scholarly journals The Positive Role of Multiplicative Noise in Complete Synchronization of Unidirectionally Coupled Ring with Three Nodes

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Yuzhu Xiao ◽  
Sufang Tang ◽  
Zhongkui Sun
Keyword(s):  
Differential Equations ◽  
Theoretical Result ◽  
Numerical Simulations ◽  
Neuron Model ◽  
Multiplicative Noise ◽  
Lorenz System ◽  
Complete Synchronization ◽  
Positive Role ◽  
The Lorenz System

The role of multiplicative noise in the synchronization of unidirectionally coupled ring with three nodes is studied. Based on the theory of stochastic differential equations, we demonstrate that noise plays a positive role in complete synchronization. In numerical simulations, the Lorenz system, Rössler like system, and Hindmarsh-Rose neuron model are employed to demonstrate the correctness of our theoretical result.

A modified Lorenz system: Definition and solution

2020 ◽  
Vol 13 (08) ◽  
pp. 2050164
Author(s):  
Biljana Zlatanovska ◽  
Donc̆o Dimovski
Keyword(s):  
Differential Equations ◽  
Difference Equations ◽  
Lorenz System ◽  
Local Approximation ◽  
System Of Differential Equations ◽  
Lorentz System ◽  
Constant Coefficients ◽  
Linear Differential ◽  
The Lorenz System ◽  
Fifth Order

Based on the approximations of the Lorenz system of differential equations from the papers [B. Zlatanovska and D. Dimovski, Systems of difference equations approximating the Lorentz system of differential equations, Contributions Sec. Math. Tech. Sci. Manu. XXXIII 1–2 (2012) 75–96, B. Zlatanovska and D. Dimovski, Systems of difference equations as a model for the Lorentz system, in Proc. 5th Int. Scientific Conf. FMNS, Vol. I (Blagoevgrad, Bulgaria, 2013), pp. 102–107, B. Zlatanovska, Approximation for the solutions of Lorenz system with systems of differential equations, Bull. Math. 41(1) (2017) 51–61], we define a Modified Lorenz system, that is a local approximation of the Lorenz system. It is a system of three differential equations, the first two are the same as the first two of the Lorenz system, and the third one is a homogeneous linear differential equation of fifth order with constant coefficients. The solution of this system is based on the results from [D. Dimitrovski and M. Mijatovic, A New Approach to the Theory of Ordinary Differential Equations (Numerus, Skopje, 1995), pp. 23–33].

scholarly journals Reconstruction of the Lorenz system using the method of perspective coefficients

2018 ◽  
Author(s):  
V. G. Gorodetskiy ◽  
N. P. Osadchuk
Keyword(s):  
Time Series ◽  
Numerical Methods ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Method ◽  
Lorenz System ◽  
Ode System ◽  
Right Hand ◽  
Observable Variable ◽  
The Lorenz System

Reconstruction of the Lorenz ordinary differential equations system is performed by using perspective coefficients method. Four systems that have structures different from Lorenz system and can reproduce time series of one variable of Lorenz system were found. In many areas of science, the problem of identifying a system of ordinary differential equations (ODE) from a time series of one observable variable is relevant. If the right-hand sides of an ODE system are polynomials, then solving such a problem only by numerical methods allows to obtain a model containing, in most cases, redundant terms and not reflecting the physics of the process. The preliminary choice of the structure of the system allows to improve the precision of the reconstruction. Since this study considers only the single time series of the observable variable, and there are no additional requirements for candidate systems, we will look only for systems of ODE's that have the least number of terms in the equations. We will look for candidate systems among particular cases of the system with quadratic polynomial right-hand sides. To solve this problem, we will use a combination of analytical and numerical methods proposed in [12, 11]. We call the original system (OS) the ODE system, which precisely describes the dynamics of the process under study. We also use another type of ODE system-standard system (SS), which has the polynomial or rational function only in one equation. The number of OS variables is equal to the number of SS variables. The observable variable of the SS coincides with the observable variable of the OS. The SS must correspond to the OS. Namely, all the SS coefficients can be analytically expressed in terms of the OS coefficients. In addition, there is a numerical method [12], which allows to determine the SS coefficients from a time series. To find only the simplest OS, one can use the perspective coefficients method [10], which means the following. Initially, the SS is reconstructed from a time series using a numerical method. Then, using analytical relations and the structure of the SS, we determine which OS coefficients are strictly zero and strictly non-zero and form the initial system (IS), which includes only strictly non-zero coefficients. After that, the IS is supplemented with OS coefficients until the corresponding SS coincides with the SS obtained by a numerical method. The result will be one or more OS’s. Using this approach, we have found 4 OS structures with 7 coefficients that differ from the Lorenz system [17], but are able to reproduce exactly the time series of X variable of the Lorenz system. Numerical values of the part of the coefficients and relations connecting the rest of the coefficients were found for each OS

Comparison Principle and Solution Bound of Fractional Differential Equations

2015 ◽  
Author(s):  
Yufeng Xu
Keyword(s):  
Theoretical Analysis ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Numerical Simulations ◽  
Fractional Differential Equations ◽  
Comparison Principle ◽  
Lorenz System ◽  
Fractional Differential ◽  
Solution Bound ◽  
The Right

The comparison principle of fractional differential equations is discussed in this paper. We obtain two kinds of comparison principle which are related to the functions in the right hand side of equations, and the order of fractional derivative, respectively. By using the comparison principle, the boundedness of fractional Lorenz system and fractional Lorenz-like system are studied numerically. Numerical simulations are carried out which demonstrate our theoretical analysis.

scholarly journals Positive role of multiplication noise in attaining complete synchronization on large complex networks of dynamical systems

2018 ◽  
Vol 54 ◽  
pp. 803-816 ◽  
Author(s):  
Yuzhu Xiao ◽  
Sufang Tang ◽  
Zhongkui Sun ◽  
Xueli Song
Keyword(s):  
Dynamical Systems ◽  
Complex Networks ◽  
Complete Synchronization ◽  
Positive Role ◽  
Large Complex

Positive role of fractional Gaussian noise in FitzHugh–Nagumo neuron model

2021 ◽  
Vol 146 ◽  
pp. 110914
Author(s):  
Fengyin Gao ◽  
Yanmei Kang
Keyword(s):  
Gaussian Noise ◽  
Neuron Model ◽  
Fractional Gaussian Noise ◽  
Positive Role

scholarly journals Stochastic parametrizations and model uncertainty in the Lorenz ’96 system

2013 ◽  
Vol 371 (1991) ◽  
pp. 20110479 ◽  
Author(s):  
H. M. Arnold ◽  
I. M. Moroz ◽  
T. N. Palmer
Keyword(s):  
Model Uncertainty ◽  
Multiplicative Noise ◽  
Lorenz System ◽  
Initial Conditions ◽  
Short Term ◽  
Climate Forecasting ◽  
Weather And Climate ◽  
The Lorenz System ◽  
Lorenz 96 ◽  
Perturbed Parameter Ensembles

Simple chaotic systems are useful tools for testing methods for use in numerical weather simulations owing to their transparency and computational cheapness. The Lorenz system was used here; the full system was defined as ‘truth’, whereas a truncated version was used as a testbed for parametrization schemes. Several stochastic parametrization schemes were investigated, including additive and multiplicative noise. The forecasts were started from perfect initial conditions, eliminating initial condition uncertainty. The stochastically generated ensembles were compared with perturbed parameter ensembles and deterministic schemes. The stochastic parametrizations showed an improvement in weather and climate forecasting skill over deterministic parametrizations. Including a temporal autocorrelation resulted in a significant improvement over white noise, challenging the standard idea that a parametrization should only represent sub-gridscale variability. The skill of the ensemble at representing model uncertainty was tested; the stochastic ensembles gave better estimates of model uncertainty than the perturbed parameter ensembles. The forecasting skill of the parametrizations was found to be linked to their ability to reproduce the climatology of the full model. This is important in a seamless prediction system, allowing the reliability of short-term forecasts to provide a quantitative constraint on the accuracy of climate predictions from the same system.

scholarly journals The influence of the Lorenz system fractionality on its recurrensivity

2019 ◽  
Vol 252 ◽  
pp. 02006
Author(s):  
Magdalena Gregorczyk ◽  
Andrzej Rysak
Keyword(s):  
Time Series ◽  
Differential Equations ◽  
System Dynamics ◽  
Phase Diagrams ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Lorenz System ◽  
Recurrence Analysis ◽  
The Lorenz System ◽  
Fractional Orders

In this work, we investigate the recurrensivity of the Lorenz system with fractional order of derivatives occurring in its all three differential equations. Several solutions of the system for varying fractional orders of individual derivatives were calculated, which was followed by an analysis of changes in the selected recurrence quantifiers occurring due to modifications of the fractional orders {q1, q2, q3}. The results of the recurrence analysis were referred to the time series plots, phase diagrams and FFT spectra. The obtained results were finally used to examine the influence of fractional derivatives on the chaos - periodicity transition of the system dynamics.

scholarly journals SYSTEMS OF DIFFERENCE EQUATIONS APPROXIMATING THE LORENZ SYSTEM OF DIFFERENTIAL EQUATIONS

2017 ◽  
Vol 33 (1-2) ◽  
Author(s):  
Biljana Zlatanovska ◽  
Dončo Dimovski
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Difference Equations ◽  
Lorenz System ◽  
System Of Differential Equations ◽  
Polynomial Approximations ◽  
Local Approximations ◽  
Lorentz System ◽  
Computer Calculations ◽  
The Lorenz System

A b s t r a c t: In this paper, starting from the Lorenz system of differential equations, some systems of difference equations are produced. Using some regularities in these systems of difference equations, polynomial approximations of their solutions are found. Taking these approximations as coefficients, three power series are obtained and by computer calculations is examined that these power series are local approximations of the solutions of the starting Lorentz system of differential equations.

The role of multiplicative noise in complete synchronization of bidirectionally coupled chain

2014 ◽  
Vol 87 (6) ◽  
Author(s):  
Yuzhu Xiao ◽  
Sufang Tang ◽  
Zhongkui Sun
Keyword(s):  
Multiplicative Noise ◽  
Complete Synchronization
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