Perturbations near zero of the leading coefficient of solutions to a nonlinear differential equation
HTML articles powered by AMS MathViewer
- by Vadim Komkov PDF
- Proc. Amer. Math. Soc. 99 (1987), 93-104 Request permission
Abstract:
The equation \[ \frac {d}{{dt}}\left ( {a\left ( t \right ) \cdot \psi \left ( x \right )\frac {{dx}}{{dt}}} \right ) + f\left ( x \right ) \cdot \frac {{dx}}{{dt}} + c\left ( t \right )x = g\left ( t \right )\] that generalizes the more commonly studied selfadjoint second order equation \[ \frac {d}{{dt}}\left ( {a\left ( t \right )\frac {{dx}}{{dt}}} \right ) + c\left ( t \right )x = g\left ( t \right )\] 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
- Oeuvres de Henri Poincaré, Gauthier-Villars, Paris, 1952 (French). Publié avec la collaboration de Jacques Lévy. Publiées sous les auspices de l’Académie des Sciences par la Section de Géométrie. Tome VII. MR 0046985
- Vadim Komkov, Continuability and estimates of solutions of $(a(t)\psi (x)x^{\prime } )^{\prime } +c(t)f(x)=0$, Ann. Polon. Math. 30 (1974), 125–137. MR 355152, DOI 10.4064/ap-30-2-125-137
- V. Komkov, Asymptotic behavior of nonlinear differential equations via nonstandard analysis. III. Boundedness and monotone behavior of the equation $(a(t)\varphi (x)x^{\prime } )^{\prime } +c(t)f(x)=q(t)$, Ann. Polon. Math. 38 (1980), no. 2, 101–108. MR 599234, DOI 10.4064/ap-38-2-101-108
- Abraham Robinson, Introduction to model theory and to the metamathematics of algebra, North-Holland Publishing Co., Amsterdam, 1963. MR 0153570 —, Non-standard analysis, North-Holland, Amsterdam, 1962. 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.
- P. Kart′e, Singular perturbations of ordinary differential equations and nonstandard analysis, Uspekhi Mat. Nauk 39 (1984), no. 2(236), 57–76 (Russian). Translated from the French by A. K. Zvonkin. MR 740000 D. P. Merkin, Gyroscopic motion, Nauka, Moscow, 1974. M. Diener, Etude générique des canards, Thèse, Strasbourg, 1981. Jane Cronin-Scanlon, Entrainment of frequency in singularly perturbed systems, Abstract Amer. Math. Soc. 6 (1985), Abstract 817-34-148. 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.
Additional Information
- © Copyright 1987 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 99 (1987), 93-104
- MSC: Primary 34C15; Secondary 03H05, 26E35, 58F05
- DOI: https://doi.org/10.1090/S0002-9939-1987-0866436-3
- MathSciNet review: 866436