Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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



Sub- and superharmonic synchronization in weakly nonlinear systems: Integral constraints and duality

Authors: Samuel A. Musa and Richard E. Kronauer
Journal: Quart. Appl. Math. 25 (1968), 399-414
DOI: https://doi.org/10.1090/qam/99873
MathSciNet review: QAM99873
Full-text PDF Free Access

Abstract | References | Additional Information

Abstract: A forced system described by the differential equation:

$\displaystyle x'' + \epsilon f\left( {x,x'} \right) + \omega _0^2x = F\cos \omega t$

is considered for cases where $ \left( {n/m} \right)\omega $ is close to $ {\omega _0}$ . (Here $ m$ and $ n$ are integers and $ n/m > 1$ denotes superharmonic while $ n/m < 1$ denotes subharmonics.) If the unforced system $ \left( {F = 0} \right)$ is conservative, the forced system is shown to possess an integral constraint and the solution is reduced to quadratures, even though the force adds or removes energy from the oscillations. Furthermore, the sub- and superharmonic cases where the $ n/m$ ratios are inverse are shown to be intimately related, and results for one can be deduced from the other by appropriate interchange of variables.

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

  • [1] R. E. Kronauer and S. A. Musa, The exchange of energy between oscillations in weakly-nonlinear conservative systems, J. Appl. Mech., ASME, 33, 2, 451-452 (1966)
  • [2] C. Hayashi, Nonlinear oscillations in physical systems, McGraw-Hill, New York, 1964 MR 0170071
  • [3] N. Minorsky, Nonlinear oscillations, D. Van Nostrand Co., Princeton, N. J., 1962 MR 0137891
  • [4] J. Hale, Oscillations in nonlinear systems, McGraw-Hill, New York, 1963 MR 0150402
  • [5] R. Struble and J. Fletcher, General perturbational solution of the harmonically forced Van der Pol equation, J. Math. Phys. 2, 880-891 (1961) MR 0136824
  • [6] R. Struble and S. Yionoulis, General perturbational solution of the harmonically forced Duffing equation, Arch. Rational Mech. Anal. 422-438 (1962) MR 0136813
  • [7] J. Cole and J. Kevorkian, Uniformly valid asymptotic approximations for certain nonlinear differential equations, pp. 113-120, International Symposium on Nonlinear Differential Equations and Nonlinear Mechanics, Colorado Springs; Academic Press, New York, 1963 MR 0147701
  • [8] S. A. Musa, Synchronized oscillations in driven nonlinear systems, Ph.D. Dissertation, Harvard University, Cambridge, Mass., 1965
  • [9] R. E. Kronauer and S. A. Musa, Necessary conditions for subharmonic and superharmonic oscillations in weakly-nonlinear systems, Quart. Appl. Math. 24, 2, 153-160 (1966) MR 0203183

Additional Information

DOI: https://doi.org/10.1090/qam/99873
Article copyright: © Copyright 1968 American Mathematical Society

American Mathematical Society