Abstract
In this paper, a fractional-order Mackey-Glass equation with constant delay is considered. The local stability of the fixed points is analyzed. Moreover, a discretization process is applied to convert the fractional-order delay equation to its discrete analog. A numerical simulation including Lyapunov exponent, phase diagrams, bifurcation, and chaos is carried out using Matlab to ensure theoretical results and to reveal more complex dynamics of the equation after discretization.
Highlights
Delay differential equations (DDEs) arise in the mathematical description of systems whose time evolution depends explicitly on a past state of the system, as for example in the case of delayed feedback
DDEs arise in many areas of mathematical modeling: for example, population dynamics, infectious diseases, physiological and pharmaceutical kinetics, chemical kinetics, the navigational control of ships and aircraft, and more general control problems
In order to study the local stability of these fixed points, we need the moduli of the eigenvalues of the Jacobian matrix evaluated at each of the fixed points [ ]
Summary
Delay differential equations (DDEs) arise in the mathematical description of systems whose time evolution depends explicitly on a past state of the system, as for example in the case of delayed feedback. Fractional calculus is a generalization of classical differentiation and integration to arbitrary (non-integer) order [ – ]. Many mathematicians and applied researchers have tried to model real processes using fractional calculus [ – ]. In recent years differential equations with fractional-order have attracted many researchers because of their applications in many areas of science and engineering. We recall the basic definitions (Caputo) and properties of fractional-order differentiation and integration. Definition The fractional integral of order β ∈ R+ of the function f (t), t > is defined by. We will show that considering a fractional-order derivative with delay in equation
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