scholarly journals Solving the problem of mathematical models overparameterization for some nonlinear oscillating systems

2021 ◽  
pp. 135-148
Author(s):  
Viktor Gorodetskyi ◽  
Mykola Osadchuk
Keyword(s):  
Time Series ◽  
Initial Conditions ◽  
Deterministic Chaos ◽  
High Sensitivity ◽  
Original System ◽  
Differential Model ◽  
Oscillating Systems ◽  
Observable Variable ◽  
Numerical Analytical Method ◽  
Derivatives Of

This study proposes a numerical-analytical method that allows us to simplify the model, which is obtained on the basis of the single observable variable of an object under the study, and which may be overparameterized. As a model, we consider a system of ordinary differential equations with polynomial right-hand sides. To solve this problem, the so-called differential model is used, that is, a system in which unknown variables are replaced by derivatives of the observed variable, and which is derived on the basis of a system under the study so that the observed variables of these systems coincide. The method of simplification of a system under the study is based on the fact that using a numerical method, a simpler differential model can be obtained. Next, an analytical transition from a simplified differential model to a simplified original system is performed. In this case, the time series error remains within given limits even for systems with deterministic chaos, despite their high sensitivity to the initial conditions.

Chaos in blood flow control in genetic and renovascular hypertensive rats

1991 ◽  
Vol 261 (3) ◽  
pp. F400-F408 ◽  
Author(s):  
K. P. Yip ◽  
N. H. Holstein-Rathlou ◽  
D. J. Marsh
Keyword(s):  
Time Series ◽  
Blood Flow ◽  
Phase Space ◽  
Renal Artery ◽  
Initial Conditions ◽  
Pressure Fluctuations ◽  
Deterministic Chaos ◽  
Tubuloglomerular Feedback ◽  
Hypertensive Rats ◽  
Statistical Measures

Hydrostatic pressure and flow in renal proximal tubules oscillate at 30–40 mHz in normotensive rats anesthetized with halothane. The oscillations originate in tubuloglomerular feedback, a mechanism that provides local blood flow regulation. Instead of oscillations, spontaneously hypertensive rats (SHR) have aperiodic tubular pressure fluctuations; the pattern is suggestive of deterministic chaos. Normal rats made hypertensive by clipping one renal artery had similar aperiodic tubular pressure fluctuations in the unclipped kidney, and the fraction of rats with irregular fluctuations increased with time after the application of the renal artery clip. Statistical measures of deterministic chaos were applied to tubular pressure data. The correlation dimension, a measure of the dimension of the phase space attractor generating the time series, indicated the presence of a low-dimension strange attractor, and the largest Lyapunov exponent, a measure of the rate of divergence in phase space, was positive, indicating sensitivity to initial conditions. These time series therefore satisfy two criteria of deterministic chaos. The measures were the same in SHR as in rats with renovascular hypertension. Since two different models of hypertension displayed similar dynamics, we suggest that chaotic behavior is a common feature of renal vascular control in the natural history of the disease.

scholarly journals Signal Processing and Sampling Method for Obtaining Time Series Corresponding to Higher Order Derivatives

2010 ◽  
Vol 2010 ◽  
pp. 1-9 ◽  
Author(s):  
Andreea Sterian ◽  
Alexandru Toma
Keyword(s):  
Time Series ◽  
Dynamic System ◽  
Sampling Procedure ◽  
Higher Order ◽  
Time Interval ◽  
Limited Time ◽  
Time Intervals ◽  
Oscillating Systems ◽  
Robust Computation ◽  
Derivatives Of

For modeling and controlling dynamic phenomena it is important to establish with higher accuracy some significant quantities corresponding to the dynamic system. For fast phenomena, such significant quantities are represented by the derivatives of the received signals. In case of advanced computer modeling, the received signal should be filtered and converted into a time series corresponding to the estimated values for the dynamic system through a sampling procedure. This paper will show that present-day methods for computing in a robust manner the first derivative of a received signal (using an oscillating system working on a limited time interval and a supplementary differentiation method) can be extended to the robust computation of higher order derivatives of the received signal by using a specific set of second-order oscillating systems (working also on limited time intervals) so as estimative values for higher-order derivatives are to be directly generated (avoiding the necessity of additional differentiation or amplifying procedures, which represent a source of supplementary errors in present-day methods).

MODELING NONLINEAR DETERMINISM IN SHORT TIME SERIES FROM NOISE DRIVEN DISCRETE AND CONTINUOUS SYSTEMS

2000 ◽  
Vol 10 (12) ◽  
pp. 2745-2766 ◽  
Author(s):  
K. H. CHON ◽  
K. P. YIP ◽  
B. M. CAMINO ◽  
D. J. MARSH ◽  
N.-H. HOLSTEIN-RATHLOU
Keyword(s):  
Time Series ◽  
Lyapunov Exponents ◽  
Initial Conditions ◽  
Deterministic Chaos ◽  
Continuous Systems ◽  
Hypertensive Rats ◽  
Free Data ◽  
Renal Tubular ◽  
Short Time ◽  
Sensitivity To Initial Conditions

Current methods for detecting deterministic chaos in a time series require long, stationary, and relatively noise-free data records. This limits the utility of these methods in most experimental and clinical settings. Recently we presented a new method for detecting determinism in a time series, and for assessing whether this determinism has chaotic attributes, i.e. sensitivity to initial conditions. The method is based on fitting a deterministic nonlinear autoregressive (NAR) model to the data [Chon et al., 1997]. This approach assumes that the noise in the model can be represented as a series of independent, identically distributed random variables. If this is not the case the accuracy of the algorithm may be compromised. To explicitly deal with the possibility of more complex noise structures, we present a method based on a stochastic NAR model. The method iteratively estimates NAR models for both the deterministic and the stochastic parts of the signal. An additional feature of the algorithm is that it includes only the significant autoregressive terms among the pool of candidate terms searched. As a result the algorithm results in a model with significantly fewer terms than a model obtained by traditional model order search criterions. Subsequently, Lyapunov exponents are calculated for the estimated models to examine if chaotic determinism (i.e. sensitivity to initial conditions) is present in the time series. The major advantages of this algorithm are: (1) it provides accurate parameter estimation with a small number of data points, (2) it is accurate for signal-to-noise ratios as low as -9 dB for discrete and -6 dB for continuous chaotic systems, and (3) it allows characterization of the dynamics of the system, and thus prediction of future states of the system, over short time scales. The stochastic NAR model is applied to renal tubular pressure data from normotensive and hypertensive rats. One form of hypertension was genetic, and the other was induced on normotensive rats by placing a restricting clip on one of their renal arteries. In both types of hypertensive rats, positive Lyapunov exponents were present, indicating that the fluctuations observed in the proximal tubular pressure were due to the operation of a system with chaotic determinism. In contrast, only negative exponents were found in the time series from normotensive rats.

scholarly journals Deterministic Chaos Detection and Simplicial Local Predictions Applied to Strawberry Production Time Series

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 3034
Author(s):  
Juan D. Borrero ◽  
Jesus Mariscal
Keyword(s):  
Time Series ◽  
Initial Conditions ◽  
Moving Average ◽  
Deterministic Chaos ◽  
Autoregressive Moving Average ◽  
Chaos Detection ◽  
Moving Average Models ◽  
Non Linear ◽  
Long Time ◽  
Novel Method

In this work, we attempted to find a non-linear dependency in the time series of strawberry production in Huelva (Spain) using a procedure based on metric tests measuring chaos. This study aims to develop a novel method for yield prediction. To do this, we study the system’s sensitivity to initial conditions (exponential growth of the errors) using the maximal Lyapunov exponent. To check the soundness of its computation on non-stationary and not excessively long time series, we employed the method of over-embedding, apart from repeating the computation with parts of the transformed time series. We determine the existence of deterministic chaos, and we conclude that non-linear techniques from chaos theory are better suited to describe the data than linear techniques such as the ARIMA (autoregressive integrated moving average) or SARIMA (seasonal autoregressive moving average) models. We proceed to predict short-term strawberry production using Lorenz’s Analog Method.

Deep Learning Based Classification of Time Series of Chaotic Systems over Graphic Images

2021 ◽  
Author(s):  
Süleyman UZUN ◽  
Sezgin KAÇAR ◽  
Burak ARICIOĞLU
Keyword(s):  
Time Series ◽  
Deep Learning ◽  
Transfer Learning ◽  
Chaotic Systems ◽  
Initial Conditions ◽  
Step Size ◽  
Learning Methods ◽  
Data Set ◽  
Graphic Images ◽  
Different Chaotic Systems

Abstract In this study, for the first time in the literature, identification of different chaotic systems by classifying graphic images of their time series with deep learning methods is aimed. For this purpose, a data set is generated that consists of the graphic images of time series of the most known three chaotic systems: Lorenz, Chen, and Rossler systems. The time series are obtained for different parameter values, initial conditions, step size and time lengths. After generating the data set, a high-accuracy classification is performed by using transfer learning method. In the study, the most accepted deep learning models of the transfer learning methods are employed. These models are SqueezeNet, VGG-19, AlexNet, ResNet50, ResNet101, DenseNet201, ShuffleNet and GoogLeNet. As a result of the study, classification accuracy is found between 96% and 97% depending on the problem. Thus, this study makes association of real time random signals with a mathematical system possible.

scholarly journals Mathematical modeling of the process of ferrum removal in a biological reactor by bacteria Gallionella and Leptothrix

2020 ◽  
pp. 51-54
Author(s):  
Olena Prysiazhniuk ◽  
Igor Prysіazhnіuk ◽  
Alexander Kvartenko
Keyword(s):  
Nonlinear Boundary ◽  
Bacterial Biomass ◽  
Computer Experiments ◽  
Inhomogeneous System ◽  
Cleaning Efficiency ◽  
Computer Prediction ◽  
Proposed Model ◽  
Filter Loading ◽  
Numerical Analytical Method ◽  
Derivatives Of

This paper proposes a mathematical model for computer prediction of the process of biological deironing of groundwater in a bioreactor, taking into account the presence of two types of iron bacteria Leptothrix and Gallionella in groundwater while maintaining a constant filtration rate. An algorithm for a numerical-analytical method for solving the corresponding nonlinear boundary value problem for an inhomogeneous system of differential equations in partial derivatives of the first order has been developed. The developed model allows to use computer experiments to predict the change in time on the depth of contact loading of cleaning efficiency, distribution of bacterial biomass values ​​in both filtered water and in filter loading, mass of stationary and mobile matrix structures. Also, the proposed model allows to predict the duration of effective operation of the biological reactor of iron deironing between its washing.

scholarly journals Determine of the wave equation in the task of electrical oscillations

2018 ◽  
Vol 184 ◽  
pp. 01023
Author(s):  
Gordana V. Jelić ◽  
Vladica Stanojević ◽  
Dragana Radosavljević
Keyword(s):  
Wave Equation ◽  
Initial Conditions ◽  
Mathematical Physics ◽  
Equation Of Motion ◽  
Independent Variables ◽  
Equations Of Mathematical Physics ◽  
Basic Equations ◽  
Derivatives Of ◽  
Two Independent Variables ◽  
Transverse Oscillations

One of the basic equations of mathematical physics (for instance function of two independent variables) is the differential equation with partial derivatives of the second order (3). This equation is called the wave equation, and is provided when considering the process of transverse oscillations of wire, longitudinal oscillations of rod, electrical oscillations in a conductor, torsional vibration at waves, etc… The paper shows how to form the equation (3) which is the equation of motion of each point of wire with abscissa x in time t during its oscillation. It is also shown how to determine the equation (3) in the task of electrical oscillations in a conductor. Then equation (3) is determined, and this solution satisfies the boundary and initial conditions.

scholarly journals ESTIMATION OF AUTOREGRESSIVE ROOTS NEAR UNITY USING PANEL DATA

2000 ◽  
Vol 16 (6) ◽  
pp. 927-997 ◽  
Author(s):  
Hyungsik R. Moon ◽  
Peter C.B. Phillips
Keyword(s):  
Time Series ◽  
Panel Data ◽  
Time Series Data ◽  
Initial Conditions ◽  
Series Data ◽  
Test Statistics ◽  
Rate Condition ◽  
Panel Models ◽  
Stochastic Trends ◽  
Efficient Extraction

Time series data are often well modeled by using the device of an autoregressive root that is local to unity. Unfortunately, the localizing parameter (c) is not consistently estimable using existing time series econometric techniques and the lack of a consistent estimator complicates inference. This paper develops procedures for the estimation of a common localizing parameter using panel data. Pooling information across individuals in a panel aids the identification and estimation of the localizing parameter and leads to consistent estimation in simple panel models. However, in the important case of models with concomitant deterministic trends, it is shown that pooled panel estimators of the localizing parameter are asymptotically biased. Some techniques are developed to overcome this difficulty, and consistent estimators of c in the region c < 0 are developed for panel models with deterministic and stochastic trends. A limit distribution theory is also established, and test statistics are constructed for exploring interesting hypotheses, such as the equivalence of local to unity parameters across subgroups of the population. The methods are applied to the empirically important problem of the efficient extraction of deterministic trends. They are also shown to deliver consistent estimates of distancing parameters in nonstationary panel models where the initial conditions are in the distant past. In the development of the asymptotic theory this paper makes use of both sequential and joint limit approaches. An important limitation in the operation of the joint asymptotics that is sometimes needed in our development is the rate condition n/T → 0. So the results in the paper are likely to be most relevant in panels where T is large and n is moderately large.

THE QNDE USING IMAGE PROCESSING OF THE MAGNETIC CAMERA

2006 ◽  
Vol 20 (25n27) ◽  
pp. 4625-4630 ◽  
Author(s):  
JINYI LEE ◽  
JISEONG HWANG ◽  
SEHO CHOI
Keyword(s):  
Image Processing ◽  
High Speed ◽  
Magnetic Hysteresis ◽  
High Sensitivity ◽  
Magnetic Sensors ◽  
Second Derivatives ◽  
Crack Volume ◽  
Lift Off ◽  
Magnetic Camera ◽  
Derivatives Of

A scan type magnetic camera was proposed to satisfy the following demands: to obtain high speed quantitative magnetic flux leakage (MFL) distribution with homogeneous lift-off by using 2-dimensionally arrayed high sensitivity magnetic sensors; to concentrate the MFL; and to ignore the residual magnetization and magnetic hysteresis by using a magnetic fluid lens. The magnetic field distribution (MFD) image obtained by using the scan type magnetic camera is inclined to the scanning direction (x-direction) because of the poles of the magnetizer. Also, the image shows a homogeneous trend relative to the x-direction, but there are small waves in the distribution in the sensor arraying direction (y-direction). The crack information in the MFD image can be extracted using image processing. The first and second derivatives of both x and y are used in this processing. These are "1st derivative of x, ∂B/∂x", "1st derivative of y, ∂B/∂y", "2nd derivative of x, ∂2B/∂x2", "2nd derivative of y, ∂2B/∂y2", and "2nd derivative of x and y, ∂2B/∂x∂y". The ∂B/∂x distribution shows the existence of the crack. Also, the crack volume can be evaluated quantitatively, regardless of the crack direction, by using ∂B/∂x and a cross type magnetic coil.

Image Encryption Algorithm Based on a Novel 4D Chaotic System

2021 ◽  
Vol 15 (4) ◽  
pp. 118-131
Author(s):  
Sadiq A. Mehdi
Keyword(s):  
Chaotic System ◽  
Initial Conditions ◽  
High Sensitivity ◽  
Encryption Algorithm ◽  
Pseudorandom Sequence ◽  
Key Space ◽  
Xor Operation ◽  
Change Intensity ◽  
And Performance ◽  
Image Cryptosystem

In this paper, a novel four-dimensional chaotic system has been created, which has characteristics such as high sensitivity to the initial conditions and parameters. It also has two a positive Lyapunov exponents. This means the system is hyper chaotic. In addition, a new algorithm was suggested based on which they constructed an image cryptosystem. In the permutation stage, the pixel positions are scrambled via a chaotic sequence sorting. In the substitution stage, pixel values are mixed with a pseudorandom sequence generated from the 4D chaotic system using XOR operation. A simulation has been conducted to evaluate the algorithm, using the standardized tests such as information entropy, histogram, number of pixel change rate, unified average change intensity, and key space. Experimental results and performance analyses demonstrate that the proposed encryption algorithm achieves high security and efficiency.

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