Simple Chaotic Systems with Specific Analytical Solutions

Abstract

In this paper, a new structure of chaotic systems is proposed. There are many examples of differential equations with analytic solutions. Chaotic systems cannot be studied with the classical methods. However, in this paper we show that a system that has a simple analytical solution can also have a strange attractor. The main goal of this paper is to show examples of chaotic systems with a simple analytical solution that is unstable so that the chaotic orbit does not track it. We believe the structures presented here are new. Two categories of chaotic systems are described, and their dynamical properties are investigated. The proposed systems have analytic solutions that exist far from the equilibrium. Of course, all strange attractors are dense in unstable periodic orbits, but mostly the equations that describe these orbits are unknown and difficult to calculate. The analytical solutions provide examples where the orbits can be calculated despite their instability.