Global Structure of Quaternion Polynomial Differential Equations

In this paper we mainly study the global structure of the quaternion Bernoulli equations q = aq + bq for q ∈ 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 H. Moreover, if n = 2 all the invariant tori are full of periodic orbits; if n = 3 there are infinitely many invariant tori fulfilling periodic orbits and also infinitely many invariant ones fulfilling dense orbits. © 2011 Springer-Verlag.