Discretization of conformable fractional differential equations by a piecewise constant approximation

Abstract

In this paper, a conformable fractional-order logistic differential equation including both discrete and continuous time is taken into account. By using a piecewise constant approximation, a discretization method which transforms a fractional-order differential equation into a difference equation is introduced. Necessary and sufficient conditions for both local and global stability of the discretized system are obtained. The control space diagrams (α,r) and (h,r) with the fractional-order parameter α, a discretization parameter (h) and the growth parameter (r) are obtained and these diagrams illustrate the regions where the solutions of the system approach to the positive equilibrium point with monotonic and damped oscillations. Finally, the existence of flip bifurcation is proved using the centre manifold theory and these theoretical results are supported by numerical calculations.