RK4 AND VECTORIAL RK4 USED TO ANALYZE A MODIFIED FORM OF FRACTIONAL ORDER LORENZ SYSTEM

Ylldrita SALIHI, Miranda XHAFERI, Krutan RASIMI, Alit IBRAIMI, Flamure SADIKI

Abstract

The dynamics of a generalized form of fractional-order Lorenz system are investigated by employing a modified version of the Runge-Kutta 4 method (RK4). The method is very simple and very effective for solving differential equations of fractional order, it may be used. To illustrate the new technique, the numerical algorithm is applied in the 3D solution of the Lorenz system by adding the fourth varied parameter, considered as a highly simplified model for the weather. Parameter fixed dynamical analysis method and chaos diagram are used. Results show that the fractional order Lorenz system has rich dynamical behavior and it is a potential model for application. Investigation of dynamics is realized by fixing the parameters (system has chaotic behavior, numerically illustrated), for , implemented with the aid of Mathematica symbolic package. The fractional derivatives are described in the Caputo sense. Based on RK4 and Vectorial RK4 algorithms, is shown that the system has rich dynamical characteristics, it changes from a non-chaotic system to a chaotic one, which is more complex for smaller fractional derivative order , closer to 0.

Pages: 425 - 431