Chaos and Bifurcations in the Dynamics of the Variable-Order Fractional Rössler System

Athar I. Ahmed (), Mohamed Elbadri (), Naseam Al-kuleab, Dalal M. AlMutairi, Nidal E. Taha and Mohammed E. Dafaalla
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Athar I. Ahmed: Department of Mathematics, College of Science, University of Ha’il, Ha’il 55473, Saudi Arabia
Mohamed Elbadri: Mathematics Department, College of Science, Jouf University, Sakaka 72388, Saudi Arabia
Naseam Al-kuleab: Department of Mathematics and Statistics, College of Science, King Faisal University, Al-Ahsa 31982, Saudi Arabia
Dalal M. AlMutairi: Unit of Natural Sciences, Applied College, Shaqra University, Shaqra 11961, Saudi Arabia
Nidal E. Taha: Department of Mathematics, College of Science, Qassim University, Buraidah 51452, Saudi Arabia
Mohammed E. Dafaalla: Department of Mathematics, College of Science, Qassim University, Buraidah 51452, Saudi Arabia

Mathematics, 2025, vol. 13, issue 22, 1-16

Abstract: This article investigates the chaotic features of a novel variable-order fractional Rössler system built with Liouville–Caputo derivatives of variable order. Variable-order fractional (VOF) operators incorporated in the system render its dynamics more flexible and richer with memory and hereditary effects. We run numerical simulations to see how different fractional-order functions alter the qualitative behavior of the system. We demonstrate this via phase portraits and time-series responses. The research analyzes bifurcation development, chaotic oscillations, and stability transition and demonstrates dynamic patterns impossible to describe with integer-order models. Lyapunov exponent analysis also demonstrates system sensitivity to initial conditions and small disturbances. The outcomes confirm that the variable-order procedure provides a faithful representation of nonlinear and intricate processes of engineering and physical sciences, pointing out the dominant role of memory effects on the transitions among periodic, quasi-periodic, and chaotic regimes.

Keywords: fractional derivatives; chaos; bifurcation; simulation; dynamic system (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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