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Journal articles on the topic 'Lyapunov exponents. eng'

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Published: 4 June 2021

Last updated: 15 February 2022

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1

Kwasniok, Frank. "Fluctuations of finite-time Lyapunov exponents in an intermediate-complexity atmospheric model: a multivariate and large-deviation perspective." Nonlinear Processes in Geophysics 26, no. 3 (July 31, 2019): 195–209. http://dx.doi.org/10.5194/npg-26-195-2019.

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Abstract. The stability properties as characterized by the fluctuations of finite-time Lyapunov exponents around their mean values are investigated in a three-level quasi-geostrophic atmospheric model with realistic mean state and variability. Firstly, the covariance structure of the fluctuation field is examined. In order to identify dominant patterns of collective excitation, an empirical orthogonal function (EOF) analysis of the fluctuation field of all of the finite-time Lyapunov exponents is performed. The three leading modes are patterns where the most unstable Lyapunov exponents fluctuate in phase. These modes are virtually independent of the integration time of the finite-time Lyapunov exponents. Secondly, large-deviation rate functions are estimated from time series of finite-time Lyapunov exponents based on the probability density functions and using the Legendre transform method. Serial correlation in the time series is properly accounted for. A large-deviation principle can be established for all of the Lyapunov exponents. Convergence is rather slow for the most unstable exponent, becomes faster when going further down in the Lyapunov spectrum, is very fast for the near-neutral and weakly dissipative modes, and becomes slow again for the strongly dissipative modes at the end of the Lyapunov spectrum. The curvature of the rate functions at the minimum is linked to the corresponding elements of the diffusion matrix. Also, the joint large-deviation rate function for the first and the second Lyapunov exponent is estimated.

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Dahal, Abinash, Deepashree Devaraj, and Dr N. Pradhan Dr. N. Pradhan. "Topological Characteristics of ECG Signal using Lyapunov Exponent and RBF Network." Indian Journal of Applied Research 1, no. 9 (October 1, 2011): 53–55. http://dx.doi.org/10.15373/2249555x/jun2012/22.

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3

Yakovleva, Tatiana V., Ilya E. Kutepov, Antonina Yu Karas, Nikolai M. Yakovlev, Vitalii V. Dobriyan, Irina V. Papkova, Maxim V. Zhigalov, et al. "EEG Analysis in Structural Focal Epilepsy Using the Methods of Nonlinear Dynamics (Lyapunov Exponents, Lempel–Ziv Complexity, and Multiscale Entropy)." Scientific World Journal 2020 (February 11, 2020): 1–13. http://dx.doi.org/10.1155/2020/8407872.

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This paper analyzes a case with the patient having focal structural epilepsy by processing electroencephalogram (EEG) fragments containing the “sharp wave” pattern of brain activity. EEG signals were recorded using 21 channels. Based on the fact that EEG signals are time series, an approach has been developed for their analysis using nonlinear dynamics tools: calculating the Lyapunov exponent’s spectrum, multiscale entropy, and Lempel–Ziv complexity. The calculation of the first Lyapunov exponent is carried out by three methods: Wolf, Rosenstein, and Sano–Sawada, to obtain reliable results. The seven Lyapunov exponent spectra are calculated by the Sano–Sawada method. For the observed patient, studies showed that with medical treatment, his condition did not improve, and as a result, it was recommended to switch from conservative treatment to surgical. The obtained results of the patient’s EEG study using the indicated nonlinear dynamics methods are in good agreement with the medical report and MRI data. The approach developed for the analysis of EEG signals by nonlinear dynamics methods can be applied for early detection of structural changes.

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Hajipour, Ahmad, and Hamidreza Tavakoli. "Dynamic Analysis and Adaptive Sliding Mode Controller for a Chaotic Fractional Incommensurate Order Financial System." International Journal of Bifurcation and Chaos 27, no. 13 (December 15, 2017): 1750198. http://dx.doi.org/10.1142/s021812741750198x.

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In this study, the dynamic behavior and chaos control of a chaotic fractional incommensurate-order financial system are investigated. Using well-known tools of nonlinear theory, i.e. Lyapunov exponents, phase diagrams and bifurcation diagrams, we observe some interesting phenomena, e.g. antimonotonicity, crisis phenomena and route to chaos through a period doubling sequence. Adopting largest Lyapunov exponent criteria, we find that the system yields chaos at the lowest order of [Formula: see text]. Next, in order to globally stabilize the chaotic fractional incommensurate order financial system with uncertain dynamics, an adaptive fractional sliding mode controller is designed. Numerical simulations are used to demonstrate the effectiveness of the proposed control method.

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WANG, Zhenzhou. "Lyapunov exponents for synchronous 12-lead ECG signals." Chinese Science Bulletin 47, no. 21 (2002): 1845. http://dx.doi.org/10.1360/02tb9403.

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Wang, Zhenzhou, Zheng Li, Yixiang Wei, Xinbao Ning, and Yuzheng Lin. "Lyapunov exponents for synchronous 12-lead ECG signals." Science Bulletin 47, no. 21 (November 2002): 1845–48. http://dx.doi.org/10.1007/bf03183855.

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Übeyli, Elif Derya. "Detecting variabilities of ECG signals by Lyapunov exponents." Neural Computing and Applications 18, no. 7 (January 8, 2009): 653–62. http://dx.doi.org/10.1007/s00521-008-0229-8.

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8

BERGAMASCO, L., M. SERIO, A. R. OSBORNE, and L. CAVALERI. "FINITE CORRELATION DIMENSION AND POSITIVE LYAPUNOV EXPONENTS FOR SURFACE WAVE DATA IN THE ADRIATIC SEA NEAR VENICE." Fractals 03, no. 01 (March 1995): 55–78. http://dx.doi.org/10.1142/s0218348x95000060.

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We study wind-driven surface wave data taken on an offshore platform in 16 m of water, about 20 km from Venice in the Northern Adriatic Sea. The data are investigated for the effects of chaos and to this end they are subjected to a variety of time series analysis techniques from the field of dynamical systems theory. For certain data sets we find a finite value for the correlation dimension (~7) and a positive value for the largest Lyapunov exponent (~1.5×10−3 bit/sec). In spite of the fact that these results suggest the possibility of chaotic behavior in the data, the correct interpretation is that the data are essentially stochastic, and that the correlation dimensions and Lyapunov exponents result from the anomalous statistical behavior of certain near-Gaussian random processes whose properties we discuss.

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9

Übeyli, Elif Derya. "Lyapunov exponents/probabilistic neural networks for analysis of EEG signals." Expert Systems with Applications 37, no. 2 (March 2010): 985–92. http://dx.doi.org/10.1016/j.eswa.2009.05.078.

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GULER, N., E. UBEYLI, and I. GULER. "Recurrent neural networks employing Lyapunov exponents for EEG signals classification." Expert Systems with Applications 29, no. 3 (October 2005): 506–14. http://dx.doi.org/10.1016/j.eswa.2005.04.011.

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11

Kutepov, Ilya E., Vitaliy V. Dobriyan, Maxim V. Zhigalov, Mikhail F. Stepanov, Anton V. Krysko, Tatyana V. Yakovleva, and Vadim A. Krysko. "EEG analysis in patients with schizophrenia based on Lyapunov exponents." Informatics in Medicine Unlocked 18 (2020): 100289. http://dx.doi.org/10.1016/j.imu.2020.100289.

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12

Ayodele, Kayode P., Wisdom O. Ikezogwo, and Anthony A. Osuntuyi. "Empirical Characterization of the Temporal Dynamics of EEG Spectral Components." International Journal of Online and Biomedical Engineering (iJOE) 16, no. 15 (December 15, 2020): 80. http://dx.doi.org/10.3991/ijoe.v16i15.16663.

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The properties of time-domain electroencephalographic data have been studied extensively. There has however been no attempt to characterize the temporal evolution of resulting spectral components when successive segments of electroencephalographic data are decomposed. We analysed resting-state scalp electroencephalographic data from 23 subjects, acquired at 256 Hz, and transformed using 64-point Fast Fourier Transform with a Hamming window. KPSS and Nason tests were administered to study the trend- and wide sense stationarity respectively of the spectral components. Their complexities were estimated using fuzzy entropy. Thereafter, the rosenstein algorithm for dynamic evolution was applied to determine the largest Lyapunov exponents of each component’s temporal evolution. We found that the evolutions were wide sense stationary for time scales up to 8 s, and had significant interactions, especially between spectral series in the frequency ranges 0-4 Hz, 12-24 Hz, and 32-128 Hz. The highest complexity was in the 12-24 Hz band, and increased monotonically with scale for all band sizes. However, the complexity in higher frequency bands changed more rapidly. The spectral series were generally non-chaotic, with average largest Lyapunov exponent of 0. The results show that significant information is contained in all frequency bands, and that the interactions between bands are complicated and time-varying.

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Osowski, Stanislaw, Bartosz Swiderski, Andrzej Cichocki, and Andrzej Rysz. "Epileptic seizure characterization by Lyapunov exponent of EEG signal." COMPEL - The international journal for computation and mathematics in electrical and electronic engineering 26, no. 5 (November 13, 2007): 1276–87. http://dx.doi.org/10.1108/03321640710823019.

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Li, J. C., W. W. Wang, Y. H. Wang, and X. Wu. "The Lyapunov Exponent of the EEG after Administration of Topiramate." Clinical EEG and Neuroscience 36, no. 3 (July 2005): 202–6. http://dx.doi.org/10.1177/155005940503600312.

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The objective of our study was to investigate the effect of topiramate (TPM) on the Lyapunov exponent of EEG by means of quantitative pharmacoelectroencephalography (QPEEG) and nonlinear analysis methods. One dose of TPM was administrated to epileptics and healthy adults. EEG samples were obtained prior to and at regular intervals (at 0.5, 1, 2, 4, 6, 8, 12, 24 hours) within the 24 hours following the administration of TPM. EEG activity was processed with nonlinear analysis methods. The Lyapunov exponent of the scalp areas was calculated through 60 s epochs without artifacts after each recording. The statistical difference between baseline predrug assessment and each postdrug control was calculated by computing the paired t test. Results showed that the Lyapunov exponent increased first, then decreased, then increased finally. We conclude that TPM can change the complexity of EEG.

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15

PARLITZ, ULRICH. "IDENTIFICATION OF TRUE AND SPURIOUS LYAPUNOV EXPONENTS FROM TIME SERIES." International Journal of Bifurcation and Chaos 02, no. 01 (March 1992): 155–65. http://dx.doi.org/10.1142/s0218127492000148.

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A new method for the identification of true and spurious Lyapunov exponents computed from time series is presented. It is based on the observation that the true Lyapunov exponents change their signs upon time reversal whereas the spurious exponents do not. Furthermore by comparison of the spectra of the original data and the reversed time series suitable values for the free parameters of the algorithm used for computing the Lyapunov exponents (e.g., the number of nearest neighbors) are determined. As an example for this general approach an algorithm using local nonlinear approximations of the flow map in embedding space by radial basis functions is presented. For noisy data a regularization method is applied in order to get smooth approximating functions. Numerical examples based on data from the Hénon map, a four-dimensional analog of the Hénon map, a quasiperiodic time series, the Lorenz model, and Duffing’s equation are given.

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Derya Übeyli, Elif. "Recurrent neural networks employing Lyapunov exponents for analysis of ECG signals." Expert Systems with Applications 37, no. 2 (March 2010): 1192–99. http://dx.doi.org/10.1016/j.eswa.2009.06.022.

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17

Xie, Xuping, Peter J. Nolan, Shane D. Ross , Changhong Mou , and Traian Iliescu. "Lagrangian Reduced Order Modeling Using Finite Time Lyapunov Exponents." Fluids 5, no. 4 (October 23, 2020): 189. http://dx.doi.org/10.3390/fluids5040189.

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There are two main strategies for improving the projection-based reduced order model (ROM) accuracy—(i) improving the ROM, that is, adding new terms to the standard ROM; and (ii) improving the ROM basis, that is, constructing ROM bases that yield more accurate ROMs. In this paper, we use the latter. We propose two new Lagrangian inner products that we use together with Eulerian and Lagrangian data to construct two new Lagrangian ROMs, which we denote α-ROM and λ-ROM. We show that both Lagrangian ROMs are more accurate than the standard Eulerian ROMs, that is, ROMs that use standard Eulerian inner product and data to construct the ROM basis. Specifically, for the quasi-geostrophic equations, we show that the new Lagrangian ROMs are more accurate than the standard Eulerian ROMs in approximating not only Lagrangian fields (e.g., the finite time Lyapunov exponent (FTLE)), but also Eulerian fields (e.g., the streamfunction). In particular, the α-ROM can be orders of magnitude more accurate than the standard Eulerian ROMs. We emphasize that the new Lagrangian ROMs do not employ any closure modeling to model the effect of discarded modes (which is standard procedure for low-dimensional ROMs of complex nonlinear systems). Thus, the dramatic increase in the new Lagrangian ROMs’ accuracy is entirely due to the novel Lagrangian inner products used to build the Lagrangian ROM basis.

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18

Zhao, Zhike, Xin Wang, and Xiaoguang Zhang. "Fault Diagnosis of Broken Rotor Bars in Squirrel-Cage Induction Motor of Hoister Based on Duffing Oscillator and Multifractal Dimension." Advances in Mechanical Engineering 6 (January 1, 2014): 849670. http://dx.doi.org/10.1155/2014/849670.

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This paper is to propose a novel fault diagnosis method for broken rotor bars in squirrel-cage induction motor of hoister, which is based on duffing oscillator and multifractal dimension. Firstly, based on the analysis of the structure and performance of modified duffing oscillator, the end of transitional slope from chaotic area to large-scale cycle area is selected as the optimal critical threshold of duffing oscillator by bifurcation diagrams and Lyapunov exponent. Secondly, the phase transformation duffing oscillator from chaos to intermittent chaos is sensitive to the signals, whose frequency difference is quite weak from the reference signal. The spectrums of the largest Lyapunov exponents and bifurcation diagrams of the duffing oscillator are utilized to analyze the variance in different parameters of frequency. Finally, this paper is to analyze the characteristics of both single fractal (box-counting dimension) and multifractal and make a comparison between them. Multifractal detrended fluctuation analysis is applied to detect extra frequency component of current signal. Experimental results reveal that the method is effective for early detection of broken rotor bars in squirrel-cage induction motor of hoister.

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Starchenkova, K. S., and L. A. Manilo. "Using a senior Lyapunov exponent to recognize biomedical signals." Issues of radio electronics, no. 3 (April 26, 2020): 23–29. http://dx.doi.org/10.21778/2218-5453-2020-3-23-29.

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The article considers the actual problem of analysis of biosignals with chaotic properties. Its solution is important for the recognition of various signals associated with a change in the functional state of the patient during continuous observation. The paper considers the possibility of recognizing atrial fibrillation of rhythmogram signals and anesthesia levels by EEG signals using Lyapunov indicators. The initial data are heart rhythm signals with a duration of 300 cardiocycles and five-second EEG signals. The work shows that the senior Lyapunov indicator allows one to recognize atrial fibrillation against the background of normal rhythm signals and frequent extrasystole, as well as the stage of anesthesia by the EEG signal. The developed algorithm is intended for medical computer systems and is implemented in the MATLAB software environment.

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Casaleggio, Aldo, and Stefano Braiotta. "Estimation of Lyapunov exponents of ECG time series—The influence of parameters." Chaos, Solitons & Fractals 8, no. 10 (October 1997): 1591–99. http://dx.doi.org/10.1016/s0960-0779(97)00040-4.

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Majumdar, Kaushik, and Mark H. Myers. "Amplitude Suppression and Chaos Control in Epileptic EEG Signals." Computational and Mathematical Methods in Medicine 7, no. 1 (2006): 53–66. http://dx.doi.org/10.1080/10273660600890032.

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In this paper we have proposed a novel amplitude suppression algorithm for EEG signals collected during epileptic seizure. Then we have proposed a measure of chaoticity for a chaotic signal, which is somewhat similar to measuring sensitive dependence on initial conditions by measuring Lyapunov exponent in a chaotic dynamical system. We have shown that with respect to this measure the amplitude suppression algorithm reduces chaoticity in a chaotic signal (EEG signal is chaotic). We have compared our measure with the estimated largest Lyapunov exponent measure by the largelyap function, which is similar to Wolf's algorithm. They fit closely for all but one of the cases. How the algorithm can help to improve patient specific dosage titration during vagus nerve stimulation therapy has been outlined.

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22

Übeyli, Elif Derya. "Adaptive neuro-fuzzy inference system for classification of ECG signals using Lyapunov exponents." Computer Methods and Programs in Biomedicine 93, no. 3 (March 2009): 313–21. http://dx.doi.org/10.1016/j.cmpb.2008.10.012.

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23

FRIEDLAND, SHMUEL. "Corrections to ‘Discrete Lyapunov exponents and Hausdorff dimension’ 20 (2000), 145–172." Ergodic Theory and Dynamical Systems 20, no. 5 (October 2000): 1551. http://dx.doi.org/10.1017/s0143385700000833.

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The function $d$ defined in (0.3) may not be a metric unless \begin{equation} \phi_p(a_1,\dots, a_p)\le \phi_{p+1} (a_1,\dots, a_{p+1}),\quad \text{for all } (a_i)^{p+1}_1 \in \Gamma^{p+1},\ p=1,\dots.\tag{A} \end{equation} Let $$ \tilde\phi_p(a_1,\dots, a_p)=\max_{1\le i\le p}\phi_1 (a_1,\dots, a_i),\quad (a_i)^p_1\in\Gamma^p,\ p=1,\dots. $$ Then $\tilde\phi$ satisfies (A) and (0.1). Note that \begin{equation} \phi_p(a_1,\dots, a_p) \le\tilde\phi_p (a_1,\dots, a_p)\le p \max_{1\le i\le n} \phi_1(a_i).\tag{B} \end{equation} In (0.3) replace the equality $d(a,b)=e^{-\phi_p(a_1,\dots,a_p)}$ by $d(a,b)=e^{-\tilde\phi_p(a_1,\dots,a_p)}$. Then $d(a,b)\le \max(d(a,c), d(c,b))$. Let \begin{gather*} \tilde\psi_p(x):=\max_{1\le i\le p} \psi_i(x),\ x\in\Gamma^\infty,\quad \tilde\alpha_p(\mu) :=\int \tilde\psi_p\,d\mu,\ p=1, \dots\\ \tilde\alpha(\mu) :=\lim_{p\rightarrow\infty} \frac{\tilde\alpha_p(\mu)}{p}. \end{gather*} Then (B) yields $$ \lim_{p\rightarrow\infty} \frac{\psi_p(x)}{p} = s\Rightarrow \lim_{p\rightarrow\infty} \frac{\tilde\psi_p(x)}{p}=s. $$

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Röschke, J., J. Fell, and P. Beckmann. "The calculation of the first positive Lyapunov exponent in sleep EEG data." Electroencephalography and Clinical Neurophysiology 86, no. 5 (May 1993): 348–52. http://dx.doi.org/10.1016/0013-4694(93)90048-z.

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25

Khoa, Truong Quang Dang, Nguyen Thi Minh Huong, and Vo Van Toi. "Detecting Epileptic Seizure from Scalp EEG Using Lyapunov Spectrum." Computational and Mathematical Methods in Medicine 2012 (2012): 1–11. http://dx.doi.org/10.1155/2012/847686.

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One of the inherent weaknesses of the EEG signal processing is noises and artifacts. To overcome it, some methods for prediction of epilepsy recently reported in the literature are based on the evaluation of chaotic behavior of intracranial electroencephalographic (EEG) recordings. These methods reduced noises, but they were hazardous to patients. In this study, we propose using Lyapunov spectrum to filter noise and detect epilepsy on scalp EEG signals only. We determined that the Lyapunov spectrum can be considered as the most expected method to evaluate chaotic behavior of scalp EEG recordings and to be robust within noises. Obtained results are compared to the independent component analysis (ICA) and largest Lyapunov exponent. The results of detecting epilepsy are compared to diagnosis from medical doctors in case of typical general epilepsy.

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Mazurek, Jiří. "The evaluation of COVID-19 prediction precision with a Lyapunov-like exponent." PLOS ONE 16, no. 5 (May 28, 2021): e0252394. http://dx.doi.org/10.1371/journal.pone.0252394.

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In the field of machine learning, building models and measuring their performance are two equally important tasks. Currently, measures of precision of regression models’ predictions are usually based on the notion of mean error, where by error we mean a deviation of a prediction from an observation. However, these mean based measures of models’ performance have two drawbacks. Firstly, they ignore the length of the prediction, which is crucial when dealing with chaotic systems, where a small deviation at the beginning grows exponentially with time. Secondly, these measures are not suitable in situations where a prediction is made for a specific point in time (e.g. a date), since they average all errors from the start of the prediction to its end. Therefore, the aim of this paper is to propose a new measure of models’ prediction precision, a divergence exponent, based on the notion of the Lyapunov exponent which overcomes the aforementioned drawbacks. The proposed approach enables the measuring and comparison of models’ prediction precision for time series with unequal length and a given target date in the framework of chaotic phenomena. Application of the divergence exponent to the evaluation of models’ accuracy is demonstrated by two examples and then a set of selected predictions of COVID-19 spread from other studies is evaluated to show its potential.

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HOMBURG, ALE JAN, and VAHATRA RABODONANDRIANANDRAINA. "On–off intermittency and chaotic walks." Ergodic Theory and Dynamical Systems 40, no. 7 (January 30, 2019): 1805–42. http://dx.doi.org/10.1017/etds.2018.142.

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We consider a class of skew product maps of interval diffeomorphisms over the doubling map. The interval maps fix the end points of the interval. It is assumed that the system has zero fiber Lyapunov exponent at one endpoint and zero or positive fiber Lyapunov exponent at the other endpoint. We prove the appearance of on–off intermittency. This is done using the equivalent description of chaotic walks: random walks driven by the doubling map. The analysis further relies on approximating the chaotic walks by Markov random walks, that are constructed using Markov partitions for the doubling map.

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Bondarenko, Vladimir E. "Analog Neural Network Model Produces Chaos Similar to the Human EEG." International Journal of Bifurcation and Chaos 07, no. 05 (May 1997): 1133–40. http://dx.doi.org/10.1142/s0218127497000935.

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The self-organization processes in an analog asymmetric neural network with the time delay were considered. It was shown that in dependence on the value of coupling constants between neurons the neural network produced sinusoidal, quasi-periodic or chaotic outputs. The correlation dimension, largest Lyapunov exponent, Shannon entropy and normalized Shannon entropy of the solutions were studied from the point of view of the self-organization processes in systems far from equilibrium state. The quantitative characteristics of the chaotic outputs were compared with the human EEG characteristics. The calculation of the correlation dimension ν shows that its value is varied from 1.0 in case of sinusoidal oscillations to 9.5 in chaotic case. These values of ν agree with the experimental values from 6 to 8 obtained from the human EEG. The largest Lyapunov exponent λ calculated from neural network model is in the range from -0.2 s -1 to 4.8 s -1 for the chaotic solutions. It is also in the interval from 0.028 s -1 to 2.9 s -1 of λ which is observed in experimental study of the human EEG.

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WANG, Nai. "Principal component cluster analysis of ECG time series based on Lyapunov exponent spectrum." Chinese Science Bulletin 49, no. 18 (2004): 1980. http://dx.doi.org/10.1360/04we0078.

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Wang, Nai, and Jiong Ruan. "Principal component cluster analysis of ECG time series based on Lyapunov exponent spectrum." Chinese Science Bulletin 49, no. 18 (September 2004): 1980–85. http://dx.doi.org/10.1007/bf03184292.

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Röschke, J., J. Fell, and P. Beckmann. "Nonlinear analysis of sleep eeg in depression: Calculation of the largest lyapunov exponent." European Archives of Psychiatry and Clinical Neuroscience 245, no. 1 (March 1995): 27–35. http://dx.doi.org/10.1007/bf02191541.

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KOWALIK, ZBIGNIEW J., and THOMAS ELBERT. "A PRACTICAL METHOD FOR THE MEASUREMENTS OF THE CHAOTICITY OF ELECTRIC AND MAGNETIC BRAIN ACTIVITY." International Journal of Bifurcation and Chaos 05, no. 02 (April 1995): 475–90. http://dx.doi.org/10.1142/s0218127495000375.

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The degree of complexity of the cortical processes related to different cognitive actions should logically produce an image in the electric and magnetic signals of the human brain. Hence, those measures of chaoticity will depend on the particular task being investigated. In many cases, the duration of a brain process does not allow for the generated signal to be treated as stationary. Therefore, the application of the standard method of nonlinear system theory is often questionable. The chaoticity of the process has to characterize the predictability of future state of the system. In the stationary case, such a quantity can be directly expressed by the largest Lyapunov exponent or by K-S entropy. In this study, we performed a test for the applicability of the local Lyapunov exponent for the description of the chaoticity of the brain processes measured in EEG and MEG experiments. We demonstrate an algorithm for computation of chaoticity based on the local Lyapunov exponent and present possible applications of this method for specific cases with a diagnosis of schizophrenia or of tinnitus. We also show that chaoticity is able to detect critical transitions (phase-transition-like phenomena) which occur in the dynamics of neural mass activity at a specific point in time.

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Röschke, Joachim, Jürgen Fell, and Peter Beckmann. "Nonlinear analysis of sleep EEG data in schizophrenia: calculation of the principal Lyapunov exponent." Psychiatry Research 56, no. 3 (April 1995): 257–69. http://dx.doi.org/10.1016/0165-1781(95)02562-b.

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Pradhan, N., and P. K. Sadasivan. "The nature of dominant lyapunov exponent and attractor dimension curves of eeg in sleep." Computers in Biology and Medicine 26, no. 5 (September 1996): 419–28. http://dx.doi.org/10.1016/0010-4825(96)00019-4.

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Geng, Yi Xiang, and Han Ze Liu. "Subharmonic Bifurcations and Transition to Chaos in a Pipe Conveying Fluid under Harmonic Excitation." Applied Mechanics and Materials 444-445 (October 2013): 791–95. http://dx.doi.org/10.4028/www.scientific.net/amm.444-445.791.

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The subharmonic and chaotic behavior of a two end-fixed fluid conveying pipe whose base is subjected to a harmonic excitation are investigated. Melnikov method is applied for the system, and Melnikov criterions for subharmonic and homoclinic bifurcations are obtained analytically. The numerical simulations (including bifurcation diagrams, maximal Lyapunov exponents, phase portraits and Poincare map) confirm the analytical predictions and exhibit the complicated dynamical behaviors.

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Liu, Ya Chong, An Kang Hu, and Feng Lei Han. "Numerical Identification of Ship-Roll Chaos Threshold." Applied Mechanics and Materials 556-562 (May 2014): 3078–83. http://dx.doi.org/10.4028/www.scientific.net/amm.556-562.3078.

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Melnikov function is currently the only way to theoretically resolve the chaotic threshold. Considering the calculation difficulties of Melnikov function, Gauss-Legendre numerical method is accepted in this paper to ascertain the chaotic threshold of a nonlinear system. Two forms of numerical technique, namely Lyapunov exponents and phase plan are adopted to validate the computation results. The method is applied to the ship-roll system and the chaos threshold is numerically computed in the end.

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Aftanas, Ljubomir I., Natalia V. Lotova, Vladimir I. Koshkarov, Vera L. Pokrovskaja, Serguei A. Popov, and Victor P. Makhnev. "Non-linear analysis of emotion EEG: calculation of Kolmogorov entropy and the principal Lyapunov exponent." Neuroscience Letters 226, no. 1 (April 1997): 13–16. http://dx.doi.org/10.1016/s0304-3940(97)00232-2.

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Kim, Dai-Jin, Jaeseung Jeong, Jeong-Ho Chae, Seongchong Park, Soo Yong Kim, Hyo Jin Go, In-Ho Paik, Kwang-Soo Kim, and Bomoon Choi. "An estimation of the first positive Lyapunov exponent of the EEG in patients with schizophrenia." Psychiatry Research: Neuroimaging 98, no. 3 (May 2000): 177–89. http://dx.doi.org/10.1016/s0925-4927(00)00052-4.

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Shayegh, F., S. Sadri, R. Amirfattahi, and K. Ansari-Asl. "A model-based method for computation of correlation dimension, Lyapunov exponents and synchronization from depth-EEG signals." Computer Methods and Programs in Biomedicine 113, no. 1 (January 2014): 323–37. http://dx.doi.org/10.1016/j.cmpb.2013.08.014.

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LEGA, ELENA, GABRIELLA DELLA PENNA, CLAUDE FROESCHLÉ, and ALESSANDRA CELLETTI. "ON THE COMPUTATION OF LYAPUNOV EXPONENTS FOR DISCRETE TIME SERIES: APPLICATIONS TO TWO-DIMENSIONAL SYMPLECTIC AND DISSIPATIVE MAPPINGS." International Journal of Bifurcation and Chaos 10, no. 12 (December 2000): 2791–805. http://dx.doi.org/10.1142/s0218127400001857.

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Many techniques have been developed for the measure of the largest Lyapunov exponent of experimental short data series. The main idea, underlying the most common algorithms, is to mimic the method of computation proposed by Benettin and Galgani [1979]. The aim of the present paper is to provide an explanation for the reliability of some algorithms developed for short time series. To this end, we consider two-dimensional mappings as model problems and we compare the results obtained using the Benettin and Galgani method to those obtained using some algorithms for the computation of the largest Lyapunov exponent when dealing with short data series. In particular we focus our attention on conservative systems, which are not widely investigated in the literature. We show that using standard techniques the results obtained for discrete series related to area-preserving mappings are often unreliable, while dissipative systems are easier to analyze. In order to overcome the problem arising with conservative systems, we develop an alternative method, which takes advantage of the existing techniques. In particular, all algorithms provide a good approximation of the largest Lyapunov exponent in the strong chaotic symplectic case and in the dissipative one. However, the application of standard algorithms provides results which are not in agreement with the theoretical expectation for weak chaotic motions, and sometimes also for regular orbits. On the contrary, the method that we propose in this paper seems to work well for the weak chaotic case. Because of the speed of computation, we suggest to use all algorithms to cross-check the results.

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LI, YONGMEI, WANYU TAN, KAIXUAN TAN, ZEHUA LIU, and YANSHI XIE. "FRACTAL AND CHAOS ANALYSIS FOR DYNAMICS OF RADON EXHALATION FROM URANIUM MILL TAILINGS." Fractals 24, no. 03 (August 30, 2016): 1650029. http://dx.doi.org/10.1142/s0218348x16500298.

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Tailings from mining and milling of uranium ores potentially are large volumes of low-level radioactive materials. A typical environmental problem associated with uranium tailings is radon exhalation, which can significantly pose risks to environment and human health. In order to reduce these risks, it is essential to study the dynamical nature and underlying mechanism of radon exhalation from uranium mill tailings. This motivates the conduction of this study, which is based on the fractal and chaotic methods (e.g. calculating the Hurst exponent, Lyapunov exponent and correlation dimension) and laboratory experiments of the radon exhalation rates. The experimental results show that the radon exhalation rate from uranium mill tailings is highly oscillated. In addition, the nonlinear analyses of the time series of radon exhalation rate demonstrate the following points: (1) the value of Hurst exponent much larger than 0.5 indicates non-random behavior of the radon time series; (2) the positive Lyapunov exponent and non-integer correlation dimension of the time series imply that the radon exhalation from uranium tailings is a chaotic dynamical process; (3) the required minimum number of variables should be five to describe the time evolution of radon exhalation. Therefore, it can be concluded that the internal factors, including heterogeneous distribution of radium, and randomness of radium decay, as well as the fractal characteristics of the tailings, can result in the chaotic evolution of radon exhalation from the tailings.

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Nayak, Suraj K., Arindam Bit, Anilesh Dey, Biswajit Mohapatra, and Kunal Pal. "A Review on the Nonlinear Dynamical System Analysis of Electrocardiogram Signal." Journal of Healthcare Engineering 2018 (2018): 1–19. http://dx.doi.org/10.1155/2018/6920420.

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Electrocardiogram (ECG) signal analysis has received special attention of the researchers in the recent past because of its ability to divulge crucial information about the electrophysiology of the heart and the autonomic nervous system activity in a noninvasive manner. Analysis of the ECG signals has been explored using both linear and nonlinear methods. However, the nonlinear methods of ECG signal analysis are gaining popularity because of their robustness in feature extraction and classification. The current study presents a review of the nonlinear signal analysis methods, namely, reconstructed phase space analysis, Lyapunov exponents, correlation dimension, detrended fluctuation analysis (DFA), recurrence plot, Poincaré plot, approximate entropy, and sample entropy along with their recent applications in the ECG signal analysis.

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Röschke, J., J. Fell, and K. Mann. "Non-linear dynamics of alpha and theta rhythm: correlation dimensions and Lyapunov exponents from healthy subject's spontaneous EEG." International Journal of Psychophysiology 26, no. 1-3 (June 1997): 251–61. http://dx.doi.org/10.1016/s0167-8760(97)00768-x.

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Parisi, J., J. Peinke, and R. P. Huebener. "Evidence of Chaotic Hierarchy in a Semiconductor Experiment." Zeitschrift für Naturforschung A 44, no. 11 (November 1, 1989): 1046–50. http://dx.doi.org/10.1515/zna-1989-1103.

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We study the cooperative spatio-temporal behavior of semiconductor breakdown via both probabilistic and dynamical characterization methods (fractal dimensions, entropies, Lyapunov exponents, and the corresponding scaling functions). Agreement between the results obtained from the different numerical concepts (e.g., verification of the Kaplan-Yorke conjecture and the Newhouse- Ruelle-Takens theorem) gives a self-consistent picture of the physical situation investigated. As a consequence, the affirmed chaotic hierarchy of generalized horseshoe-type strange attractors may be ascribed to weak nonlinear coupling between competing localized oscillation centers intrinsic to the present semiconductor system

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Nejad, Hadi Chahkandi, Omid Khayat, and Javad Razjouyan. "CHAOTIC FEATURE EXTRACTION AND NEURO-FUZZY CLASSIFIER FOR ECG SIGNAL CHARACTERIZATION." Biomedical Engineering: Applications, Basis and Communications 26, no. 03 (March 17, 2014): 1450038. http://dx.doi.org/10.4015/s1016237214500380.

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In this paper, a neuro-fuzzy network is employed to classify the ECG beats based on the extracted chaotic features. Six groups of ECG beats (MIT-BIH Normal Sinus rhythm, BIDMC congestive heart failure, CU ventricular tachyarrhythmia, MIT-BIH atrial fibrillation, MIT-BIH Malignant Ventricular Arrhythmia and MIT-BIH supraventricular arrhythmia) are characterized by the six chaotic parameters including the largest Lyapunov exponent and average of the Lyapunov spectrum (related to the chaoticity of the signal), time lag and embedding dimension (related to the phase space reconstruction) and correlation dimension and approximate entropy of the signal (related to the complexity of the signal). Finally, six structures of the neuro-fuzzy network (in terms of the type of fuzzy set, the number of fuzzy sets per variable and the number of learning epochs) were employed to perform the ECG beats classification based on all extracted features for two lengths of the signals. It was found that all respective chaotic features are discriminative and they improve the classification rate of ECG beats. Also, it is shown that a minimum length of the signal is needed for exhibitive feature extraction and for the higher lengths of the signal (in time) no significant improvement is achieved in feature extraction and calculations. The criteria for the classification task are considered as accuracy, specificity and sensitivity which all together comprehensively demonstrate the capability and performance of the classification. Some conclusions are drawn and they are discussed at the end of the paper.

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Krystal, Andrew D., Craig Zaidman, Henry S. Greenside, Richard D. Weiner, and C. Edward Coffey. "The largest Lyapunov exponent of the EEG during ECT seizures as a measure of ECT seizure adequacy." Electroencephalography and Clinical Neurophysiology 103, no. 6 (December 1997): 599–606. http://dx.doi.org/10.1016/s0013-4694(97)00062-x.

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MARTIS, ROSHAN JOY, JEN HONG TAN, CHUA KUANG CHUA, TOO CHEAH LOON, SHARON WAN JIE YEO, and LOUIS TONG. "EPILEPTIC EEG CLASSIFICATION USING NONLINEAR PARAMETERS ON DIFFERENT FREQUENCY BANDS." Journal of Mechanics in Medicine and Biology 15, no. 03 (June 2015): 1550040. http://dx.doi.org/10.1142/s0219519415500402.

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Epilepsy is a chronic neurological disorder with considerable incidence and affects the population everywhere in the world. It occurs due to recurrent unprovoked seizures which can be noninvasively diagnosed using electroencephalograms (EEGs) which are the neuronal electrical activity recorded on the scalp. The EEG signal is highly random, nonlinear, nonstationary and non-Gaussian in nature. The nonlinear features characterize the EEG more accurately than linear models. EEG comprsises of different activities like delta, theta, lower alpha, upper alpha, lower beta, upper beta and lower gamma which are correlated to the brain anatomy and its function. In the current study, the nonlinear features such as Hurst exponent (HE), Higuchi fractal dimension (HFD), largest Lyapunov exponent (LLE) and sample entropy (SE) are extracted on these individual activities to provide improved discrimination. The ranked features are classified using support vector machine (SVM) with different kernel functions, decision tree (DT) and k-nearest neighbor (KNN) to select the best classifier. It is observed that SVM with radial basis function (RBF) kernel provides highest accuracy of 98%, sensitivity and specificity of 99.5% and 100%, respectively using five features. The developed methodology is ready for epilepsy screening and can be deployed in many programmes.

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NEWMAN, JULIAN. "Necessary and sufficient conditions for stable synchronization in random dynamical systems." Ergodic Theory and Dynamical Systems 38, no. 5 (January 24, 2017): 1857–75. http://dx.doi.org/10.1017/etds.2016.109.

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For a composition of independent and identically distributed random maps or a memoryless stochastic flow on a compact space$X$, we find conditions under which the presence of locally asymptotically stable trajectories (e.g. as given by negative Lyapunov exponents) implies almost-sure mutual convergence of any given pair of trajectories (‘synchronization’). Namely, we find that synchronization occurs and is ‘stable’ if and only if the system exhibits the following properties: (i) there is asmallestnon-empty invariant set$K\subset X$; (ii) any two points in$K$are capable of being moved closer together; and (iii) $K$admits asymptotically stable trajectories.

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AGOP, M., V. P. PAUN, and T. DANDU-BIBIRE. "CHAOS VIA FRACTALITY IN GRAVITATIONAL SYSTEMS DYNAMICS: A NEW APPROACH (I)." International Journal of Bifurcation and Chaos 22, no. 12 (December 2012): 1250299. http://dx.doi.org/10.1142/s0218127412502999.

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The neutral particle motion in a composite field (gravitoelectromagnetic field overlapping a constant external gravitomagnetic field) is analyzed. A detailed nonlinear dynamics description, using temporal series, Poincaré sections, phase space, Lyapunov exponents, bifurcation diagrams and fractal analysis, is performed. New phenomena, e.g. the gravitational gun-type, gravitational chaotic gun-type and gravitational multigun-type effects are discovered, if the amplitude of the gravitoelectromagnetic field is sufficiently large. These effects induce a high acceleration of the neutral particles which lead to sudden jumps between different Larmor-type orbits. It results both in a chaotic behavior and patterns formation in gravitational systems.

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Xin, Baogui, Junhai Ma, and Qin Gao. "Complex Dynamics of an Adnascent-Type Game Model." Discrete Dynamics in Nature and Society 2008 (2008): 1–12. http://dx.doi.org/10.1155/2008/467972.

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The paper presents a nonlinear discrete game model for two oligopolistic firms whose products are adnascent. (In biology, the term adnascent has only one sense, “growing to or on something else,” e.g., “moss is an adnascent plant.” See Webster's Revised Unabridged Dictionary published in 1913 by C. & G. Merriam Co., edited by Noah Porter.) The bifurcation of its Nash equilibrium is analyzed with Schwarzian derivative and normal form theory. Its complex dynamics is demonstrated by means of the largest Lyapunov exponents, fractal dimensions, bifurcation diagrams, and phase portraits. At last, bifurcation and chaos anticontrol of this system are studied.

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