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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abraham R., Marsden J.E.: Foundations of Mechanics. 2nd Ed. Redwood City, CA, Addison–Wesley (1987)
Adler S.L.: Quaternionic quantum field theory. Commun. Math. Phys. 104, 611–656 (1986)
Adler S.L.: Quaternionic Quantum Mechanics and Quantum Fields. Oxford University Press, New York (1995)
Álvarez M.J., Gasull A., Prohens R.: Configurations of critical points in complex polynomial differential equations. Nonlin. Anal. 71, 923–934 (2009)
Arnold V.I.: Mathematial Methods of Classical Mechanics. Springer-Verlag, New York (1978)
Bruschi M., Calogero F.: Integrable systems of quartic oscillators. II. Phys. Lett. A 327, 320–326 (2004)
Calogero F., Degasperis A.: New integrable PDEs of boomeronic type. J. Phys. A 39, 8349–8376 (2006)
Calogero F., Degasperis A.: New integrable equations of nonlinear Schrödinger type. Stud. Appl. Math 113, 91–137 (2004)
Chen C., Cao J., Zhang X.: The topological structure of the Rabinovich system having an invariant algebraic surface. Nonlinearity 21, 211–220 (2008)
Campos J., Mawhin J.: Periodic solutions of quaternionic-values ordinary differential equations. Ann. di Mat. 185, S109–S127 (2006)
Cima A., Gasull A., Mañosa V.: Some properties of the k–dimensional Lyness’s map. J. Phys. A: Math. Theor. 41, 285205 (2008)
Finkelstein D., Jauch J.M., Schiminovich S., Speiser D.: Foundations of quaternion quantum mechanics . J. Math. Phys. 3, 207–220 (1962)
Frobenius F.G.: Ueber lineare Substitutionen und bilineare Formen. J. Reine Angew. Math 84, 1–63 (1878)
Gasull A., Llibre J., Mãosa V., Mãosas F.: The focus-centre problem for a type of degenerate system. Nonlinearity 13, 699–729 (2000)
Gasull A., Llibre J., Zhang X.: One–dimensional quaternion homogeneous polynomial differential equations. J. Math. Phys. 50, 082705 (2009)
Gibbon J.D.: A quaternionic structure in the three–dimensional Euler and ideal magneto–hydrodynamics equation. Physica D 166, 17–28 (2002)
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)
Giné J., Llibre J.: Integrability and algebraic limit cycles for polynomial differential systems with homogeneous nonlinearities. J. Diff. Eqs. 197, 147–161 (2004)
Hamilton S.W.R.: Lectures on Quaternions. Royal Irish Academy, Dublin, Hodges and Smith (1853)
Hanson A.J.: Quaternions. Elsevier, San Francisco (2006)
Iona S., Calogero F.: Integrable systems of quartic oscillators in ordinary (three-dimensional) space. J. Phys. A 35, 3091–3098 (2002)
Lang S.: Differential and Riemannian Manifolds. Springer-Verlag, New York (1995)
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)
Llibre J.: Handbook of Differential Equations, pp. 437–532. Elsevier/North–Holland, Amsterdam (2004)
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)
Llibre J., Yu J.: On the periodic orbits of the static, spherically symmetric Einstein-Yang-Mills equations. Commun. Math. Phys. 286, 277–281 (2009)
Llibre J., Zhang X.: Invariant algebraic surfaces of the Lorenz systems. J. Math. Phys. 43, 1622–1645 (2002)
Llibre J., Zhang X.: Darboux theory of integrability in C n taking into account the multiplicity. J. Diff. Eqs. 246, 541–551 (2009)
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)
Roubtsov V.N., Roulstone I.: Holomorphic structures in hydrodynamical models of nearly geostrophic flow. Proc. R. Soc. London A 457, 1519–1531 (2001)
Wilczynski P.: Quaternionic–valued ordinary differential equations. The Riccati equation. J. Diff. Eqs. 247, 2163–2187 (2009)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by G. Gallavotti
The author is partially supported by NNSF of China grant 10831003 and Shanghai Pujiang Program grant 09PJD013.
Rights and permissions
About this article
Cite this article
Zhang, X. Global Structure of Quaternion Polynomial Differential Equations. Commun. Math. Phys. 303, 301–316 (2011). https://doi.org/10.1007/s00220-011-1196-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-011-1196-y