Global structure of quaternion polynomial differential equations

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.