scholarly journals FOR ONE SYSTEM OF DIFFERENTIAL EQUATIONS TAKEN AS DUAL OF THE LORENZ SYSTEM

2016 ◽  
pp. 37-44
Author(s):  
Boro M Piperevski
Keyword(s):  
Differential Equations ◽  
Lorenz System ◽  
System Of Differential Equations ◽  
The Lorenz System

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 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.

Dynamic analysis of the dual Lorenz system

2020 ◽  
Vol 13 (08) ◽  
pp. 2050171
Author(s):  
Biljana Zlatanovska ◽  
Boro Piperevski
Keyword(s):  
Lyapunov Function ◽  
Differential Equations ◽  
Dynamic Analysis ◽  
Fixed Points ◽  
Autonomous System ◽  
Lorenz System ◽  
Mathematical Software ◽  
System Of Differential Equations ◽  
Graphical Visualization ◽  
The Lorenz System

The dual Lorenz system as an autonomous system of three differential equations is obtained by using the Lorenz system of differential equations in the paper [B. M. Piperevski, For one system of differential equations taken as dual of the Lorenz system, Bull. Math. 40(1) (2014) 37–44]. In this paper, we will do a comparison of the dual Lorenz system with the Lorenz system for different values of parameters. The dynamic analysis of its behavior will be done. The basic properties of the dual Lorenz system are analyzed by means of the symmetry of the system, dissipativity of the system, the Lyapunov function, the behavior of the system in the neighborhood of fixed points, etc. By using mathematical software Mathematica, we will give a graphical visualization of the dual Lorenz system for some values of parameters via examples.

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

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.

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.

ACCURATE PARALLEL INTEGRATION OF LARGE SPARSE SYSTEMS OF DIFFERENTIAL EQUATIONS

1996 ◽  
Vol 06 (04) ◽  
pp. 535-568 ◽  
Author(s):  
DONALD J. ESTEP ◽  
ROY D. WILLIAMS
Keyword(s):  
Differential Equations ◽  
Error Estimation ◽  
Lorenz System ◽  
Sparse Matrix ◽  
Posteriori Error ◽  
Two Dimensions ◽  
Partial Differential ◽  
Sparse Systems ◽  
The Lorenz System ◽  
Large Sparse Systems

We describe a MIMD parallel code to solve a general class of ordinary differential equations, with particular emphasis on the large, sparse systems arising from space discretization of systems of parabolic partial differential equations. The main goals of this work are sharp bounds on the accuracy of the computed solution and flexibility of the software. We discuss the sources of error in solving differential equations, and the resulting constraints on time steps. We also discuss the theory of a posteriori error analysis for the Galerkin finite element methods, and its implementation in error control and estimation. The software is designed in a matrix-free fashion, which enables the solver to effectively tackle large sparse systems with minimal memory consumption and an easy and natural transition to MIMD (distributed memory) parallelism. In addition, there is no need for the choice of a particular representation of a sparse matrix. All memory is dynamically allocated, with a new expandable array object used for archiving results. The implicit solution of the discrete equations is carried out by replaceable modules: the nonlinear solver module may be a full Newton scheme or a quasi-Newton; these in turn are implemented with a linear solver, for which we have used both a direct solver and QMR, an iterative (Krylov space) method. Three computations are presented: the Lorenz system, which has dimension three and the discretized versions of the (partial-differential) bistable equation in one and two dimensions. The Lorenz system demonstrates the quality of the error estimation. The discretized bistable examples provide large sparse systems, and our precise error estimation shows, contrary to standard error estimates, that reliable computation is possible for large times.

THE POINTWISE COMPUTABILITY OF THE LORENZ SYSTEM

1998 ◽  
Vol 08 (08) ◽  
pp. 1277-1305 ◽  
Author(s):  
DON ESTEP ◽  
CLAES JOHNSON
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Lorenz System ◽  
Adaptive Finite Element Method ◽  
Adaptive Finite Element ◽  
Time Intervals ◽  
The Lorenz System ◽  
Moderate Length

We give evidence that most trajectories of the Lorenz system of ordinary differential equations are pointwise computable on time intervals of moderate length using an adaptive finite element method. Based on accurate computation, we present a description of the structure of the solutions of the Lorenz system as a repeated process of cutting-expanding-flipping-interlacing. We also give some general remarks on issues of computability and the relation between discrete and continuous dynamical systems.

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