Global structure of quaternion polynomial differential equations

TL;DR

This paper investigates the global structure of quaternion Bernoulli differential equations, revealing the existence of invariant tori with diverse dynamical behaviors, including periodic and dense orbits, using topological and integrability methods.

Contribution

It demonstrates the presence of invariant tori in quaternion Bernoulli equations and characterizes their dynamical properties based on the degree n, extending understanding of quaternion differential systems.

Findings

Invariant tori exist with full measure in quaternion Bernoulli equations.

For n=2, all invariant tori contain periodic orbits.

For n=3, there are infinitely many invariant tori with periodic and dense orbits.

Abstract

In this paper we mainly study the global structure of the quaternion Bernoulli equations $\dot q=aq+bq^n$ for $q\in \mathbb H$ the quaternion field and also some other form of cubic quaternion differential equations. By using the Liouvillian theorem of integrability and the topological characterization of $2$--dimensional torus: orientable compact connected surface of genus one, we prove that the quaternion Bernoulli equations may have invariant tori, which possesses a full Lebesgue measure subset of $\mathbb H$. Moreover, if $n=2$ all the invariant tori are full of periodic orbits; if $n=3$ there are nfiinitely many invariant tori fulfilling periodic orbits and also infinitely many invariant ones fulfilling dense orbits.