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Finding Chaos in Noisy Systems
, 1991
"... In the past twenty years there has been much interest in the physical and biological sciences in nonlinear dynamical systems that appear to have random, unpredictable behavior. One important parameter of a dynamic system is the dominant Lyapunov exponent (LE). When the behavior of the system is comp ..."
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Cited by 70 (2 self)
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In the past twenty years there has been much interest in the physical and biological sciences in nonlinear dynamical systems that appear to have random, unpredictable behavior. One important parameter of a dynamic system is the dominant Lyapunov exponent (LE). When the behavior 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 behavior. 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 consistently estimate the LE. The properties of these methods have been studied using simulated data and are applied to a biological time series: marten fur returns for the Hudson Bay Company (18201900). Based on a nonparametric analysis there is little evidence for lowdimensional 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.
Phase Space Topography and the Lyapunov Exponent of Electrocorticograms in Partial Seizures
, 1990
"... over a period of time (10 minutes before to 10 minutes after the seizure outburst) revealed a remarkable coherence of theabrupt transient drops of L* for the electrodes that showed the initial ictal onset. The L* values for the electrodes away from the focus exhibited less abrupt transient drops. T ..."
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Cited by 28 (5 self)
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over a period of time (10 minutes before to 10 minutes after the seizure outburst) revealed a remarkable coherence of theabrupt transient drops of L* for the electrodes that showed the initial ictal onset. The L* values for the electrodes away from the focus exhibited less abrupt transient drops. These results indicate that the largest average Lyapunov exponent L can be useful in seizure detection as well as a discriminatory factor for focus localization in multielectrocle analysis. Key words: phase space; chaos; Lyapunov exponents; ECoG; partial epileptic seizures; epileptogenic focus localization. Introduction Longterm recordings of brain electrical activity recorded from scalp and sphenoidal electrodes, depth electrodes or subdural electrodes are employed in our clinical laboratories to localize the origin of seizure discharges in patients with partial (focal) seizures who are candidates for surgical removal of the seizure focus. Currently, in clinical practice, th
Estimating Lyapunov Exponents with Nonparametric Regression
, 1990
"... We discuss procedures based on nonparametric regression for estimating the dominant Lyapunov exponent Al from timeseries data generated by a system x t ..."
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Cited by 11 (1 self)
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We discuss procedures based on nonparametric regression for estimating the dominant Lyapunov exponent Al from timeseries data generated by a system x t
Resonance like phenomena in Lyapunovcalculations from data reconstructed by the timedelaymethod
, 1994
"... The timedelaymethod introduces a certain structure in form of symmetries into reconstructed data sets. We show, that this structure may cause resonancelike phenomena in Lyapunovcalculations. The "resonancee#ects" can be avoided by a small modification of the algorithms. Key words: Ti ..."
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Cited by 1 (1 self)
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The timedelaymethod introduces a certain structure in form of symmetries into reconstructed data sets. We show, that this structure may cause resonancelike phenomena in Lyapunovcalculations. The "resonancee#ects" can be avoided by a small modification of the algorithms. Key words: Timedelaymethod; reconstruction; Lyapunovexponent The timedelaymethod is a widly used technique for the reconstruction of data sets based on the works of Takens [1] and more recently Sauer, Yorke and Casdagli [2]. Inspite of its simple implementation naive application of the timedelay method can produce very misleading results. The reconstruction essentially depends on two parameters: the embedding dimension and the socalled timedelay. According to Whitney's theorem the minimal dimension of the embedding space is 2m, where m is the lowest possible dimension of a manifold that contains the original attractor. Takens [1] could show that embedding into R 2m+1 is generically su#cient. Anyhow, in t...
Linear and Nonlinear Dynamical Systems . . .
, 1996
"... This work presents a methodology for analyzing developmental and physiological time series from the perspective of dynamical systems. An overview of recent advances in nonlinear techniques for time series analysis is presented. Methods for generating a nonlinear dynamical systems analog to a covaria ..."
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This work presents a methodology for analyzing developmental and physiological time series from the perspective of dynamical systems. An overview of recent advances in nonlinear techniques for time series analysis is presented. Methods for generating a nonlinear dynamical systems analog to a covariance matrix are proposed. A novel application of structural equation modeling is proposed in which structural expectations can be fit to these nonlinear dependency matrices. A data set has been selected to demonstrate an application of some of these linear and nonlinear descriptive analyses, a suurogate data null hypothesis test, and nonlinear dependency analysis. The dynamical systems methods are evaluated in the light of (a) whether the techniques can be successfully applied to the example data and if so, (b) whether the results of these analyses provide insight into the processes under study which was not provided by other analyses.
Phase Space Topography and the Lyapunov Exponent of ElectrocorlJcograms in Partial Seizures
"... Summary: Electrocorticograms (ECoG's) from 16 of 68 chronically implanted subdural electrodes, placed over the right temporal cortex in a patient with a right medial temporal focus, were analyzed using methods from nonlinear dynamics. A time series provides information about a large number of p ..."
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Summary: Electrocorticograms (ECoG's) from 16 of 68 chronically implanted subdural electrodes, placed over the right temporal cortex in a patient with a right medial temporal focus, were analyzed using methods from nonlinear dynamics. A time series provides information about a large number of pertinent variables, which may be used to explore and characterize the system's dynamics. These variables and their evolution in time produce the phase portrait of the system. The phase spaces for each of 16 electrodes were constructed and from these the largest average Lyapunov exponents (L's), measures of chaoticity of the system (the larger the L, the more chaotic the system is), were estimated over time for every electrode before, in and after the epileptic seizure for three seizures of the same patient. The start of the seizure corresponds to a simultaneous drop in L values obtained at the electrodes nearest the focus. L values for the rest of the electrodes follow. The mean values of L for all electrodes in the postictal state are larger than the ones in the preictal state, denoting a more chaotic state postictally. The lowest values of L occur during the seizure but they are still positive denoting the presence of a chaotic attractor. Based on the procedure for the estimation of L we were able to develop a methodology for detecting prominent spikes in the ECoG. These measures (L*) calculated over a period of time (10 minutes before to 10 minutes after the seizure outburst) revealed a remarkable coherence of the abrupt transient drops of L * for the electrodes that showed the inital ictal onset. The L * values for the electrodes away from the focus exhibited less abrupt transient drops. These results indicate that the largest average Lyapunov exponent L can be useful in seizure detection as well as a discriminatory factor for focus localization in multielectrode analysis. Key words: phase space; chaos; Lyapunov exponents; ECoG; partial epileptic seizures; epileptogenic focus localization.
The evolution with time of the spatial distribution of the largest Lyapunov exponent on the human epileptic cortex
, 1991
"... The topic of this presentation is the investigation of the epileptic human brain as a nonlinear system that undergoes a phase transition (epileptic seizure). The estimated values of the largest Lyapunov exponent L over time indicated a more chaotic state postictally than ictally or preictally. The s ..."
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The topic of this presentation is the investigation of the epileptic human brain as a nonlinear system that undergoes a phase transition (epileptic seizure). The estimated values of the largest Lyapunov exponent L over time indicated a more chaotic state postictally than ictally or preictally. The start of a seizure corresponds to a simultaneous drop in the values of L at the focal electrode sites. The observed slow cyclic variations in the temporal Lyapunov profiles imply attempts of the system to undergo a phase transition minutes before the seizure's onset. The analysis of the maximum rate of entropy production over space revealed an initial phase difference of minutes preictally at the sites overlying the seizure focus, which progressed to phase locking with a slow entrainment of the rest of the cortical sites shortly before the onset of a seizure. It is also conjectured that the abnormal spiking electrical activity of the brain plays a major role in the unfolding of the phenomeno...
Hyperbolic Trajectories of Time Discretizations
"... A new paradigm for numerically approximating trajectories of an ODE is espoused. We ask for a one to one correspondence between trajectories of an ODE and its discrete approximation. The results enable one, in principal, to compute a trajectory of a discrete approximation, and to use this computatio ..."
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A new paradigm for numerically approximating trajectories of an ODE is espoused. We ask for a one to one correspondence between trajectories of an ODE and its discrete approximation. The results enable one, in principal, to compute a trajectory of a discrete approximation, and to use this computation to rigorously prove the existence of a trajectory of the ODE near the discrete trajectory. More precisely, we formulate an appropriate notion of hyperbolicity for a bounded trajectory of a discrete approximation to an ODE. Our definition is motivated from the continuous case. An example of a discrete trajectory satisfying our definition of hyperbolicity is given. We show that the underlying ODE inherits a unique nearby trajectory which is hyperbolic in the continuous sense. Periodicity is also inherited. The discrete trajectory converges to the continuous trajectory with the `correct' order of convergence. The strength of the hyperbolicity may tend to zero as the accuracy of the approximat...
Applied Mathematics Computing Project
"... this paper. As an additional verication we used Poincare sections, which gave us a geometrical interpretation of the systems considered. ..."
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this paper. As an additional verication we used Poincare sections, which gave us a geometrical interpretation of the systems considered.