A Novel Fractional-Order System Described by the Caputo Derivative, Its Numerical Discretization, and Qualitative Properties

Authored by: Ndolane Sene

Handbook of Fractional Calculus for Engineering and Science

Print publication date:  February  2022
Online publication date:  February  2022

Print ISBN: 9781032047799
eBook ISBN: 9781003263517
Adobe ISBN:


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This chapter considers a new fractional chaotic system described by the Caputo derivative. The Lyapunov exponents give the nature of the behaviors of the solutions of the fractional system under consideration: chaotic and hyperchaotic behavior. Phase portraits are obtained after implementation of the numerical scheme. Note that the numerical scheme presented in this chapter includes the numerical discretization of the Riemann–Liouville fractional integral. The stability of the equilibrium points is illustrated using the Matignon criterion. The impacts of the initial conditions on the behaviors of the solutions are discussed. The associated Lyapunov exponents are calculated because the initial conditions significantly influence the nature of the proposed model’s dynamics. The chapter shows that the bifurcation diagrams and the Lyapunov exponents play a crucial role in the nature of the dynamics. The coexistence of attractors in the same and different planes of our proposed fractional system is also discussed. The circuit schematic should be implemented for applications in real-world science and engineering problems. Chaotic circuits for science and engineering applications are simulated in this chapter using Multisim software.

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