On Solving the Lorenz System by Differential Transformation Method (original) (raw)

A multi-step differential transform method and application to non-chaotic or chaotic systems

2010

The differential transform method (DTM) is an analytical and numerical method for solving a wide variety of differential equations and usually gets the solution in a series form. In this paper, we propose a reliable new algorithm of DTM, namely multi-step DTM, which will increase the interval of convergence for the series solution. The multi-step DTM is treated as an algorithm in a sequence of intervals for finding accurate approximate solutions for systems of differential equations. This new algorithm is applied to Lotka-Volterra, Chen and Lorenz systems. Then, a comparative study between the new algorithm, multistep DTM, classical DTM and the classical Runge-Kutta method is presented. The results demonstrate reliability and efficiency of the algorithm developed.

Several Chaotic Analysis of Lorenz System

This paper aim to describe a number of simplifications that can be made to the Lorenz system that preserve its dynamics as well as a number of chaotic systems. The butterfly effect was proven. The solution of differential equation and Lorenz Attractor were investigated. This study compares the dynamical behaviors of the Lorenz system with complex variables to that of the standard Lorenz system involving real variables. Different methodologies, including the Lyapunov exponents spectrum, the bifurcation diagram, the first return map to the Poincare section, topological entropy, periodic and quasi-periodic phase portraits, and chaotic behavior of the resulting system were discussed in Matlab.

A novel approach to obtain analytical-numerical solutions of nonlinear Lorenz system

Numerical Algorithms, 2013

In this research we apply an analytic approach to solve the well-known Lorenz system in the non-chaotic regime. The proposed approach is based on modal expansion by infinite series. The analytical-numerical results show that for real initial conditions and under the non-convective regime the modal expansion series reproduce correctly the dynamical behavior of the solution of the Lorenz system. The validity and reliability of the proposed analytical approach with few terms is tested by its application to the convective and non-convective regime with various parameter values. The main advantage is that the obtained solution is global and is presented in analytical form.

A Efficient Analytical Approach for Nonlinear System of Advanced Lorenz Model

Science Proceedings Series, 2020

This work proposed a new analytical approach for solving a famous model from mathematical physics, namely, advanced Lorenz system. The method combines the Natural transform and Homotopy analysis method, and it’s have been suggested for the solution of different types of nonlinear systems of delay differential equations. This technique gives solution in a series form where the He’s polynomial is adjusted for the series calculation of nonlinear terms of Lorenz system. By choosing an optimal value of auxiliary parameters the more precise approximate Solution of this model is obtained from only three iterations number of terms. Some figures are used to demonstrate the accuracy of the result based on the residual error function. Therefore, the approach gives rise to an easy and straightforward means of solving these models analytically. Hence, it can be used in finding solutions to other forms of nonlinear problems.

Dynamical Analysis of a Modified Lorenz System

Journal of Mathematics, 2013

This paper presents another new modified Lorenz system which is chaotic in a certain range of parameters. Besides that, this paper also presents explanations to solve the new modified Lorenz system. Furthermore, some of the dynamical properties of the system are shown and stated. Basically, this paper shows the finding that led to the discovery of fixed points for the system, dynamical analysis using complementary-cluster energy-barrier criterion (CCEBC), finding the Jacobian matrix, finding eigenvalues for stability, finding the Lyapunov functions, and finding the Lyapunov exponents to investigate some of the dynamical behaviours of the system. Pictures and diagrams will be shown for the chaotic systems using the aide of MAPLE in 2D and 3D views. Nevertheless, this paper is to introduce the new modified Lorenz system.

Analytical prediction of the transition to chaos in Lorenz equations

Applied Mathematics Letters, 2010

The apparent failure of the linear stability analysis to predict accurately the transition point from steady to chaotic solutions in Lorenz equations motivates this study. A weak nonlinear solution to the problem is shown to produce an accurate analytical expression for the transition point as long as the condition of validity and consequent accuracy of the latter solution is fulfilled. The analytical results are compared to accurate computational solutions, showing an excellent fit within the validity domain of the analytical solution.

Systems of differential equations approximating the Lorenz system

2017

By using modified Lorenz system from [1] as the system of differential equations of seventh order which approximated the Lorenz system, we obtained four new systems of differential equations of third, fourth, fifth and sixth order. Every new system of differential equations is obtained using the solutions of the third differential equation from the modified Lorenz system. The third differential equation of modified Lorenz system is homogeneous linear differential equation of fifth order with constant coefficients which can be solved. By computer simulations we compare the local behavior of modified systems of differential equations with the global behavior of the Lorenz system

Dynamics of the Modified n-Degree Lorenz System

Applied Mathematics and Nonlinear Sciences

The Lorenz model is one of the most studied dynamical systems. Chaotic dynamics of several modified models of the classical Lorenz system are studied. In this article, a new chaotic model is introduced and studied computationally. By finding the fixed points, the eigenvalues of the Jacobian, and the Lyapunov exponents. Transition from convergence behavior to the periodic behavior (limit cycle) are observed by varying the degree of the system. Also transiting from periodic behavior to the chaotic behavior are seen by changing the degree of the system.

Investigation of Lorenz Equation System with Variable Step Size Strategy

2019

In this study, variable step size strategy has been considered to analyze the numerical solution of the Lorenz system with chaotic structure. Phase portraits have been obtained for this chaotic system. The effectiveness of the variable step size strategy for the solution of this chaotic system has been discussed.