Chaotic Dynamics of Electroencephalogram Parameters

2015 ◽  
Vol 22 (2) ◽  
pp. 38-73
Author(s):  
Горбунов ◽  
D. Gorbunov ◽  
Вохмина ◽  
Yu. Vokhmina ◽  
Еськов ◽  
...  
Keyword(s):  
Phase Space ◽  
Chaotic Dynamics ◽  
Clinical Medicine ◽  
Mental States ◽  
Rate Of Change ◽  
Estimation Of Parameters ◽  
Brain Potentials ◽  
Dimensional Phase Space ◽  
The Matrix ◽  
Entropy Parameter

A clinical medicine usually uses the amplitude-frequency characteristics of recorded biopotentials. The paper presents still chaotic dynamics, which takes place for the autocorrelation function A(t) and statistic function of distribution fix). It is proposed two new approaches for estimation of parameters of electroencephalograms. The first is based on the construction of the matrix of pairwise comparisons of recorded samples of brain potentials in subjects in different physiological (mental) states: with photostimulation and without it. The second method is based on calculating the parameters of quasi-attractors, which are built on the plane in the coordinates x1=U(t) - a function that changes due to registration of biopotential in current point and x2=dx1/dt - rate of change of x1. The quasi-attractors in this two-dimensional phase space are quantitatively different options for patients (epilepsy) and healthy subjects. It is possible to construct a three-compartmental model quasi-attractor in the phase space of states, which is also discussed. It was demonstrated the stochastic of function fix) for normal and pathological patients. It was proved the nonfictions of entropy parameter for the EEG modeling.

Stochastic and chaotic evaluation of parameters of lectroencephalogram

2015 ◽  
Vol 4 (1) ◽  
pp. 54-59 ◽  
Author(s):  
Горбунов ◽  
D. Gorbunov ◽  
Вохмина ◽  
Yu. Vokhmina ◽  
Еськов ◽  
...  
Keyword(s):  
Phase Space ◽  
Autocorrelation Function ◽  
Compartmental Model ◽  
Mental States ◽  
Rate Of Change ◽  
Pairwise Comparisons ◽  
Estimation Of Parameters ◽  
Brain Potentials ◽  
The Matrix ◽  
Evaluation Of Parameters

Traditionally parameters of amplitude-frequency characteristics of recorded biopotentials are used in EEG. In this paper we show that they are still chaotic, which takes place for the autocorrelation function A(t). It is proposed the two new approaches for estimation of parameters of electroencephalograms. The first is based on the construction of the matrix of pairwise comparisons of recorded samples of brain potentials in subjects in different physiological (mental) states. The second method is based on calculating the parameters of quasi-attractors, which are built on the plane in the coordinates x1=U(t) – a function that changes due to registration of biopotential in cur-rent point and x2=dx1/dt - rate of change of x1. It is possible to construct a three-compartmental model quasi-attractor in the phase space of states, which is also discussed

Statistical and evaluation chaotic parameters under cardio exertion

2015 ◽  
Vol 4 (2) ◽  
pp. 5-10
Author(s):  
Филатова ◽  
D. Filatova ◽  
Карпин ◽  
Vladimir Karpin ◽  
Еськов ◽  
...  
Keyword(s):  
Phase Space ◽  
Cardiovascular System ◽  
Chaos Theory ◽  
Chaotic Dynamics ◽  
Stochastic Approach ◽  
Dynamic Exercise ◽  
Self Organization ◽  
Dimensional Phase Space ◽  
Classical Statistics ◽  
Fitness Indicators

Methods of classical statistics and the theory of chaos and self-organization studied the behavior of the vector of the cardiovascular system in groups of students trained and untrained in response to dosed physical stress .It was found that students without physical fitness indicators of cardio area quasi-attractors increased after exercise . The study had shown significant changes in the dynamics of the behavior of the parameters of functional systems of the human body compared to the stochastic approach based on the histogram and Shannon entropy . It is shown the feasibility of application of chaos theory, self-organization in the evaluation of the reaction of the cardiovascular system of the person on the dynamic exercise. As a measure of the cardiovascular system of the person (to load and after the load) used quasi-attractor motion of the state vector of the system in the two-dimensional phase space of states. Within the framework of the theory of chaos and self-organization may determine the parameters of the spacecraft for individual subjects and their groups to compare their chaotic dynamics in time or in the phase space of states.

A one-dimensional self-gravitating stellar gas

1966 ◽  
Vol 25 ◽  
pp. 46-48 ◽  
Author(s):  
M. Lecar
Keyword(s):  
Gravitational Field ◽  
Phase Space ◽  
Motion Picture ◽  
Space Distribution ◽  
Two Dimensional ◽  
One Dimensional ◽  
Phase Space Distribution ◽  
Dimensional Phase Space ◽  
The Mean

“Dynamical mixing”, i.e. relaxation of a stellar phase space distribution through interaction with the mean gravitational field, is numerically investigated for a one-dimensional self-gravitating stellar gas. Qualitative results are presented in the form of a motion picture of the flow of phase points (representing homogeneous slabs of stars) in two-dimensional phase space.

scholarly journals Geometric particle-in-cell methods for the Vlasov–Maxwell equations with spin effects

2021 ◽  
Vol 87 (3) ◽  
Author(s):  
Nicolas Crouseilles ◽  
Paul-Antoine Hervieux ◽  
Yingzhe Li ◽  
Giovanni Manfredi ◽  
Yajuan Sun
Keyword(s):  
Phase Space ◽  
Maxwell Equations ◽  
Stimulated Raman Scattering ◽  
Extended Phase ◽  
Five Dimensions ◽  
Particle In Cell ◽  
Spin Effects ◽  
Dimensional Phase Space ◽  
Spin Polarized ◽  
Underdense Plasma

We propose a numerical scheme to solve the semiclassical Vlasov–Maxwell equations for electrons with spin. The electron gas is described by a distribution function $f(t,{\boldsymbol x},{{{\boldsymbol p}}}, {\boldsymbol s})$ that evolves in an extended 9-dimensional phase space $({\boldsymbol x},{{{\boldsymbol p}}}, {\boldsymbol s})$ , where $\boldsymbol s$ represents the spin vector. Using suitable approximations and symmetries, the extended phase space can be reduced to five dimensions: $(x,{{p_x}}, {\boldsymbol s})$ . It can be shown that the spin Vlasov–Maxwell equations enjoy a Hamiltonian structure that motivates the use of the recently developed geometric particle-in-cell (PIC) methods. Here, the geometric PIC approach is generalized to the case of electrons with spin. Total energy conservation is very well satisfied, with a relative error below $0.05\,\%$ . As a relevant example, we study the stimulated Raman scattering of an electromagnetic wave interacting with an underdense plasma, where the electrons are partially or fully spin polarized. It is shown that the Raman instability is very effective in destroying the electron polarization.

scholarly journals Dynamical properties and extremes of Northern Hemisphere climate fields over the past 60 years

2017 ◽  
Vol 24 (4) ◽  
pp. 713-725 ◽  
Author(s):  
Davide Faranda ◽  
Gabriele Messori ◽  
M. Carmen Alvarez-Castro ◽  
Pascal Yiou
Keyword(s):  
Phase Space ◽  
Sea Level ◽  
Northern Hemisphere ◽  
Atmospheric Dynamics ◽  
Complex Nature ◽  
Human Welfare ◽  
Sea Level Pressure ◽  
Level Pressure ◽  
Dimensional Phase Space ◽  
Dynamical Properties

Abstract. Atmospheric dynamics are described by a set of partial differential equations yielding an infinite-dimensional phase space. However, the actual trajectories followed by the system appear to be constrained to a finite-dimensional phase space, i.e. a strange attractor. The dynamical properties of this attractor are difficult to determine due to the complex nature of atmospheric motions. A first step to simplify the problem is to focus on observables which affect – or are linked to phenomena which affect – human welfare and activities, such as sea-level pressure, 2 m temperature, and precipitation frequency. We make use of recent advances in dynamical systems theory to estimate two instantaneous dynamical properties of the above fields for the Northern Hemisphere: local dimension and persistence. We then use these metrics to characterize the seasonality of the different fields and their interplay. We further analyse the large-scale anomaly patterns corresponding to phase-space extremes – namely time steps at which the fields display extremes in their instantaneous dynamical properties. The analysis is based on the NCEP/NCAR reanalysis data, over the period 1948–2013. The results show that (i) despite the high dimensionality of atmospheric dynamics, the Northern Hemisphere sea-level pressure and temperature fields can on average be described by roughly 20 degrees of freedom; (ii) the precipitation field has a higher dimensionality; and (iii) the seasonal forcing modulates the variability of the dynamical indicators and affects the occurrence of phase-space extremes. We further identify a number of robust correlations between the dynamical properties of the different variables.

scholarly journals Effects of Vegetation and Topography on the Boundary Layer Structure above the Amazon Forest

2020 ◽  
Vol 77 (8) ◽  
pp. 2941-2957
Author(s):  
Marcelo Chamecki ◽  
Livia S. Freire ◽  
Nelson L. Dias ◽  
Bicheng Chen ◽  
Cléo Quaresma Dias-Junior ◽  
...  
Keyword(s):  
Boundary Layer ◽  
Phase Space ◽  
Vertical Structure ◽  
Layer Structure ◽  
Roughness Sublayer ◽  
Large Contribution ◽  
Amazon Forest ◽  
Two Dimensional ◽  
Dimensional Phase Space ◽  
Field Campaigns

Abstract Observational data from two field campaigns in the Amazon forest were used to study the vertical structure of turbulence above the forest. The analysis was performed using the reduced turbulent kinetic energy (TKE) budget and its associated two-dimensional phase space. Results revealed the existence of two regions within the roughness sublayer in which the TKE budget cannot be explained by the canonical flat-terrain TKE budgets in the canopy roughness sublayer or in the lower portion of the convective ABL. Data analysis also suggested that deviations from horizontal homogeneity have a large contribution to the TKE budget. Results from LES of a model canopy over idealized topography presented similar features, leading to the conclusion that flow distortions caused by topography are responsible for the observed features in the TKE budget. These results support the conclusion that the boundary layer above the Amazon forest is strongly impacted by the gentle topography underneath.

One-Dimensional Harmonic Oscillator in Quantum Phase Space

2011 ◽  
Vol 110-116 ◽  
pp. 3750-3754
Author(s):  
Jun Lu ◽  
Xue Mei Wang ◽  
Ping Wu
Keyword(s):  
Phase Space ◽  
Harmonic Oscillator ◽  
Quantum Phase ◽  
Space Representation ◽  
Wave Mechanics ◽  
One Dimensional ◽  
Matrix Mechanics ◽  
The Matrix ◽  
Quantum Wave ◽  
The One

Within the framework of the quantum phase space representation established by Torres-Vega and Frederick, we solve the rigorous solutions of the stationary Schrödinger equations for the one-dimensional harmonic oscillator by means of the quantum wave-mechanics method. The result shows that the wave mechanics and the matrix mechanics are equivalent in phase space, just as in position or momentum space.

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