2023 Volume 13 Issue 1
Article Contents
Article navigation > Journal of Applied Analysis & Computation > 2023 > 13(1): 575-608
Next Article Previous Article

Mengda Wu, Yonghui Xia, Ziyi Xu. FLOQUET MULTIPLIERS AND THE STABILITY OF PERIODIC LINEAR DIFFERENTIAL EQUATIONS: A UNIFIED ALGORITHM AND ITS COMPUTER REALIZATION[J]. Journal of Applied Analysis & Computation, 2023, 13(1): 575-608. doi: 10.11948/20220518
Citation: Mengda Wu, Yonghui Xia, Ziyi Xu. FLOQUET MULTIPLIERS AND THE STABILITY OF PERIODIC LINEAR DIFFERENTIAL EQUATIONS: A UNIFIED ALGORITHM AND ITS COMPUTER REALIZATION[J]. Journal of Applied Analysis & Computation, 2023, 13(1): 575-608. doi: 10.11948/20220518
shu

FLOQUET MULTIPLIERS AND THE STABILITY OF PERIODIC LINEAR DIFFERENTIAL EQUATIONS: A UNIFIED ALGORITHM AND ITS COMPUTER REALIZATION

  • 1.

    School of Mathematical Science, Zhejiang Normal University, Jinhua, 321004, China

    Abstract

  • Floquet multipliers (characteristic multipliers) play significant role in the stability of the periodic equations. Based on the iterative method, we provide a unified algorithm to compute the Floquet multipliers (characteristic multipliers) and determine the stability of the periodic linear differential equations on time scales unifying discrete, continuous, and hybrid dynamics. Our approach is based on calculating the value of $\mathcal{A}$ and $\mathcal{B}$ (see Theorem 3.1), which are the sum and product of all Floquet multipliers (characteristic multipliers) of the system, respectively. We obtain an explicit expression of $\mathcal{A}$ (see Theorem 4.1) by the method of variation and approximation theory (iterative method), and an explicit expression of $\mathcal{B}$ by Liouville's formula. Furthermore, a computer program is designed to realize our algorithm. Specifically, you can determine the stability of a second order periodic linear system, whether they are discrete, continuous or hybrid, as long as you enter the program codes associated with the parameters of the equation. In fact, few literatures have dealt with the algorithm to compute the Floquet multipliers, not mention to design the program for its computer realization. Our algorithm gives the explicit expressions of all Floquet multipliers and our computer program is based on the approximations of these explicit expressions. In particular, on an arbitrary discrete periodic time scale, we can do a finite number of calculations to get the explicit value of Floquet multipliers (see Theorem 4.2). Therefore, for any discrete periodic system, we can accurately determine the stability of the system by our algorithm even without computer! Finally, in Section 6, several examples are presented to illustrate the effectiveness of our algorithm.

    MSC: 34N05, 34L16, 26E70, 34L15, 65L15
    • FullText(HTML)
  • 加载中
    • References
  • [1] R. P. Agarwal and M. Bohner, Basic calculus on time scales and some of its applications, Results Math., 1999, 35(1–2), 3–22.

    Google Scholar

    [2] R. P. Agarwal, M. Bohner, A. Domoshnitsky and Y. Goltser, Floquet theory and stability of nonlinear integro-differential equations, Acta Math. Hungar., 2005, 109(4), 305–330. doi: 10.1007/s10474-005-0250-7

    CrossRef Google Scholar

    [3] R. P. Agarwal, M. Bohner and D. O'Regan, Dynamic equations on time scales: a survey, J. Comput Appl. Math., 2002, 141(1–2), 1–26.

    Google Scholar

    [4] M. Adivar and H. C. Koyuncuoğlu, Floquet theory based on new periodicity concept for hybrid systems involving $ q$-difference equations, Appl. Math. Comput., 2016, 273, 1208–1233.

    $ q$-difference equations" target="_blank">Google Scholar

    [5] C. D. Ahlbrandt and J. Ridenhour, Floquet theory for time scales and Putzer representations of matrix logarithms, J. Difference Equ. Appl., 2003, 9(1), 77–92. doi: 10.1080/10236100309487536

    CrossRef Google Scholar

    [6] B. Aulbach and C. Pötzsche, Reducibility of linear dynamic equations on measure chains, J. Comput. Appl. Math., 2002, 141(1–2), 101–115.

    Google Scholar

    [7] S. R. Barone, M. A. Narcowich and F. J. Narcowich, Floquet theory and applications, Phys. Rev. A, 1977, 15(3), 1119–1125.

    Google Scholar

    [8] L. C. Becker, T. A. Burton and T. Krisztin, Floquet theory for a Volterra equation, J. London Math. Soc., 1988, 37(2), 141–147.

    Google Scholar

    [9] R. Benterki and J. Llibre, Periodic solutions of a class of duffing differential equations, J. Nonlinear Modeling Anal., 2019, 1(2), 167–177.

    Google Scholar

    [10] R. Benterki and J. Llibre, Periodic solutions of the duffing differential equation revisited via the averaging theory, J. Nonlinear Modeling Anal., 2019, 1(1), 11–26.

    Google Scholar

    [11] M. Bohner, G. Guseinov and A. Peterson, Introduction to the Time Scales Calculus: Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, 2003.

    Google Scholar

    [12] M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, 2001.

    Google Scholar

    [13] D. Breda, S. Mast and R. Vermiglio, Numerical computation of characteristic multipliers for linear time periodic coefficients delay differential equations, IFIC Proceedings Volumes, 2006, 39, 163–168.

    Google Scholar

    [14] H. Broer, Resonance tongues in Hill's equations: a geometric approach, J. Differential Equations, 2000, 166(2), 290–327. doi: 10.1006/jdeq.2000.3804

    CrossRef Google Scholar

    [15] R. Carlson, Eigenvalue estimates and trace for the matrix Hill's equation, J. Differential Equations, 2000, 167(1), 211–244. doi: 10.1006/jdeq.2000.3785

    CrossRef Google Scholar

    [16] D. Cheng, K. I. Kou and Y. Xia, A unified annlysis of linear quaternion dynamic equations on time scales, J. Appl. Anal. Comput., 2020, 10(5), 1869–1877.

    Google Scholar

    [17] C. Chicone, Ordinary Differential Equations with Applications, Springer-Verlag, New York, 1999.

    Google Scholar

    [18] S. N. Chow, K. Lu and J. Mallet-Paret, Floquet theory for parabolic differential equations, J. Differential Equations, 1994, 109(1), 147–200. doi: 10.1006/jdeq.1994.1047

    CrossRef Google Scholar

    [19] S. N. Chow and H. O. Walther, Characteristic multipliers and stability of symmetric periodic solutions of $ \dot{x}(t)=g(x(t-1))$, Trans. Amer. Math. Soc., 1988, 307(1), 127–142.

    $ \dot{x}(t)=g(x(t-1))$" target="_blank">Google Scholar

    [20] J. J. DaCunha, Lyapunov Stability and Floquet Theory for Nonautonomous Linear Dynamic Systems on Time Scales, Ph. D. Thesis, Baylor University, 2004. ISBN: 978-0496-82618-6.

    Google Scholar

    [21] J. J. DaCunha and J. M. Davis, A unified Floquet theory for discrete, continuous, and hybrid periodic linear systems, J. Differential Equations, 2011, 251(11), 2987–3027. doi: 10.1016/j.jde.2011.07.023

    CrossRef Google Scholar

    [22] A. Demir, Floquet theory and non-linear perturbation analysis for oscillators with differential-algebraic equations, Int. J. Circuit Theory Appl., 2000, 28(2), 163–185. doi: 10.1002/(SICI)1097-007X(200003/04)28:2<163::AID-CTA101>3.0.CO;2-K

    CrossRef Google Scholar

    [23] T. S. Doan, A. Kalauch, S. Siegmund and F. R. Wirth, Stability radii for positive linear time-invariant systems on time scales, Systems Control Lett., 2010, 59(3–4), 173–179.

    Google Scholar

    [24] T. S. Doan, A. Kalauch and S. Siegmund, Exponential stability of linear time-invariant systems on time scales, Nonlinear Dyn. Syst. Theory, 2009, 9(1), 37–50.

    Google Scholar

    [25] P. Dormayer, A. F. Ivanov and B. Lani-Wayda, Floquet multipliers of symmetric rapidly oscillating solutions of differential delay equations, Tohoku Math. J., 2002, 54(3), 419–441.

    Google Scholar

    [26] L. Erbe and A. Peterson, Green's functions and comparison theorems for differential equations on measure chains, Dynam. Contin. Discrete Impuls. Systems, 1999, 6(1), 121–137.

    Google Scholar

    [27] M. Federson, J. G. Mesquita and A. Slavík, Measure functional differential equations and functional dynamic equations on time scales, J. Differential Equations, 2012, 252(6), 3816–3847. doi: 10.1016/j.jde.2011.11.005

    CrossRef Google Scholar

    [28] M. Federson, R. Grau, J. G. Mesquita and E. Toon, Lyapunov stability for measure differential equations and dynamic equations on time scales, J. Differential Equations, 2019, 267(7), 4192–4223. doi: 10.1016/j.jde.2019.04.035

    CrossRef Google Scholar

    [29] H. I. Freedman, Almost Floquet systems, J. Differential Equations, 1971, 10, 345–354. doi: 10.1016/0022-0396(71)90057-X

    CrossRef Google Scholar

    [30] F. Gesztesy and R. Weikard, Floquet theory revisited, in: Differential Equations and Mathematical Physics, Proceedings of the International Conference, Univ. of Alabama at Birmingham, March 13–17, 1994, International Press, Boston, MA, 1995.

    Google Scholar

    [31] S. Hilger, Ein Maßkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten, Ph. D. thesis, Universität Würzburg, 1988.

    Google Scholar

    [32] S. Hilger, Analysis on measure chains-a unified approach to continuous and discrete calculus, Result Math., 1990, 18(1–2), 18–56.

    Google Scholar

    [33] S. Hilger, Differential and difference calculus-unified!, Nonlinear Anal., 1997, 30(5), 2683–2694. doi: 10.1016/S0362-546X(96)00204-0

    CrossRef Google Scholar

    [34] R. A. Johnson, On a Floquet theory for almost-periodic, two-dimensional linear systems, J. Differential Equations, 1980, 37(2), 184–205. doi: 10.1016/0022-0396(80)90094-7

    CrossRef Google Scholar

    [35] W. G. Kelley and A. C. Peterson, Difference Equations: An Introduction with Applications, Academic Press, San Diego, 2001.

    Google Scholar

    [36] D. Kotsis, The approximation of the characteristic multipliers of periodic differential equations, Alkalmaz. Mat. Lapok, 1976, 2(3–4), 269–276.

    Google Scholar

    [37] P. Kuchment, On the behavior of Floquet exponents of a kind of periodic evolution problems, J. Differential Equations, 1994, 109(2), 309–324. doi: 10.1006/jdeq.1994.1052

    CrossRef Google Scholar

    [38] R. Lamour, R. März and R. Winkler, How Floquet theory applies to index 1 differential algebraic equations, J. Math. Anal. Appl., 1998, 217(2), 372–394. doi: 10.1006/jmaa.1997.5714

    CrossRef Google Scholar

    [39] Z. Li, C. Wang, R. P. Agarwal and D. O'Regan, Commutativity of quaternion-matrix-valued functions and quaternion matrix dynamic equations on time scales, Stud. Appl. Math., 2021, 146(1), 139–210. doi: 10.1111/sapm.12344

    CrossRef Google Scholar

    [40] Y. Li, X. Wang and N. Huo, Weyl almost automorphic solutions in distribution sense of Clifford-valued stochastic neural networks with time-varying delays, Proc. A, 2022, 478(2257), 20210719.

    Google Scholar

    [41] T. Luzyanina and K. Engelborghs, Computing Floquet multipliers for functional differential equations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2002, 12(12), 2977–2989. doi: 10.1142/S0218127402006291

    CrossRef Google Scholar

    [42] J. Mallet-Paret and G. R. Sell, Systems of differential delay equations: Floquet multipliers and discrete Lyapunov functions, J. Differential Equations, 1996, 125(2), 385–440. doi: 10.1006/jdeq.1996.0036

    CrossRef Google Scholar

    [43] V. G. Papanicolaou and D. Kravvaritis, The Floquet theory of the periodic Euler-Bernoulli equation, J. Differential Equations, 1998, 150(1), 24–41. doi: 10.1006/jdeq.1998.3474

    CrossRef Google Scholar

    [44] C. Pötzsche, Langsame Faserbündel dynamischer Gleichungen auf Maßketten, Logos, Berlin, 2002.

    Google Scholar

    [45] C. Pötzsche, Exponential dichotomies for dynamic equations on measure chains, Nonlinear Anal., 2001, 47(2), 873–884. doi: 10.1016/S0362-546X(01)00230-9

    CrossRef Google Scholar

    [46] C. Pötzsche, Topological decoupling, linearization and perturbation on inhomogeneous time scales, J. Differential Equation, 2008, 245(5), 1210–1242. doi: 10.1016/j.jde.2008.06.011

    CrossRef Google Scholar

    [47] C. Pötzsche, S. Siegmund and F. Wirth, A spectral characterization of exponential stability for linear time-invariant systems on time scales, Discrete Contin. Dyn. Syst., 2003, 9(5), 1223–1241. doi: 10.3934/dcds.2003.9.1223

    CrossRef Google Scholar

    [48] J. Shi, On stability of two order linear differential equation with periodic coefficient, Acta Math. Sci. Ser. A, 2000 20(1), 130–139.

    Google Scholar

    [49] J. Shi, M. Lin and J. Chen, The calculation for characteristic multiplier of Hill's equation, Appl. Math. Comput., 2004, 159(1), 57–77.

    Google Scholar

    [50] S. Siegmund, A spectral notion for dynamic equations on time scales, J. Comput. Appl. Math., 2002, 141(1–2), 255–265.

    Google Scholar

    [51] C. Simmendinger, A. Wunderlin and A. Pelster, Analytical approach for the Floquet theory of delay differential equations, Phys. Rev. E, 1999, 59(5), 5344–5353. doi: 10.1103/PhysRevE.59.5344

    CrossRef Google Scholar

    [52] A. L. Skubachevskii and H. O. Walther, On the Floquet multipliers of periodic solutions to non-linear functional differential equations, J. Dynam. Differential Equations, 2006 18(2), 257–355. doi: 10.1007/s10884-006-9006-5

    CrossRef Google Scholar

    [53] Y. V. Teplinskiĭ and A. Y. Teplinskiĭ, On the Erugin and Floquet-Lyapunov theorems for countable systems of difference equations, Ukrainian Math. J., 1996, 48(2), 314–321. doi: 10.1007/BF02372054

    CrossRef Google Scholar

    [54] K. O. Val'ter and A. L. Skubachevskii, On Floquet multipliers for slowly oscillating periodic solutions of nonlinear functional differential equations, Tr. Mosk. Mat. Obs., 2003, 64, 3–53.

    Google Scholar

    [55] H. O. Walther, Hyperbolic periodic solutions, heteroclinic connections and transversal homoclinic points in autonomous differential delay equations, Mem. Amer. Math. Soc., 1989, 79(402). DOI: 10.1090/memo/0402.

    CrossRef Google Scholar

    [56] C. Wang, R. P. Agarwal and D. O'Regan, Calculus of fuzzy vector-valued functions and almost periodic fuzzy vector-valued functions on time scales, Fuzzy Sets and Systems, 2019, 375, 1–52. doi: 10.1016/j.fss.2018.12.008

    CrossRef Google Scholar

    [57] C. Wang, R. P. Agarwal and D. O'Regan, Almost periodic fuzzy multidimensional dynamic systems and applications on time scales, Chaos Solitons Fractals, 2022, 156, 111781. doi: 10.1016/j.chaos.2021.111781

    CrossRef Google Scholar

    [58] C. Wang, R. P. Agarwal, D. O'Regan and R. Sakthivel, Theory of Translation Closedness for Time Scales, Springer-Verlag, Switzerland, 2020.

    Google Scholar

    [59] C. Wang and R. P. Agarwal, Almost automorphic functions on semigroups induced by complete-closed time scales and application to dynamic equations, Discrete Contin. Dyn. Syst. Ser. B, 2020, 25(2), 781–798.

    Google Scholar

    [60] C. Wang, Z. Li and R. P. Agarwal, Hyers-Ulam-Rassias stability of high-dimensional quaternion impulsive fuzzy dynamic equations on time scales, Discrete Contin. Dyn. Syst. Ser. S, 2022, 15(2), 359–386. doi: 10.3934/dcdss.2021041

    CrossRef Google Scholar

    [61] R. Weikard, Floquet theory for linear differential equations with meromorphic solutions, Electron. J. Qual. Theory Differ. Equ., 2000, 8, 1–6. DOI: 10.14232/ejqtde.2000.1.8.

    CrossRef Google Scholar

    [62] J. Zhang, M. Fan and H. Zhu, Existence and roughness of exponential dichotomies of linear dynamic equations on time scales, Comput. Math. Appl., 2010, 59(8), 2658–2675. doi: 10.1016/j.camwa.2010.01.035

    CrossRef Google Scholar

    [63] J. Zhang, M. Fan and H. Zhu, Necessary and sufficient criteria for the existence of exponential dichotomy on time scales, Comput. Math. Appl., 2010, 60(8), 2387–2398. doi: 10.1016/j.camwa.2010.08.034

    CrossRef Google Scholar

    • Available online Date: December 2022
    Supplements
    • Cited by

Tables(2)

Export Citation
PDF
XML

Article Metrics

Article views(2613) PDF downloads(614) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Published by Wilmington Scientific Publisher