Hostname: page-component-5cf477f64f-bmd9x Total loading time: 0 Render date: 2025-04-04T07:28:10.667Z Has data issue: false hasContentIssue false
  • English
  • Français

On Linear Perturbation of Non-Linear Differential Equations

Published online by Cambridge University Press:  20 November 2018

F. V. Atkinson
F. V. Atkinson*
Affiliation:
University College, Ibadan, Nigeria
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In the theory of the asymptotic solution or stability of ordinary differential equations most attention has been given to linear or nearly-linear cases. Investigations in this field, starting primarily with those of Kneser (7) on the equation y″ + f(x)y = 0, have by now mostly been summed up in results on the vector-matrix system dy/dx = Ay + f (y, x), where y and f denote n-vectors of functions, and A an n-by-n matrix, frequently assumed constant.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 6 , 1954 , pp. 561 - 571
Copyright
Copyright © Canadian Mathematical Society 1954

References

1. Ascoli, G., Sopra un caso di stabilit per Vequazione y″ + A(x)y = 0, Ann. Mat. Pur. Appl. (4) 26 (1947), 199–206.Google Scholar
2. Atkinson, F. V., On second-order linear oscillators, Revista (Série A) Mat. y Fis. Teor. (Tucumán), 8 (1951), 71–87.Google Scholar
3. Atkinson, F. V., On second-order non-linear oscillations (to appear in Pacific J. Math.).Google Scholar
4. Demidovič, B. P., On the stability in Lyapunov's sense of a system of ordinary differential equations, Mat. Sbornik (N.S.), 28(70) (1951), 659–684.Google Scholar
5. Fowler, R. H., On Emderi's and similar differential equations, Quart. J. Math. (Oxford), 2 (1931), 259–288.Google Scholar
6. Grobman, D. M., On the characteristic indices of systems near to linear ones, Doklady Akad. Nauk SSSR, 75 (1950), 157–160.Google Scholar
7. Kneser, A., Untersuchung und asymptotische Darstellung der Intégrale gewisser linearer Differentialgleichungen bei grossen Werthen des Arguments. I, Jour, für Math., 116 (1896), 178–212.Google Scholar
8. Levinson, N., The asymptotic behaviour of a system of linear differential equations, Amer. J. Math., 68 (1946), 1–6.Google Scholar
9. Levinson, N., The asymptotic nature of solutions of linear systems of differential equations, Duke Math. J., 15 (1948), 111–126.Google Scholar
10. Weyl, H., Comment on the preceding paper, Amer. J. Math., 68 (1946), 7–12.Google Scholar
11. Wintner, A., Asymptotic integrations of the adiabatic oscillator, Amer. J. Math., 69 (1947), 251–272.Google Scholar
12. Yakubovič, V. A., On the asymptotic behaviour of a system of differential equations, Mat. Sbornik (N.S.), 28(70) (1951), 217–240.Google Scholar