Abstract
Sufficient conditions are obtained for uniform stability and asymptotic stability of the zero solution of two-dimensional quasi-linear systems under the assumption that the zero solution of linear approximation is not always uniformly attractive. A class of quasi-linear systems considered in this paper includes a planar system equivalent to the damped pendulum x′′ + h(t)x′ + sin x = 0, where h(t) is permitted to change sign. Some suitable examples are included to illustrate the main results.
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Bellman R.: Stability Theory of Differential Equations. McGraw-Hill, New York, Toronto, London (1953) Dover, Mineola, New York (2008)
Brauer F., Nohel J.A.: The Qualitative Theory of Ordinary Differential Equations. Benjamin, New York (1969) Dover, New York (1989)
Cesari L.: Asymptotic Behavior and Stability Problems in Ordinary Differential Equations. 3rd edn. Springer-Verlag, New York, Heidelberg (1971)
Coddington E.A., Levinson N.: The Theory of Ordinary Differential Equations. McGraw-Hill, New York, Toronto, London (1955) Krieger, Malabar (1984)
Coppel W.A.: Stability and Asymptotic Behavior of Differential Equations. Heath, Boston (1965)
Güvenilir A.F., Çelebi A.O.: A note on asymptotic stability of a class of functional differential equations. Indian J. Math. 42, 37–41 (2000)
Haddock J.R.: A remark on a stability theorem of M. Marachkoff. Proc. Amer. Math. Soc. 31, 209–212 (1972)
Halanay A.: Differential Equations: Stability, Oscillations, Time Lags. Academic Press, New York, London (1966)
Hale J.K.: Ordinary differential equations. Wiley-Interscience, New York, London, Sydney (1969) Krieger, Malabar (1980)
Hatvani, L.: A generalization of the Barbashin-Krasovskij theorems to the partial stability in nonautonomous systems. Qualitative Theory of Differential Equations I. (ed. M. Farkas), Colloq. Math. Soc. János Bolyai, vol. 30, pp. 381–409. North-Holland, Amsterdam and New York (1981)
Hatvani L.: On partial asymptotic stability and instability. Acta Sci. Math. (Szeged) 49, 157–167 (1985)
Hatvani L.: On the asymptotic stability for a two-dimensional linear nonautonomous differential system. Nonlinear Anal. 25, 991–1002 (1995)
Hatvani L., Totik V.: Asymptotic stability of the equilibrium of the damped oscillator. Diff. Integral Eqns. 6, 835–848 (1993)
Ko Y.: An asymptotic stability and a uniform asymptotic stability for functional-differential equations. Proc. Amer. Math. Soc. 119, 535–545 (1993)
LaSalle J.P., Lefschetz S.: Stability by Liapunov’s Direct Method with Applications. Mathematics in Science and Engineering 4. Academic Press, New York, London (1961)
Lefschetz S.: Differential Equations: Geometric Theory, 2nd ed. Interscience, New York (1963) Dover, New York (1977)
Maratschkow M.: Über einen Liapounoffschen Satz. Bull. Soc. Phys.-Math. Kazan 12(3), 171–174 (1940)
Massera J.L.: On Liapounoff’s conditions of stability. Ann. of Math. 50(2), 705–721 (1949)
Matrosov V.M.: On the stability of motion. Prikl. Mat. Meh. 26, 885–895 (1962) translated as J. Appl. Math. Mech. 26, 1337–1353 (1963)
Merkin D.R.: Introduction to the Theory of Stability. Texts in Applied Mathematics 24. Springer-Verlag, New York, Berlin, Heidelberg (1997)
Perron O.: Die Stabilitätsfrage bei Differentialgleichungen. Math. Zeits. 32, 703–728 (1930)
Persidski K.P.: Über die Stabilität einer Bewegung nach der ersten Näherung. Mat. Sb. 40, 284–293 (1933)
Rouche N., Habets P., Laloy M.: Stability Theory by Liapunov’s Direct Method. Applied Mathematical Sciences 22. Springer-Verlag, New York, Heidelberg, Berlin (1977)
Sansone G., Conti R.: Non-linear Differential Equations. Macmillan, New York (1964)
Sugie J., Onitsuka M.: Global asymptotic stability for half-linear differential equations with coefficients of indefinite sign. Arch. Math. (Brno) 44, 317–334 (2008)
Sugie J., Hata S., Onitsuka M.: Global attractivity for half-linear differential systems with periodic coefficients. J. Math. Anal. Appl. 371, 95–112 (2010)
Verhulst F.: Nonlinear Differential Equations and Dynamical Systems. Springer-Verlag, Berlin, Heidelberg (1990)
Wang T.-X.: Asymptotic stability and the derivatives of solutions of functional-differential equations. Rocky Mountain J. Math. 24, 403–427 (1994)
Wilson H.K.: Ordinary Differential Equations. Introductory and Intermediate Courses Using Matrix Methods. Addison-Wesley, Massachusetts, California, London, Ontario (1971)
Yoshizawa T.: Stability Theory by Liapunov’s Second Method. Math. Soc., Japan, Tokyo (1966)
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Sugie, J., Ogami, Y. & Onitsuka, M. Asymptotic stability for quasi-linear systems whose linear approximation is not assumed to be uniformly attractive. Annali di Matematica 190, 409–425 (2011). https://doi.org/10.1007/s10231-010-0156-z
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DOI: https://doi.org/10.1007/s10231-010-0156-z
Keywords
- Asymptotic stability
- Uniform stability
- Quasi-linear systems
- Weakly integrally positive
- Discontinuous coefficients