Communications in Mathematical Physics

, Volume 303, Issue 2, pp 301–316 | Cite as

Global Structure of Quaternion Polynomial Differential Equations

  • Xiang ZhangEmail author


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 infinitely many invariant tori fulfilling periodic orbits and also infinitely many invariant ones fulfilling dense orbits.


Periodic Orbit Invariant Torus Heteroclinic Orbit Liouvillian Theorem Invariant Plane 
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© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Department of MathematicsShanghai Jiaotong UniversityShanghaiP. R. China

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