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Attractivity for two-dimensional linear systems whose anti-diagonal coefficients are periodic

Authors: Jitsuro Sugie and Ayano Endo
Journal: Proc. Amer. Math. Soc. 137 (2009), 4117-4127
MSC (2000): Primary 34D05, 34D20; Secondary 37B25, 37C75
Published electronically: June 19, 2009
MathSciNet review: 2538573
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Abstract: This paper deals with the linear system $ \mathbf{x}' = A(t)\mathbf{x}$ with $ A(t)$ being a $ 2\times2$ matrix. The anti-diagonal components of $ A(t)$ are assumed to be periodic, but the diagonal components are not necessarily periodic. Our concern is to establish sufficient conditions for the zero solution to be attractive. Floquet theory is of no use in solving our problem, because not all components are periodic. Another approach is adopted. Some simple examples are included to illustrate the main result.

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  • 1. A. Bacciotti and L. Rosier, Liapunov Functions and Stability in Control Theory, 2nd ed., Springer-Verlag, Berlin-Heidelberg-New York, 2005. MR 2146587 (2005m:93001)
  • 2. F. Brauer and J. Nohel, The Qualitative Theory of Ordinary Differential Equations, W. A. Benjamin, New York and Amsterdam, 1969; (revised) Dover, New York, 1989.
  • 3. J. Cronin, Differential Equations: Introduction and Qualitative Theory, 2nd ed., Monographs and Textbooks in Pure and Applied Mathematics 180, Marcel Dekker, New York-Basel-Hong Kong, 1994. MR 1275827 (95b:34001)
  • 4. A. Halanay, Differential Equations: Stability, Oscillations, Time Lags, Academic Press, New York-London, 1966. MR 0216103 (35:6938)
  • 5. J. K. Hale, Ordinary Differential Equations, Wiley-Interscience, New York-London-Sydney, 1969; (revised) Krieger, Malabar, 1980. MR 0419901 (54:7918)
  • 6. L. Hatvani, On the asymptotic stability by nondecrescent Ljapunov function, Nonlinear Anal., 8 (1984), 67-77. MR 732416 (85k:34117)
  • 7. L. Hatvani, On the asymptotic stability for a two-dimensional linear nonautonomous differential system, Nonlinear Anal., 25 (1995), 991-1002. MR 1350721 (96k:34105)
  • 8. D. W. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An Introduction to Dynamical Systems, 3rd ed., Oxford Texts in Applied and Engineering Mathematics 2, Oxford University Press, Oxford, 1999. MR 1743361 (2000j:34001)
  • 9. J. LaSalle and S. Lefschetz, Stability by Liapunov's Direct Method, with Applications, Mathematics in Science and Engineering 4, Academic Press, New York-London, 1961. MR 0132876 (24:A2712)
  • 10. D. R. Merkin, Introduction to the Theory of Stability, Texts in Applied Mathematics 24, Springer-Verlag, New York-Berlin-Heidelberg, 1997. MR 1418401 (98f:34074)
  • 11. A. N. Michel, L. Hou and D. Liu, Stability of Dynamical Systems: Continuous, Discontinuous, and Discrete Systems, Birkhäuser, Boston-Basel-Berlin, 2008. MR 2351563 (2008i:93001)
  • 12. N. Rouche, P. Habets and M. Laloy, Stability Theory by Liapunov's Direct Method, Applied Mathematical Sciences 22, Springer-Verlag, New York-Heidelberg-Berlin, 1977. MR 0450715 (56:9008)
  • 13. J. Sugie, Convergence of solutions of time-varying linear systems with integrable forcing term, Bull. Austral. Math. Soc., 78 (2008), 445-462. MR 2472280
  • 14. J. Sugie, Influence of anti-diagonals on the asymptotic stability for linear differential systems, Manatsh. Math., 157 (2009), 163-176.
  • 15. J. Sugie and Y. Ogami, Asymptotic stability for three-dimensional linear differential systems with time-varying coefficients, to appear in Quart. Appl. Math.
  • 16. F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Springer-Verlag, New York-Berlin-Heidelberg, 1990. MR 1036522 (91b:34002)
  • 17. T. Yoshizawa, Note on the boundedness and the ultimate boundedness of solutions of $ x' = F(t,x)$, Mem. Coll. Sci. Univ. Kyoto Ser. A Math., 29 (1955), 275-291. MR 0098217 (20:4679)
  • 18. T. Yoshizawa, Stability Theory by Liapunov's Second Method, Math. Soc. Japan, Tokyo, 1966. MR 0208086 (34:7896)
  • 19. T. Yoshizawa, Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions, Applied Mathematical Sciences 14, Springer-Verlag, New York-Heidelberg-Berlin, 1975. MR 0466797 (57:6673)

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Additional Information

Jitsuro Sugie
Affiliation: Department of Mathematics and Computer Science, Shimane University, Matsue 690-8504, Japan

Ayano Endo
Affiliation: Department of Mathematics and Computer Science, Shimane University, Matsue 690-8504, Japan

Keywords: Attractivity, linear systems, weakly integrally positive, Floquet theory
Received by editor(s): March 17, 2009
Published electronically: June 19, 2009
Additional Notes: The first author was supported in part by a Grant-in-Aid for Scientific Research, No. 19540182, from the Japan Society for the Promotion of Science
Communicated by: Yingfei Yi
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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