Complexified Dynamical Systems

TL;DR

This paper explores PT symmetric complex solutions in dynamical systems like predator-prey models and rigid body rotation, revealing that PT symmetry ensures solutions are periodic, extending understanding beyond real solutions.

Contribution

It identifies and analyzes PT symmetric complex solutions in classical dynamical systems, highlighting their periodic nature and expanding the scope of solution analysis.

Findings

PT symmetric solutions are a subset of complex solutions.

PT symmetry ensures solutions are periodic.

Real solutions are included within PT symmetric solutions.

Abstract

Many dynamical systems, such as the Lotka-Volterra predator-prey model and the Euler equations for the free rotation of a rigid body, are PT symmetric. The standard and well-known real solutions to such dynamical systems constitute an infinitessimal subclass of the full set of complex solutions. This paper examines a subset of the complex solutions that contains the real solutions, namely, those having PT symmetry. The condition of PT symmetry selects out complex solutions that are periodic.