Complex dynamics and control of a novel physical model using nonlocal fractional differential operator with singular kernel

A E Matouk; I Khan

doi:10.1016/j.jare.2020.05.003

Complex dynamics and control of a novel physical model using nonlocal fractional differential operator with singular kernel

J Adv Res. 2020 Jun 2:24:463-474. doi: 10.1016/j.jare.2020.05.003. eCollection 2020 Jul.

Authors

A E Matouk^{1

2}, I Khan¹

Affiliations

¹ Department of Mathematics, College of Science, Al-Zulfi, Majmaah University, Al-Majmaah 11952, Saudi Arabia.
² College of Engineering, Majmaah University, Al-Majmaah 11952, Saudi Arabia.

Abstract

Fractional calculus (FC) is widely used in many interdisciplinary branches of science due to its effectiveness in describing and investigating complicated phenomena. In this work, nonlinear dynamics for a new physical model using nonlocal fractional differential operator with singular kernel is introduced. New Routh-Hurwitz stability conditions are derived for the fractional case as the order lies in [0,2). The new and basic Routh-Hurwitz conditions are applied to the commensurate case. The local stability of the incommensurate orders is also discussed. A sufficient condition is used to prove that the solution of the proposed system exists and is unique in a specific region. Conditions for the approximating periodic solution in this model via Hopf bifurcation theory are discussed. Chaotic dynamics are found in the commensurate system for a wide range of fractional orders. The Lyapunov exponents and Lyapunov spectrum of the model are provided. Suppressing chaos in this system is also achieved via two different methods.

Keywords: Chaos; Chaos control; Hopf bifurcation; Nonlocal fractional differential operator; Stability.