Bifurcation of periodic solutions in a nonlinear difference-differential equations of neutral type
Author:
Robert K. Brayton
Journal:
Quart. Appl. Math. 24 (1966), 215-224
MSC:
Primary 34.75; Secondary 34.45
DOI:
https://doi.org/10.1090/qam/204800
MathSciNet review:
204800
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Abstract: The existence of a self-sustained periodic solution in the autonomous equation \[ u’\left ( \tau \right ) - \alpha u’\left ( {\tau - h} \right ) + \beta u\left ( \tau \right ) + \alpha \gamma u\left ( {\tau - h} \right ) = \epsilon f\left ( {u\left ( \tau \right )} \right )\] is proved under appropriate assumptions on $\alpha ,\beta ,\gamma ,f$ and $h$. The method of proof consists in converting this equation into an equivalent nonlinear integral equation and demonstrating the convergence of an appropriate iteration scheme.
- Richard Bellman and Kenneth L. Cooke, Differential-difference equations, Academic Press, New York-London, 1963. MR 0147745
- N. N. Krasovsky, On periodical solutions of differential equations involving a time lag, Dokl. Akad. Nauk SSSR (N.S.) 114 (1957), 252–255 (Russian). MR 0090733
- S. N. Šimanov, Almost periodic oscillations in nonlinear systems with retardation., Dokl. Akad. Nauk SSSR 125 (1959), 1203–1206 (Russian). MR 0106317
- S. N. Šimanov, On the vibration theory of quasilinear systems with lag, J. Appl. Math. Mech. 23 (1959), 1198–1208 (Russian). MR 0112293, DOI https://doi.org/10.1016/0021-8928%2859%2990124-8
- Jack K. Hale, Linear functional-differential equations with constant coefficients, Contributions to Differential Equations 2 (1963), 291–317 (1963). MR 152725
R. K. Brayton, Nonlinear oscillations in a distributed network. To appear in a forthcoming issue of Quart, of Applied Math.
R. Bellman and K. L. Cooke, Differential-difference equations, Academic Press, New York, 1963.
N. Krasovskii, On periodic solutions of differential equations involving a time lag, Dokl. Acad. Nauk (N. S.), 114 252-255 (1957).
N. Shimanov, Almost periodic solutions in nonlinear systems with retardation, Dokl. Acad. Nauk, S. S. S. R., 125 1203-1206 (1959).
N. Shimanov, On the vibration theory of quasilinear systems with lags, Prikl. Mat. Meh. 23 836-844 (1959); TPMM, 1198-1208.
J. K. Hale, Linear functional-differential equations with constant coefficients, Contributions to Differential Equations (2), 291-317 (1963).
R. K. Brayton, Nonlinear oscillations in a distributed network. To appear in a forthcoming issue of Quart, of Applied Math.
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Article copyright:
© Copyright 1966
American Mathematical Society