On the Logistic Equation in the Complex Plane

Abstract

The famous logistic differential equation is studied in the complex plane. The method used is based on a functional analytic technique which provides a unique solution of the ordinary differential equation (ODE) under consideration in H 2 ( ) or H 1 ( ) and gives rise to an equivalent difference equation for which a unique solution is established in 2 or 1 . For the derivation of the solution of the logistic differential equation this discrete equivalent equation is used. The obtained solution is analytic in z ∈ : |z| < T , T > 0. Numerical experiments were also performed using the classical 4th order Runge-Kutta method. The obtained results were compared for real solutions as well as for solutions of the form y(t ) = u(t ) + iv(t ), t ∈ . For t ∈ the solution derived by the present method, seems to have singularities, that is, points where it ceases to be analytic, in certain sectors of the complex plane. These sectors, depending on the values of the involved parameters, can move at different directions, join forming common sectors, or pass through each other and continue moving independently. Moreover, the real and imaginary part of the solution seem to exhibit oscillatory behavior near these sectors.

An efficient numerical approach for delayed logistic models

“…Remark. Before we proceed with a conclusion, it is good to mention recent works concerning discrete equivalent of continuous logistic equation without delay ( [19], [20]). Although the authors used more general approach involving Hilbert spaces, the derived transformation rules are identical to DTM formulas.…”

A functional-analytic technique for the study of analytic solutions of PDEs

“…[13]. Recently, this technique was utilized in the framework of finding discrete equivalents of ODEs and combined with a similar technique for ordinary difference equations, in order to be proposed as a "discretization" technique for the solution of ODEs, see [15], [17]. Moreover, although this "discretization" technique is intended for the study of initial value problems of ODEs, it can be easily combined with a standard "shooting" method and give more than satisfactory results, in the numerical approximation of solutions of boundary value problems of ODEs as well, see [16].…”