Abstract
This study investigates the use of the differential transform method (DTM) for integrating the Rössler system of the fractional order. Preliminary studies of the integer-order Rössler system, with reference to other well-established integration methods, made it possible to assess the quality of the method and to determine optimal parameter values that should be used when integrating a system with different dynamic characteristics. Bifurcation diagrams obtained for the Rössler fractional system show that, compared to the RK4 scheme-based integration, the DTM results are more resistant to changes in the fractionality of the system.
Highlights
Real technical and physical systems are usually best described by differential equations with nonlinear terms
The integration of the fractional-order differential equation (FODE) using the method based on the definition of fractional derivative requires considering, in each step, the values obtained in the previous steps; theoretically, these are all values obtained from the beginning of motion of the system
To assess the quality of solutions obtained with the use of the differential transform method (DTM), a number of simulations of the chaotic Rössler system (a = 0.4079) were performed, and their results were compared with results obtained with other methods
Summary
Real technical and physical systems are usually best described by differential equations with nonlinear terms. The integration of the fractional-order differential equation (FODE) using the method based on the definition of fractional derivative requires considering, in each step, the values obtained in the previous steps; theoretically, these are all values obtained from the beginning of motion of the system. This makes such calculations very time-consuming and requires a great deal of computing power. Arikoglu and Ozkol [11] extended the DTM by introducing a fractional derivative transform, which made it possible to obtain good quality numerical solutions of fractional equations by using simple iterative DTM algorithms
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