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Communications in Mathematical Physics

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

Global Structure of Quaternion Polynomial Differential Equations

  • Xiang ZhangEmail author
Article

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

Keywords

Periodic Orbit Invariant Torus Heteroclinic Orbit Liouvillian Theorem Invariant Plane 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Abraham R., Marsden J.E.: Foundations of Mechanics. 2nd Ed. Redwood City, CA, Addison–Wesley (1987)Google Scholar
  2. 2.
    Adler S.L.: Quaternionic quantum field theory. Commun. Math. Phys. 104, 611–656 (1986)CrossRefzbMATHADSGoogle Scholar
  3. 3.
    Adler S.L.: Quaternionic Quantum Mechanics and Quantum Fields. Oxford University Press, New York (1995)zbMATHGoogle Scholar
  4. 4.
    Álvarez M.J., Gasull A., Prohens R.: Configurations of critical points in complex polynomial differential equations. Nonlin. Anal. 71, 923–934 (2009)CrossRefzbMATHGoogle Scholar
  5. 5.
    Arnold V.I.: Mathematial Methods of Classical Mechanics. Springer-Verlag, New York (1978)Google Scholar
  6. 6.
    Bruschi M., Calogero F.: Integrable systems of quartic oscillators. II. Phys. Lett. A 327, 320–326 (2004)CrossRefzbMATHADSMathSciNetGoogle Scholar
  7. 7.
    Calogero F., Degasperis A.: New integrable PDEs of boomeronic type. J. Phys. A 39, 8349–8376 (2006)CrossRefzbMATHADSMathSciNetGoogle Scholar
  8. 8.
    Calogero F., Degasperis A.: New integrable equations of nonlinear Schrödinger type. Stud. Appl. Math 113, 91–137 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Chen C., Cao J., Zhang X.: The topological structure of the Rabinovich system having an invariant algebraic surface. Nonlinearity 21, 211–220 (2008)CrossRefzbMATHADSMathSciNetGoogle Scholar
  10. 10.
    Campos J., Mawhin J.: Periodic solutions of quaternionic-values ordinary differential equations. Ann. di Mat. 185, S109–S127 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Cima A., Gasull A., Mañosa V.: Some properties of the k–dimensional Lyness’s map. J. Phys. A: Math. Theor. 41, 285205 (2008)CrossRefGoogle Scholar
  12. 12.
    Finkelstein D., Jauch J.M., Schiminovich S., Speiser D.: Foundations of quaternion quantum mechanics . J. Math. Phys. 3, 207–220 (1962)CrossRefADSMathSciNetGoogle Scholar
  13. 13.
    Frobenius F.G.: Ueber lineare Substitutionen und bilineare Formen. J. Reine Angew. Math 84, 1–63 (1878)CrossRefGoogle Scholar
  14. 14.
    Gasull A., Llibre J., Mãosa V., Mãosas F.: The focus-centre problem for a type of degenerate system. Nonlinearity 13, 699–729 (2000)CrossRefzbMATHADSMathSciNetGoogle Scholar
  15. 15.
    Gasull A., Llibre J., Zhang X.: One–dimensional quaternion homogeneous polynomial differential equations. J. Math. Phys. 50, 082705 (2009)CrossRefADSMathSciNetGoogle Scholar
  16. 16.
    Gibbon J.D.: A quaternionic structure in the three–dimensional Euler and ideal magneto–hydrodynamics equation. Physica D 166, 17–28 (2002)CrossRefzbMATHADSMathSciNetGoogle Scholar
  17. 17.
    Gibbon J.D., Holm D.D., Kerr R.M., Roulstone I.: Quaternions and particle dynamics in the Euler fluid equations. Nonlinearity 19, 1969–1983 (2006)CrossRefzbMATHADSMathSciNetGoogle Scholar
  18. 18.
    Giné J., Llibre J.: Integrability and algebraic limit cycles for polynomial differential systems with homogeneous nonlinearities. J. Diff. Eqs. 197, 147–161 (2004)CrossRefzbMATHGoogle Scholar
  19. 19.
    Hamilton S.W.R.: Lectures on Quaternions. Royal Irish Academy, Dublin, Hodges and Smith (1853)Google Scholar
  20. 20.
    Hanson A.J.: Quaternions. Elsevier, San Francisco (2006)Google Scholar
  21. 21.
    Iona S., Calogero F.: Integrable systems of quartic oscillators in ordinary (three-dimensional) space. J. Phys. A 35, 3091–3098 (2002)CrossRefzbMATHADSMathSciNetGoogle Scholar
  22. 22.
    Lang S.: Differential and Riemannian Manifolds. Springer-Verlag, New York (1995)zbMATHCrossRefGoogle Scholar
  23. 23.
    Li C., Li W., Llibre J., Zhang Z.: On the limit cycles of polynomial differential systems with homogeneous nonlinearities. Proc. Edinburgh Math. Soc. 43(2), 529–543 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Llibre J.: Handbook of Differential Equations, pp. 437–532. Elsevier/North–Holland, Amsterdam (2004)Google Scholar
  25. 25.
    Llibre J., Valls C.: Classification of the centers, their cyclicity and isochronicity for a class of polynomial differential systems generalizing the linear systems with cubic homogeneous nonlinearities. J. Diff. Eqs. 246, 2192–2204 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Llibre J., Yu J.: On the periodic orbits of the static, spherically symmetric Einstein-Yang-Mills equations. Commun. Math. Phys. 286, 277–281 (2009)CrossRefzbMATHADSMathSciNetGoogle Scholar
  27. 27.
    Llibre J., Zhang X.: Invariant algebraic surfaces of the Lorenz systems. J. Math. Phys. 43, 1622–1645 (2002)CrossRefzbMATHADSMathSciNetGoogle Scholar
  28. 28.
    Llibre J., Zhang X.: Darboux theory of integrability in C n taking into account the multiplicity. J. Diff. Eqs. 246, 541–551 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Roubtsov V.N., Roulstone I.: Examples of quaternionic and Kähler structures in Hamiltonian models of nearly geostrophic flow. J. Phys. A: Math. Gen. 30, L63–L68 (1997)CrossRefADSGoogle Scholar
  30. 30.
    Roubtsov V.N., Roulstone I.: Holomorphic structures in hydrodynamical models of nearly geostrophic flow. Proc. R. Soc. London A 457, 1519–1531 (2001)CrossRefzbMATHADSMathSciNetGoogle Scholar
  31. 31.
    Wilczynski P.: Quaternionic–valued ordinary differential equations. The Riccati equation. J. Diff. Eqs. 247, 2163–2187 (2009)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

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

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