RÖSSLER SYSTEM AND ITS CHAOS, NUMERICALLY STUDIED USING FRACTIONAL ADAMS BASHFORTH MOULTON METHOD
Ylldrita SALIHI, Krutan RASIMI
Abstract
The dynamics of the nonlinear fractional-order Rössler system are investigated through numerical simulations using the Fractional Adams–Bashforth–Moulton Method (FABM). To demonstrate the computational approach, the algorithm is applied to obtain a three-dimensional solution of the Rössler system, originally conceptualized by Otto Rössler as a mathematical model of a taffy-pulling machine. A fixed-parameter dynamical analysis, along with a chaos diagram, is conducted. The findings reveal that the fractional-order Rössler system exhibits complex and diverse dynamical behaviors, highlighting its potential for various applications. The fractional derivatives are described in the Caputo sense. Firstly, investigation of dynamics is realized by fixing the parameters (system has chaotic behavior), implemented with the aid of Mathematica symbolic package, system shows chaotic behavior for . By varying with parameter and fixed , v=0.9 based on FABM, is shown that the system has rich dynamical characteristics with different types of chaos! For a=0.01 and a=0.56231, chaos changes from periodic cycles to random chaos (deterministic chaos) for the intervals and respectively and . An active control law is applied to the incommensurate fractional order Rössler system (a=0.5, b=2, c=4), using only one input. This indicates that the proposed controller can linearize the system and has stabilized it for the found value of where the system exhibits chaotic behavior.
Pages: 452 - 460