THE FINITE-DIFFERENCE METHOD OF ONE-DIMENSIONAL NONLINEAR SYSTEMS ANALYSIS

Fractals ◽  
2000 ◽  
Vol 08 (01) ◽  
pp. 49-65 ◽  
Author(s):  
A. YU. SHAHVERDIAN
Keyword(s):  
Time Series ◽  
Systems Analysis ◽  
Computational Study ◽  
Brain Activity ◽  
Logistic Map ◽  
Cantor Sets ◽  
Point Of View ◽  
One Dimensional ◽  
Time Flow ◽  
The One

The paper introduces one-dimensional analogy of Poincare "section" method. It reduces the one-dimensional nonlinear system orbit's study to consideration of some special conjugate orbit's "asymptotical" intersections with a thin arithmetical space of zero Lebesgue measure. The application of this approach to analysis of the logistic map orbits, earthquake time-series, and the sequences of fractional parts, is considered. Through computational study of these time-series, the existence of some Cantor sets, to which the conjugate orbits are attracted, is established. A fractal dynamical system, describing these different systems from a unified point of view, is introduced. The inner differential Cantorian structure of brain activity and time flow is discussed.

COMPUTING THE DISTRIBUTION OF THE LYAPUNOV EXPONENT FROM TIME SERIES: THE ONE-DIMENSIONAL CASE STUDY

1995 ◽  
Vol 05 (06) ◽  
pp. 1721-1726 ◽  
Author(s):  
DEJIAN LAI ◽  
GUANRONG CHEN
Keyword(s):  
Time Series ◽  
Monte Carlo ◽  
Lyapunov Exponent ◽  
Logistic Map ◽  
Observation Data ◽  
Block Bootstrap ◽  
One Dimensional ◽  
The One ◽  
True Values

In this paper, a simple and direct statistical method is proposed for estimating the Lyapunov exponent of an unknown dynamic system using its time series of observation data. It is shown that the asymptotic distribution of the estimates obtained from the proposed method is normal. Monte Carlo and block bootstrap methods are used to simulate the estimation for the logistic map, in which they both provide the expectation and variance for the estimates. Computer simulations show that our estimates are very close to the true values of the exponent for the logistic map with different parameters.

RELAXATION AND TRANSIENTS IN A TIME-DEPENDENT LOGISTIC MAP

2002 ◽  
Vol 12 (07) ◽  
pp. 1667-1674 ◽  
Author(s):  
EDSON D. LEONEL ◽  
J. KAMPHORST LEAL DA SILVA ◽  
S. OLIFFSON KAMPHORST
Keyword(s):  
Control Parameter ◽  
Logistic Map ◽  
Period Doubling ◽  
Time Dependent ◽  
One Dimensional ◽  
Stability Period ◽  
The One ◽  
Exchange Of Stability ◽  
Small Periodic ◽  
Scaling Arguments

We study the one-dimensional logistic map with control parameter perturbed by a small periodic function. In the pure constant case, scaling arguments are used to obtain the exponents related to the relaxation of the trajectories at the exchange of stability, period-doubling and tangent bifurcations. In particular, we evaluate the exponent z which describes the divergence of the relaxation time τ near a bifurcation by the relation τ ~ | R - Rc |-z. Here, R is the control parameter and Rc is its value at the bifurcation. In the time-dependent case new attractors may appear leading to a different bifurcation diagram. Beside these new attractors, complex attractors also arise and are responsible for transients in many trajectories. We obtain, numerically, the exponents that characterize these transients and the relaxation of the trajectories.

Predictable Model for Characteristics of One-Dimensional Solid-Gas-Liquid Three-Phase Mixtures Flow Along a Vertical Pipeline With an Abrupt Enlargement in Diameter

1999 ◽  
Vol 121 (2) ◽  
pp. 330-342 ◽  
Author(s):  
Natsuo Hatta ◽  
Masaaki Omodaka ◽  
Fumitaka Nakajima ◽  
Takahiro Takatsu ◽  
Hitoshi Fujimoto ◽  
...  
Keyword(s):  
Gas Flow ◽  
Sudden Change ◽  
Flow Characteristics ◽  
Point Of View ◽  
Solid Particles ◽  
Governing Equations ◽  
One Dimensional ◽  
Three Phase ◽  
Experimental Values ◽  
The One

This paper treats the numerical analysis of the rising process of a solid-gas-liquid three-phase mixture along a vertical pipeline with an abrupt enlargement in diameter. The system of governing equations used is based upon the one-dimensional multifluid model and the transitions of gas flow pattern are taken into account in the system of governing equations. For the case of a sudden enlargement in diameter in a coaxial pipeline, the procedure of the numerical calculation to obtain the flow characteristics in the pipeline section after a sudden change in diameter has been established here. Furthermore, in order to confirm the validity of the present theoretical model by the comparison between the calculated and experimental values, the experiments have been made using four kinds of lifting pipes, including the straight one. Thereby, it has been found that the numerical model proposed here gives good fit to the prediction of the flow rates of lifted water and solid particles against that of air supplied for the case of a sudden change in diameter. In addition, the flowing process for each phase has been investigated from a photographic point of view. As a result, we found that the moving process of the solid particles depends strongly upon the volumetric flux of gas-phase as well as the submergence ratio.

scholarly journals Spatial components dependence for bidimensional time-constant AR(1) model with $$\alpha $$-stable noise and triangular coefficients matrix

2021 ◽  
Author(s):  
Aleksandra Grzesiek
Keyword(s):  
Time Series ◽  
Asymptotic Behaviour ◽  
Time Constant ◽  
Autoregressive Model ◽  
Asymptotic Relation ◽  
Autoregressive Time Series ◽  
One Dimensional ◽  
The Cross ◽  
The One ◽  
Stable Noise

AbstractIn this paper, we examine the bidimensional time-constant autoregressive model of order 1 with $$\alpha $$ α -stable noise. We focus on the case of the triangular coefficients matrix for which one of the spatial components of the model simplifies to the one-dimensional autoregressive time series. We study the asymptotic behaviour of the cross-codifference and the cross-covariation applied to describe the dependence in time between the spatial components of the model. As a result, we formulate the theorem about the asymptotic relation between both measures, which is consistent with the result that is correct for the case of the non-triangular coefficients matrix.

scholarly journals Multi-dimensional connectivity: a conceptual and mathematical review

2020 ◽  
Author(s):  
Hamed Nili ◽  
Alessio Basti ◽  
Olaf Hauk ◽  
Laura Marzetti ◽  
Richard Henson
Keyword(s):  
Time Series ◽  
Region Of Interest ◽  
Principal Component ◽  
Point Of View ◽  
Fmri Data ◽  
Multiple Time ◽  
Multiple Time Series ◽  
One Dimensional ◽  
Mathematical Review ◽  
The Brain

The estimation of functional connectivity between regions of the brain, for example based on statistical dependencies between the time series of activity in each region, has become increasingly important in neuroimaging. Typically, multiple time series (e.g. from each voxel in fMRI data) are first reduced to a single time series that summarises the activity in a region of interest, e.g. by averaging across voxels or by taking the first principal component; an approach we call one-dimensional connectivity. However, this summary approach ignores potential multi-dimensional connectivity between two regions, and a number of recent methods have been proposed to capture such complex dependencies. Here we review the most common multi-dimensional connectivity methods, from an intuitive perspective, from a formal (mathematical) point of view, and through a number of simulated and real (fMRI and MEG) data examples that illustrate the strengths and weaknesses of each method. The paper is accompanied with both functions and scripts, which implement each method and reproduce all the examples.

scholarly journals Modeling of stochastic brain function in artificial intelligence

2019 ◽  
Vol 4 (3) ◽  
pp. 8-14
Author(s):  
Andrei N. Volobuev ◽  
Vasiliy F. Pyatin ◽  
Natalya P. Romanchuk ◽  
Petr I. Romanchuk ◽  
Svetlana V. Bulgakova
Keyword(s):  
Mathematical Modeling ◽  
Artificial Intelligence ◽  
Brain Function ◽  
Brain Activity ◽  
Necessary Condition ◽  
Alpha Wave ◽  
Brain Functioning ◽  
One Dimensional ◽  
The One ◽  
Stochastic Mode

Objectives -research of stochastic brain function in respect to creation of artificial intelligence. Material and methods. Mathematical modeling principles were used for simulation of brain functioning in a stochastic mode. Results. Two types of brain activity were considered: determinated type, usually modeled using the perceptron, and stochastic type. It is shown, that stochastic brain function modeling is the necessary condition for AI to become capable of creativity, generation of new knowledge. Mathematical modeling of a neural network of the cerebral cortex, consisting of the set of the cyclic neuronal circuits (memory units), was performed for the stochastic mode of brain functioning. Models of "two-dimensional" and "one-dimensional" brain were analyzed. The pattern of excitation in memory units was calculated in the "one-dimensional" brain model. Conclusion. Relying on the knowledge of the stochastic mode of brain function, a way of creation of AI can be offered. а-rhythm of a patient is a recommended focus of the therapist's attention in diagnostics and treatment of brain disorders. It was noted, that the alpha wave amplitude and frequency could indicate the cognitive, creative and intuitive abilities of a person.

TWO-DIMENSIONAL COUPLED PARAMETRICALLY FORCED MAP

2013 ◽  
Vol 23 (02) ◽  
pp. 1350031 ◽  
Author(s):  
HIRONORI KUMENO ◽  
DANIÈLE FOURNIER-PRUNARET ◽  
ABDEL-KADDOUS TAHA ◽  
YOSHIFUMI NISHIO
Keyword(s):  
Newton Method ◽  
Saddle Points ◽  
Logistic Map ◽  
Two Dimensional ◽  
Bifurcation Structure ◽  
One Dimensional ◽  
Stable Order ◽  
Critical Curves ◽  
Cusp Points ◽  
The One

A two-dimensional parametrically forced system constructed from two identical one-dimensional subsystems, whose parameters are forced into periodic varying, with mutually influencing coupling is proposed. We investigate bifurcations and basins in the parametrically forced system when logistic map is used for the one-dimensional subsystem. On a parameter plane, crossroad areas centered at fold cusp points for several orders are detected. From the investigation, a foliated bifurcation structure is drawn, and existence domains of stable order cycles with synchronization or without synchronization are detected. Moreover, evolution of bifurcation curves with respect to a coupling intensity is analyzed. Basin bifurcations and preimages with respect to critical curves are described. Basins where boundary depends on the invariant manifold of saddle points are numerically analyzed by considering second order iteration and using superposition with Newton method, although the system has discontinuity regarding parameters.

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