Degree, quaternions and periodic solutions

References

• 1.
  Campos J, Mawhin J. 2006Periodic solutions of quaternionic-valued ordinary differential equations. Annali Mat. Pura Appl. 185, 109–127. (doi:10.1007/s10231-004-0139-z) Crossref, ISI, Google Scholar
• 2.
  Zoladek H. 2009Classification of diffeomorphisms of ${\mathbb{S}}^4$ induced by quaternionic Riccati equations with periodic coefficients. Topol. Methods Nonlinear Anal. 33, 205–215. (doi:10.12775/TMNA.2009.014) Crossref, ISI, Google Scholar
• 3.
  Wilczynski P. 2009Quaternionic valued ordinary differential equations. The Riccati equation. J. Differ. Equ. 247, 2163–2187. (doi:10.1016/j.jde.2009.06.015) Crossref, ISI, Google Scholar
• 4.
  Wilczynski P. 2012Quaternionic-valued ordinary differential equations II: coinciding sectors. J. Differ. Equ. 252, 4503–4528. (doi:10.1016/j.jde.2012.01.005) Crossref, ISI, Google Scholar
• 5.
  Cai ZF, Kou KI. 2018Laplace transform : a new approach in solving linear quaternion differential equations. Math. Methods Appl. Sci. 41, 4033–4048. (doi:10.1002/mma.4415) Crossref, ISI, Google Scholar
• 6.
  Cai ZF, Kou KI. 2018Solving quaternion ordinary differential equations with two-sided coefficients. Qual. Theory Dyn. Syst. 17, 441–462. (doi:10.1007/s12346-017-0246-z) Crossref, ISI, Google Scholar
• 7.
  Cheng D, Kou KI, Xia YH. 2020Floquet theory for quaternion-valued differential equations. Qual. Theory Dyn. Syst. 19, 23. (doi:10.1007/s12346-020-00341-0) Crossref, ISI, Google Scholar
• 8.
  Kou KI, Xia YH. 2018Linear quaternion differential equations: basic theory and fundamental results. Stud. Appl. Math. 141, 3–45. (doi:10.1111/sapm.12211) Crossref, ISI, Google Scholar
• 9.
  Gasull A, Llibre J, Zhang X. 2009One-dimensional quaternion homogeneous polynomial differential equations. J. Math. Phys. 50, Article ID 082705. (doi:10.1063/1.3139115) Crossref, ISI, Google Scholar
• 10.
  Zhang X. 2011Global structure of quaternion polynomial differential equations. Commun. Math. Phys. 303, 301–316. (doi:10.1007/s00220-011-1196-y) Crossref, ISI, Google Scholar
• 11.
  Huo N, Li B, Li Y. 2019Existence and exponential stability of anti-periodic solutions for inertial quaternion-valued high-order Hopfield neural networks with state-dependent delays. IEEE Access 7, 60010–60019. (doi:10.1109/ACCESS.2019.2915935) Crossref, ISI, Google Scholar
• 12.
  Li YK, Li B, Yao S, Xiong L. 2018The global exponential pseudo almost periodic synchronization of quaternion-valued cellular neural networks with time-varying delays. Neurocomputing 303, 75–87. (doi:10.1016/j.neucom.2018.04.044) Crossref, ISI, Google Scholar
• 13.
  Li YK, Qin JL. 2018Existence and global exponential stability of periodic solutions for quaternion-valued cellular neural networks with time-varying delays. Neurocomputing 292, 91–103. (doi:10.1016/j.neucom.2018.02.077) Crossref, ISI, Google Scholar
• 14.
  Li YK, Qin JL, Li B. 2019Anti-periodic solutions for quaternion-valued high-order Hopfield neural networks with time-varying delays. Neural Process. Lett. 49, 1217–1237. (doi:10.1007/s11063-018-9867-8) Crossref, ISI, Google Scholar
• 15.
  Li YK, Qin JL, Li B. 2019Existence and global exponential stability of anti-periodic solutions for delayed quaternion-valued cellular neural networks with impulsive effects. Math. Methods Appl. Sci. 42, 5–23. (doi:10.1002/mma.5318) Crossref, ISI, Google Scholar
• 16.
  Li YK, Xiang JL. 2020Existence and global exponential stability of anti-periodic solutions for quaternion-valued cellular neural networks with time-varying delays. Adv. Differ. Equ. 2020, 47. (doi:10.1186/s13662-020-2523-4) Crossref, ISI, Google Scholar
• 17.
  Mawhin J. 2005Simple proofs of various fixed point and existence theorems based on exterior calculus. Math. Nachr. 278, 1607–1614. (doi:10.1002/mana.200410325) Crossref, ISI, Google Scholar
• 18.
  Eilenberg S, Niven I. 1944The ‘fundamental theorem of algebra’ for quaternions. Bull. Am. Math. Soc. 50, 246–248. (doi:10.1090/S0002-9904-1944-08125-1) Crossref, Google Scholar
• 19.
  Mawhin J1979Topological degree methods in nonlinear boundary value problems. CBMS Series, No. 40. Providence, RI: American Math. Soc. Crossref, Google Scholar
• 20.
  Mawhin J. 1993Topological degree and boundary value problems for nonlinear differential equations. In Topological Methods for Ordinary Differential Equations (eds M Furi, P Zecca), CIME Montecatini Terme 1991. Lecture Notes in Math. No. 1537, pp. 74–142. Berlin, Germany: Springer. Google Scholar
• 21.
  Ebbinghaus HD, Hermes H, Hirzebruch F, Koecher M, Mainzer K, Neukirch J, Prestel A, Remmert R. 1991Numbers. New York, NY: Springer. Crossref, Google Scholar
• 22.
  Mawhin J. 2004A simple approach to Brouwer degree based on differential forms. Adv. Nonlinear Stud. 4, 501–514. (doi:10.1515/ans-2004-0409) Crossref, ISI, Google Scholar
• 23.
  Pumplün S, Walcher S. 2002On the zeros of polynomials over quaternions. Commun. Algebra 30, 4007–4018. (doi:10.1081/AGB-120005832) Crossref, ISI, Google Scholar