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Transactions of the American Mathematical Society

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Nonstandard analysis of dynamical systems. I. Limit motions, stability.

Author: A. E. Hurd
Journal: Trans. Amer. Math. Soc. 160 (1971), 1-26
MSC: Primary 34.65
MathSciNet review: 0280827
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Abstract: The methods of nonstandard analysis are applied to the study of the qualitative theory of dynamical systems. The nonstandard notions connected with limiting behavior of motions (limit sets, etc.) are developed, and then applied to the study of stability theory, including stability of sets, attracting properties, first prolongations and stability of motions.

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  • [1] N. P. Bhatia and G. P. Szegő, Dynamical systems: Stability theory and applications, Lecture Notes in Mathematics, No. 35, Springer-Verlag, Berlin-New York, 1967. MR 0219843
  • [2] Applications of model theory to algebra, analysis, and probability., Edited by W. A. J. Luxemburg, Holt, Rinehart and Winston, New York-Montreal, Que.-London, 1969. MR 0234829
  • [3] -, A general theory of monads, in [2].
  • [4] -, Non-standard analysis, California Institute of Technology, Pasadena, Calif., 1964.
  • [5] Moshé Machover and Joram Hirschfeld, Lectures on non-standard analysis, Lecture Notes in Mathematics, Vol. 94, Springer-Verlag, Berlin-New York, 1969. MR 0249285
  • [6] V. V. Nemytskii and V. V. Stepanov, Qualitative theory of differential equations, Princeton Mathematical Series, No. 22, Princeton University Press, Princeton, N.J., 1960. MR 0121520
  • [7] Abraham Robinson, Non-standard analysis, Nederl. Akad. Wetensch. Proc. Ser. A 64 = Indag. Math. 23 (1961), 432–440. MR 0142464
  • [8] Abraham Robinson, Non-standard analysis, North-Holland Publishing Co., Amsterdam, 1966. MR 0205854
  • [9] Taro Ura, Sur les courbes définies par les équations différentielles dans l’espace à 𝑚 dimensions, Ann. Sci. Ecole Norm. Sup. (3) 70 (1953), 287–360 (French). MR 0064227
  • [10] Taro Ura, On the flow outside a closed invariant set; stability, relative stability and saddle sets, Contributions to Differential Equations 3 (1964), 249–294. MR 0163018
  • [11] V. I. Zubov, Methods of A. M. Lyapunov and their application, Translation prepared under the auspices of the United States Atomic Energy Commission; edited by Leo F. Boron, P. Noordhoff Ltd, Groningen, 1964. MR 0179428

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Keywords: Dynamical systems, limit motions, stability, nonstandard analysis
Article copyright: © Copyright 1971 American Mathematical Society

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