Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • 1. Andrea Bacciotti and Lionel Rosier, Liapunov functions and stability in control theory, 2nd ed., Communications and Control Engineering Series, Springer-Verlag, Berlin, 2005. MR 2146587
  • 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. Jane Cronin, Differential equations, 2nd ed., Monographs and Textbooks in Pure and Applied Mathematics, vol. 180, Marcel Dekker, Inc., New York, 1994. Introduction and qualitative theory. MR 1275827
  • 4. A. Halanay, Differential equations: Stability, oscillations, time lags, Academic Press, New York-London, 1966. MR 0216103
  • 5. Jack K. Hale, Ordinary differential equations, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1969. Pure and Applied Mathematics, Vol. XXI. MR 0419901
  • 6. L. Hatvani, On the asymptotic stability by nondecrescent Ljapunov function, Nonlinear Anal. 8 (1984), no. 1, 67–77. MR 732416,
  • 7. L. Hatvani, On the asymptotic stability for a two-dimensional linear nonautonomous differential system, Nonlinear Anal. 25 (1995), no. 9-10, 991–1002. MR 1350721,
  • 8. D. W. Jordan and P. Smith, Nonlinear ordinary differential equations, 3rd ed., Oxford Texts in Applied and Engineering Mathematics, vol. 2, Oxford University Press, Oxford, 1999. An introduction to dynamical systems. MR 1743361
  • 9. Joseph LaSalle and Solomon Lefschetz, Stability by Liapunov’s direct method, with applications, Mathematics in Science and Engineering, Vol. 4, Academic Press, New York-London, 1961. MR 0132876
  • 10. David R. Merkin, Introduction to the theory of stability, Texts in Applied Mathematics, vol. 24, Springer-Verlag, New York, 1997. Translated from the third (1987) Russian edition, edited and with an introduction by Fred F. Afagh and Andrei L. Smirnov. MR 1418401
  • 11. Anthony N. Michel, Ling Hou, and Derong Liu, Stability of dynamical systems, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 2008. Continuous, discontinuous, and discrete systems. MR 2351563
  • 12. Nicolas Rouche, P. Habets, and M. Laloy, Stability theory by Liapunov’s direct method, Springer-Verlag, New York-Heidelberg, 1977. Applied Mathematical Sciences, Vol. 22. MR 0450715
  • 13. Jitsuro Sugie, Convergence of solutions of time-varying linear systems with integrable forcing term, Bull. Aust. Math. Soc. 78 (2008), no. 3, 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. Ferdinand Verhulst, Nonlinear differential equations and dynamical systems, Universitext, Springer-Verlag, Berlin, 1990. Translated from the Dutch. MR 1036522
  • 17. Taro 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,
  • 18. Taro Yoshizawa, Stability theory by Liapunov’s second method, Publications of the Mathematical Society of Japan, No. 9, The Mathematical Society of Japan, Tokyo, 1966. MR 0208086
  • 19. T. Yoshizawa, Stability theory and the existence of periodic solutions and almost periodic solutions, Springer-Verlag, New York-Heidelberg, 1975. Applied Mathematical Sciences, Vol. 14. MR 0466797

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 34D05, 34D20, 37B25, 37C75

Retrieve articles in all journals with MSC (2000): 34D05, 34D20, 37B25, 37C75

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.

American Mathematical Society