On the Second Lyapunov Method for Quaternionic Differential Equations

• Huang, Song ^[1] ; He, Fuli ^[1]
  1. [1] Central South University

     Central South University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 20, Nº 2, 2021
• Idioma: inglés
• DOI: 10.1007/s12346-021-00476-8
• Enlaces
  • Texto completo
• Resumen

  • In this paper we mainly study the stability of quaternion-valued differential equations. We first establish the second Lyapunov method over quaternion field and then we use this theory to study the stability in both autonomous systems and periodic nonautonomous systems cases.

• Referencias bibliográficas
  • Alder, S.L.: Quaternionic quantum field theory. Commun. Math. Phys. 104(4), 611–656 (1986)
  • Adler, S.L.: Quaternionic QuantumMechanics and Quantum Fields. Oxford University Press, NewYork (1995)
  • Finkelstein, D., Jauch, J.M., Schiminovich, S., Speiser, D.: Foundations of quaternion quantum mechanics. J. Math. Phys. 3(2), 207–220 (1962)
  • Roubtsov, V.N., Roulstone, I.: Examples of quaternionic and Khler structures in Hamiltonian models of nearly geostrophic flow. J. Phys. A...
  • Roubtsov, V.N., Roulstone, I.: Holomorphic structures in hydrodynamical models of nearly geostrophic flow. Proc. R. Soc. London A 457, 1519–1531...
  • Gibbon, J.D.: A quaternionic structure in the three-dimensional Euler and ideal magnetohydrodynamics equation. Physica D. 166(1–2), 17–28...
  • Chou, J.C.: Quaternion kinematic and dynamic differential equations. IEEE Trans. Robot. Autom. 8(1), 53–64 (1992)
  • Gupta, S.: Linear quaternion equations with application to spacecraft attitude propagation. In: Aerospace Conference, vol. 1, 69-76. IEEE...
  • De Leo, S., Ducati, G.C.: Solving simple quaternionic differential equations. J. Math. Phys. 44(5), 2224–2233 (2003)
  • Campos, J., Mawhin, J.: Periodic solutions of quaternionic-valued ordinary differential equations. Ann. Mat. Pura Appl. 185(s5), S109–S127...
  • Wilczy ´nski, P.: Quaternionic-valued ordinary differential equations. The Riccati equation. J. Differ. Equ. 247(7), 2163–2187 (2009)
  • Wilczy ´nski, P.: Quaternionic-valued ordinary differential equations II. Coinciding sectors. J. Differ. Equ. 252(8), 4503–4528 (2012)
  • Gasull, A., Llibre, J., Zhang, X.: One-dimensional quaternion homogeneous polynomial differential equations. J. Math. Phys. 50(8), (2009)
  • Zhang, X.: Global structure of quaternion polynomial differential equations. Commun. Math. Phys. 303(2), 301–316 (2011)
  • Walcher, S., Zhang, X.: Polynomial differential equations over the quaternions. J. Differ. Equ. 282, 566–595 (2011)
  • Li, Z., Wang, C.: Cauchy matrix and liouville formula of quaternion impulsive dynamic equations on time scales. Open Math. 18(1), 353–377...
  • Liu, Y., Zhang, D., Lou, J., Lu, J., Cao, J.: Stability analysis of quaternion-valued neural networks: decomposition and direct approaches....
  • Kou, K.I., Liu, W.K., Xia, Y.H.: Solve the linear quaternion-valued differential equations having multiple eigenvalues. J. Math. Phys. 60(2),...
  • Kou, K.I., Xia, Y.H.: Linear quaternion differential equations: basic theory and fundamental results. Stud. Appl. Math. 141(1), 3–45 (2018)
  • Cheng, D., Kou, K.I., Xia, Y.H.: Floquet theory for quaternion-valued differential equations. Qual. Theory Dyn. Syst. 19, 14 (2020)
  • Sudbery, A.: Quaternionic analysis. In: Mathematical proceedings of the Cambridge philosophical society. Cambridge University Press, Cambridge....
  • Kyrchei, I.I.: Cramer’s rule for quaternion systems of linear equations. J. Math. Sci. 155(6), 839–858 (2008)
  • Kyrchei, I.I.: Cramer’s rule for some quaternion matrix equations. App. Math. Comp. 217, 2024–2030 (2010)
  • Zhang, F.: Quaternions and matrices of quaternions. Linear Algebr. Appl. 251(8), 21–57 (1997)
  • Rodman, L.: Topics in Quaternion Linear Algebra. Princeton University Press, Princeton (2014)
  • Zhang, F., Wei, Y.: Jordan canonical form of a partitioned complex matrix and its applications to real quaternion matrices. Commun. Alg. 29(6),...