Analyzing the inverted fractional rossler system through two approaches: numerical scheme and LHAM

This paper aims to develop the scenario to find out how two approaches can be employed in estimating the numerical results in systems. The first technique is the application of the Caputo Fabrizio fractional schema, which is a method for overcoming the challenges of singular calculus while providing a generalized approach in a more flexible framework. The second approach involves the Laplace transform and the Homotopy analysis method (LHAM) to give an analytical solution that inherits good features of both methods, capable of tackling nonlinearity and yielding fast convergence. In the present work, we use these techniques to investigate the inverted Rössler system where the equations are defined within the Caputo Fabrizio derivative. By applying this system, which is associated with behaviour analysis using both methods, we conclude that Caputo Fabrizio is more appropriate for deriving solutions, whereas LHAM is preferable since it converges rapidly and is computationally efficient. Comparisons are provided for the validity of the outlined approaches, which is corroborated by the potential use of such methods in solving systems that recognize ‘chaotic’ patterns and in extending the applications of the forecasting approaches identified to other branches of science and engineering. Besides, an additional analysis focuses on the strengths and weaknesses of each method to enable the user to understand how the method works and how it produces results.