The Verhulst-Like Equations: Integrable OΔE and ODE with Chaotic Behavior

Abstract

In this paper, we study various variants of Verhulst-like ordinary differential equations (ODE) and ordinary difference equations (OΔE). Usually Verhulst ODE serves as an example of a deterministic system and discrete logistic equation is a classic example of a simple system with very complicated (chaotic) behavior. In our paper we present examples of deterministic discretization and chaotic continualization. Continualization procedure is based on Padé approximants. To correctly characterize the dynamics of obtained ODE we measured such characteristic parameters of chaotic dynamical systems as the Lyapunov exponents and the Lyapunov dimensions. Discretization and continualization lead to a change in the symmetry of the mathematical model (i.e., group properties of the original ODE and OΔE). This aspect of the problem is the aim of further research.