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Floquet theory for quaternion-valued differential equations. (English) Zbl 1439.34019

Summary: This paper describes the Floquet theory for quaternion-valued differential equations (QDEs). The Floquet normal form of fundamental matrix for linear QDEs with periodic coefficients is presented and the stability of quaternionic periodic systems is accordingly studied. As an important application of Floquet theory, we give a discussion on the stability of quaternion-valued Hill’s equation. Examples are presented to illustrate the proposed results.

MSC:

34A30 Linear ordinary differential equations and systems
34B30 Special ordinary differential equations (Mathieu, Hill, Bessel, etc.)
20G20 Linear algebraic groups over the reals, the complexes, the quaternions
34D20 Stability of solutions to ordinary differential equations
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[1] Chou, Jc, Quaternion kinematic and dynamic differential equations, IEEE Trans. Robot. Autom., 8, 1, 53-64 (1992)
[2] Gupta, S.: Linear quaternion equations with application to spacecraft attitude propagation. In: Aerospace Conference, , vol. 1, pp. 69-76. IEEE (1998)
[3] Gibbon, J., A quaternionic structure in the three-dimensional Euler and ideal magneto-hydrodynamics equations, Phys. D Nonlinear Phenom., 166, 1, 17-28 (2002) · Zbl 1011.35110
[4] Gibbon, Jd; Holm, Dd; Kerr, Rm; Roulstone, I., Quaternions and particle dynamics in the euler fluid equations, Nonlinearity, 19, 8, 1969 (2006) · Zbl 1145.76012
[5] Alder, Sl, Quaternionic quantum field theory, Commun. Math. Phys., 104, 4, 611-656 (1986) · Zbl 0594.58059
[6] Adler, Sl, Quaternionic Quantum Mechanics and Quantum Fields (1995), Oxford: Oxford University Press, Oxford · Zbl 0885.00019
[7] De Leo, S.; Ducati, Gc, Solving simple quaternionic differential equations, J. Math. Phys., 44, 5, 2224-2233 (2003) · Zbl 1062.81026
[8] Campos, J.; Mawhin, J., Periodic solutions of quaternionic-valued ordinary differential equations, Ann. Mat. Pura Appl., 185, S109-S127 (2006) · Zbl 1162.34324
[9] Wilczyński, P., Quaternionic-valued ordinary differential equations. The Riccati equation, J. Differ. Equ., 247, 7, 2163-2187 (2009) · Zbl 1191.34054
[10] Wilczyński, P., Quaternionic-valued ordinary differential equations ii. coinciding sectors, J. Differ. Equ., 252, 8, 4503-4528 (2012) · Zbl 1259.34032
[11] Gasull, A.; Llibre, J.; Zhang, X., One-dimensional quaternion homogeneous polynomial differential equations, J. Math. Phys., 50, 8, 082705 (2009) · Zbl 1223.34060
[12] Zhang, X., Global structure of quaternion polynomial differential equations, Commun. Math. Phys., 303, 2, 301-316 (2011) · Zbl 1251.34019
[13] Kou, Ki; Xia, Y-H, Linear quaternion differential equations: basic theory and fundamental results, Stud. Appl. Math., 141, 1, 3-45 (2018) · Zbl 1406.34036
[14] Kou K.I., Liu, W. K., Xia, Y. H.: Linear quaternion differential equations: basic theory and fundamental results II. arXiv preprint arXiv:1602.01660 (2016)
[15] Kou, Ki; Liu, Wk; Xia, Yh, Solve the linear quaternion-valued differential equations having multiple eigenvalues, J. Math. Phys., 60, 2, 023510 (2019) · Zbl 1410.34039
[16] Cheng, D.; Kou, Ki; Xia, Yh, A unified analysis of linear quaternion dynamic equations on time scales, J. Appl. Anal. Comput., 8, 1, 172-201 (2018) · Zbl 1459.34196
[17] Eilenberg, S.; Niven, I., The “fundamental theorem of algebra” for quaternions, Bull. Am. Math. Soc., 50, 4, 246-248 (1944) · Zbl 0063.01228
[18] Serôdio, R.; Siu, L-S, Zeros of quaternion polynomials, Appl. Math. Lett., 14, 2, 237-239 (2001) · Zbl 0979.30030
[19] Pogorui*, A.; Shapiro, M., On the structure of the set of zeros of quaternionic polynomials, Complex Var. Theory Appl. Int. J., 49, 6, 379-389 (2004) · Zbl 1160.30353
[20] Zhang, F., Quaternions and matrices of quaternions, Linear Algebra Appl., 251, 21-57 (1997) · Zbl 0873.15008
[21] Rodman, L., Topics in Quaternion Linear Algebra (2014), Princeton: Princeton University Press, Princeton · Zbl 1304.15004
[22] Wang, Q-W; Chang, H-X; Ning, Q., The common solution to six quaternion matrix equations with applications, Appl. Math. Comput., 198, 1, 209-226 (2008) · Zbl 1141.15016
[23] Wang, Q-W; Li, C-K, Ranks and the least-norm of the general solution to a system of quaternion matrix equations, Linear Algebra Appl., 430, 5, 1626-1640 (2009) · Zbl 1158.15010
[24] Sudbery, A., Quaternionic analysis, Mathematical Proceedings of the Cambridge Philosophical Society, 85, 2, 199-225 (1979) · Zbl 0399.30038
[25] Chicone, C., Ordinary Differential Equations with Applications (2006), Berlin: Springer, Berlin · Zbl 1120.34001
[26] Hale, Jk, Ordinary Differential Equations (2009), New York: Dover Publications, New York
[27] Johnson, Ra, On a Floquet theory for almost-periodic, two-dimensional linear systems, J. Differ. Equ., 37, 2, 184-205 (1980) · Zbl 0508.34031
[28] Chow, S-N; Lu, K.; Malletparet, J., Floquet theory for parabolic differential equations, J. Differ. Equ., 109, 1, 147-200 (1994) · Zbl 0804.35045
[29] Kuchment, P., Floquet Theory for Partial Differential Equations (1993), Berlin: Springer, Berlin · Zbl 0789.35002
[30] Kuchment, P., On the behavior of Floquet exponents of a kind of periodic evolution problems, J. Differ. Equ., 109, 2, 309-324 (1994) · Zbl 0792.35026
[31] Ahlbrandt, Cd; Ridenhour, J., Floquet theory for time scales and Putzer representations of matrix logarithms, J. Differ. Equ. Appl., 9, 1, 77-92 (2003) · Zbl 1032.39005
[32] Dacunha, Jj; Davis, Jm, A unified Floquet theory for discrete, continuous, and hybrid periodic linear systems, J. Differ. Equ., 251, 11, 2987-3027 (2011) · Zbl 1233.34039
[33] Agarwal, R.; Lupulescu, V.; O’Regan, D.; Younus, A., Floquet theory for a Volterra integro-dynamic system, Appl. Anal., 93, 9, 2002-2013 (2014) · Zbl 1326.34139
[34] Adivar, M.; Koyuncuoglu, Hc, Floquet theory based on new periodicity concept for hybrid systems involving q-difference equations, Appl. Math. Comput., 273, 1208-1233 (2016) · Zbl 1410.34120
[35] Aslaksen, H., Quaternionic determinants, Math. Intell., 18, 3, 57-65 (1996) · Zbl 0881.15007
[36] Baker, A., Right eigenvalues for quaternionic matrices: a topological approach, Linear Algebra Appl., 286, 1, 303-309 (1999) · Zbl 0941.15013
[37] Zhang, F.; Wei, Y., Jordan canonical form of a partitioned complex matrix and its applications to real quaternion matrices, Commun. Algebra, 29, 6, 2363-2375 (2001) · Zbl 0999.15015
[38] Afanasiev, Vn; Kolmanovskii, V.; Nosov, Vr, Mathematical Theory of Control Systems Design (2013), Berlin: Springer, Berlin
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