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Perturbations near zero of the leading coefficient of solutions to a nonlinear differential equation

Author: Vadim Komkov
Journal: Proc. Amer. Math. Soc. 99 (1987), 93-104
MSC: Primary 34C15; Secondary 03H05, 26E35, 58F05
MathSciNet review: 866436
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Abstract: The equation

$\displaystyle \frac{d}{{dt}}\left( {a\left( t \right) \cdot \psi \left( x \righ... ...eft( x \right) \cdot \frac{{dx}}{{dt}} + c\left( t \right)x = g\left( t \right)$

that generalizes the more commonly studied selfadjoint second order equation

$\displaystyle \frac{d}{{dt}}\left( {a\left( t \right)\frac{{dx}}{{dt}}} \right) + c\left( t \right)x = g\left( t \right)$

occurs in celestial dynamics, in the study of gyroscopic systems, and in other problems of nonlinear mechanics. Using techniques of nonstandard analysis we derive some properties of the "duck"-type cycles for the solutions of this equation.

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

  • [1] H. Poincaré, Oeuvres de Henri Poincaré, vol. VII, Gauthier-Villars, Paris, 1952. MR 0046985 (13:810s)
  • [2] V. Komkov, Continuability and estimates of solutions of $ \left( {a\left( t \right) \cdot \psi \left( x \right)x'} \right) + c\left( t \right) \cdot f\left( t \right) = 0$, Ann. Polon. Math. 30 (1974), 125-137. MR 0355152 (50:7629)
  • [3] -, Asymptotic behaviour of non-linear differential equations via non-standard analysis. Part III, Boundedness and monotone behaviour of the equation $ \left( {a\left( t \right) \cdot \psi \left( x \right)x'} \right) + c\left( t \right)f\left( x \right) = q\left( t \right)$, Ann. Polon. Math. 38 (1980), 101-108. MR 599234 (82d:34044)
  • [4] A. Robinson, Introduction to model theory and to metamathematics of algebra, North-Holland, Amsterdam, 1966. MR 0153570 (27:3533)
  • [5] -, Non-standard analysis, North-Holland, Amsterdam, 1962.
  • [6] W. A. J. Luxemberg, What is non-standard analysis, Papers in the Foundations of Mathematics, Math. Assoc. Amer., Slaught Memorial Papers No. 13, 1966, pp. 38-67.
  • [7] P. Cartier, Singular perturbations of ordinary differential equations and non-standard analysis, Uspehi Mat. Nauk. 39 (1984), 57-76. MR 740000 (85j:34114)
  • [8] D. P. Merkin, Gyroscopic motion, Nauka, Moscow, 1974.
  • [9] M. Diener, Etude générique des canards, Thèse, Strasbourg, 1981.
  • [10] Jane Cronin-Scanlon, Entrainment of frequency in singularly perturbed systems, Abstract Amer. Math. Soc. 6 (1985), Abstract 817-34-148.
  • [11] A. L. Hodgkin and A. F. Huxley, A qualitative description of membrane current ant its application to conduction and excitation in nerves, J. Physiol. 171 (1952), 500-544.

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Article copyright: © Copyright 1987 American Mathematical Society

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