IJAM: Volume 38, No. 1 (2025)

DOI: 10.12732/ijam.v38i1.7

FRACTIONAL MULTISTEP DIFFERENTIAL TRANFORMATION

METHOD USED TO ANALYZE A MODIFIED FORM OF FRACTIONAL

ORDER LORENZ SYSTEM

Ylldrita Salihi 1 , Krutan Rasimi 2, Gjorgji Markoski 3

1 University of Tetova, Str.101, Sllatino

Tetovo - 1201, Republic NORTH MACEDONIA

2 University of Tetova, Str. of Luboten 96

Tetovo - 1201, Republic NORTH MACEDONIA

3 Ss.Cyril and Methodius University, Arhimedova 3

Skopje-1000, Republic NORTH MACEDONIA

Abstract. Dynamics of nonlinear fractional-order Lorenz system is investigated by employing Fractional Multistep Differential Transformation Method (FMDTM). In order to illustrate the new technique, the numerical algorithm is applied in the 3D solution of modified Lorenz system by adding the forth varied parameter a = 40, =3, =10, 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 behaviour and it is a potential model for application. Investigation of dynamics

is realized by fixing the parameters (system has chaotic behaviour) and by changing the added parameter d ∈ [5, 38], implemented with the aid of Mathematica

symbolic package. For d = 25, the minimal fractional order, for which the system shows chaotic behaviour is v = 0.8726, for v = 0.998, the minimal value of d, for which system shows chaotic behaviour is d > 12.05219. The fractional derivatives are described in the Caputo sense. Based on FMDTM, is shown that the system has rich dynamical characteristics, it changes from a non-chaotic system to a chaotic one, using fractional order v ∈ (0, 1] . The method deals with the approximated solutions to integer-order differential equations and is based on polynomial approximations, with good results (based on numerical experiments) for fractional order closed to 1.

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How to cite this paper?
DOI: 10.12732/ijam.v38i1.7
Source: International Journal of Applied Mathematics
ISSN printed version: 1311-1728
ISSN on-line version: 1314-8060
Year: 2025
Volume: 38
Issue: 1

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