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.

scholarly journals Conditions for Factorization of Linear Differential-Difference Equations

2013 ◽  
Vol 54 (1) ◽  
pp. 93-99
Author(s):  
Klara R. Janglajew ◽  
Kim G. Valeev
Keyword(s):  
Linear System ◽  
Small Parameter ◽  
Difference Equations ◽  
Integral Manifold ◽  
Sufficient Conditions ◽  
System Of Differential Equations ◽  
Deviating Argument ◽  
Constant Coefficients ◽  
Complex Deviating Argument ◽  
Linear Differential

Abstract The paper deals with a linear system of differential equations of the form with constant coefficients, a small parameter and complex deviating argument. Sufficient conditions for factorizing of this system are presented. These conditions are obtained by construction of an integral manifold of solutions to the considered system.

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 Monte-Carlo for solving large linear systems of ordinary differential equations

2021 ◽  
Vol 8 (1) ◽  
pp. 37-48
Author(s):  
Sergey M. Ermakov ◽  
◽  
Maxim G. Smilovitskiy ◽  
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Differential Equations ◽  
Integral Equations ◽  
Virtual Machines ◽  
Variable Coefficients ◽  
Fredholm Type ◽  
Type Integral ◽  
Constant Coefficients ◽  
Linear Differential

Monte-Carlo approach towards solving Cauchy problem for large systems of linear differential equations is being proposed in this paper. Firstly, a quick overlook of previously obtained results from applying the approach towards Fredholm-type integral equations is being made. In the main part of the paper, a similar method is being applied towards a linear system of ODE. It is transformed into an equivalent system of Volterra-type integral equations, which relaxes certain limitations being present due to necessary conditions for convergence of majorant series. The following theorems are being stated. Theorem 1 provides necessary compliance conditions that need to be imposed upon initial and transition distributions of a required Markov chain, for which an equality between estimate’s expectation and a desirable vector product would hold. Theorem 2 formulates an equation that governs estimate’s variance, while theorem 3 states a form for Markov chain parameters that minimise the variance. Proofs are given, following the statements. A system of linear ODEs that describe a closed queue made up of ten virtual machines and seven virtual service hubs is then solved using the proposed approach. Solutions are being obtained both for a system with constant coefficients and time-variable coefficients, where breakdown intensity is dependent on t. Comparison is being made between Monte-Carlo and Rungge Kutta obtained solutions. The results can be found in corresponding tables.

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