Global Structure of Quaternion Polynomial Differential Equations

By Xiang Zhang. 2014

50 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/e3af94e8e51c900ec0783fe5c98424166b48cc52

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In Applied Mathematics

1. Kit Ian Kou
2. Yonghui Xia
3. Tao Qian

In Geometry and Topology

1. Kit Ian Kou
2. Yonghui Xia

In Statistical and Nonlinear Physics

1. Kit Ian Kou
2. Yonghui Xia

Citations in the corpus

• Linear Quaternion Differential Equations: Basic Theory and Fundamental Results (2015, background): “Zhang [12] studied the global structure of the one-dimensional quaternion Bernoulli equations q̇ = aq + aq.”