Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Nonlinear oscillations in a distributed network

Author: Robert K. Brayton
Journal: Quart. Appl. Math. 24 (1967), 289-301
DOI: https://doi.org/10.1090/qam/99914
MathSciNet review: QAM99914
Full-text PDF Free Access

Abstract | References | Additional Information

Abstract: The oscillations of small amplitude in a lossless transmission line terminated with a nonlinear circuit are studied by perturbation theory. The equations describing this system are reduced to a difference-differential equation with one delay. A general procedure is given for equations of this type for finding the expansion of the oscillation to any order in terms of the coefficient of the fundamental frequency. The frequency-amplitude relations are obtained to second order and compared with results found on the computer. Both the autonomous and forced cases are studied. It is indicated in the forced case that the frequency-amplitude relation gives approximately the range of ``locking in".

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

  • [1] N. N. Bogoliubov and Y. A. Mitropolsky, Asymptotic methods in the theory of nonlinear oscillations, Hindustan Publishing Corporation, Delhi, India, 1961 MR 0141845
  • [2] J. J. Stoker, Nonlinear vibrations, Interscience, New York, 1950 MR 0034932
  • [3] J. K. Hale, Oscillations in nonlinear systems, McGraw-Hill, New York, 1963 MR 0150402
  • [4] R. Bellman and K. L. Cooke, Differential-difference equations, Academic Press, New York, 1963 MR 0147745
  • [5] L. S. Pontryagin, On the zeros of some elementary Transcedental functions, Amer. Math. Soc. Transl., (2), 1, 95-110 (1955) MR 0073686
  • [6] R. K. Brayton, Bifurcation of periodic solutions in a nonlinear difference-differential equation of neutral type, Quart. Appl. Math. 24, No. 3, 215-224 (1966) MR 0204800
  • [7] A. Stokes, A Floquet theory for functional differential equations, Proc. Nat. Acad. Sci. (8) 48, 1330-1334 (1962) MR 0141858
  • [8] W. Hahn, On difference differential equations with periodic coefficients, J. Math. Anal. Appl. 3, 70-101 (1961) MR 0155063
  • [9] W. L. Miranker, Existence, uniqueness, and stability of solutions of systems of nonlinear difference-differential equations, J. Math. Mech. 11, 101-108 (1962) MR 0140787

Additional Information

DOI: https://doi.org/10.1090/qam/99914
Article copyright: © Copyright 1967 American Mathematical Society

American Mathematical Society