep stages I, II, III, IV and REM (according to Rechtschaffen and Kales). Our investigations support the assumption that EEG signals are neither quasiperiodic waves nor simple noise. Moreover…
Nonlinear analysis of sleep EEG data in schizophrenia: calculation of the principal Lyapunov exponent
1995
The generating mechanism of the electroencephalogram (EEG) points to the hypothesis that EEG signals derive from a nonlinear dynamic system. Hence, the unpredictability of the EEG might be considered as a phenomenon exhibiting its chaotic character. The essential property of chaotic dynamics is the so-called sensitive dependence on initial conditions. This property can be quantified by calculating the system's first positive Lyapunov exponent, L1. We calculated L1 for sleep EEG segments of 13 schizophrenic patients and 13 control subjects that corresponded to sleep stages I, II, III, IV and REM (rapid eye movement), as defined by Rechtschaffen and Kales, for the lead positions Cz and Pz. Du…
Nonlinear analysis of sleep eeg in depression: Calculation of the largest lyapunov exponent
1995
Conventional sleep analysis according to Rechtschaffen and Kales (1968) has provided meaningful contributions to the understanding of disturbed sleep architecture in depression. However, there is no characteristic alteration of the sleep cycle, which could serve as a highly specific feature for depressive illness. Therefore, we started to investigate nonlinear properties of sleep electroencephalographic (EEG) data in order to elucidate functional alterations other than those obtained from classical sleep analysis. The application of methods from nonlinear dynamical system theory to EEG data has led to the assumption that the EEG can be treated as a deterministic chaotic process. Chaotic sys…
Chaotic dynamics and partial hyperbolicity
2017
The dynamics of hyperbolic systems is considered well understood from topological point of view as well as from stochastic point of view. S. Smale and R. Abraham gave an example showing that, in general, the hyperbolic systems are not dense among all differentiable systems. In 1970s, M. Brin and Y. Pesin proposed a new notion: partial hyperbolicity to release the notion of hyperbolicity. One aim of this thesis is to understand the dynamics of certain partially hyperbolic systems from stochastic point of view as well as from topological point of view. From stochastic point of view, we prove the following results: — There exists an open and dense subset U of robustly transitive nonhyperbolic …
Exchange rates expectations and chaotic dynamics: a replication study
2018
Abstract In this paper the author analyzes the behavior of exchange rates expectations for four currencies, by considering a re-calculation and an extension of Resende and Zeidan (Expectations and chaotic dynamics: empirical evidence on exchange rates, Economics Letters, 2008). Considering Lyapunov exponent-based tests results, they are not supportive of chaos in exchange rates expectations, although the so-called 0–1 test strongly supports the chaos hypothesis.