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
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Abstract: A forced system described by the differential equation: \[ x” + \epsilon f\left ( {x,x’} \right ) + \omega _0^2x = F\cos \omega t\] 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.
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)
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S. A. Musa, Synchronized oscillations in driven nonlinear systems, Ph.D. Dissertation, Harvard University, Cambridge, Mass., 1965
- Richard E. Kronauer and Samuel A. Musa, Necessary conditions for subharmonic and superharmonic synchronization in weakly nonlinear systems, Quart. Appl. Math. 24 (1966), 153–160. MR 203183, DOI https://doi.org/10.1090/S0033-569X-1966-0203183-5
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)
C. Hayashi, Nonlinear oscillations in physical systems, McGraw-Hill, New York, 1964
N. Minorsky, Nonlinear oscillations, D. Van Nostrand Co., Princeton, N. J., 1962
J. Hale, Oscillations in nonlinear systems, McGraw-Hill, New York, 1963
R. Struble and J. Fletcher, General perturbational solution of the harmonically forced Van der Pol equation, J. Math. Phys. 2, 880–891 (1961)
R. Struble and S. Yionoulis, General perturbational solution of the harmonically forced Duffing equation, Arch. Rational Mech. Anal. 422–438 (1962)
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
S. A. Musa, Synchronized oscillations in driven nonlinear systems, Ph.D. Dissertation, Harvard University, Cambridge, Mass., 1965
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)
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Article copyright:
© Copyright 1968
American Mathematical Society