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- W2497892639 abstract "We analyze perturbation solutions to some problems in nonlinear oscillations. The method we use involves a numerical extension of the perturbation solution, followed by an analysis of the analytical structure of the solution as a function of the perturbation parameter. As a first example, we study the limit cycle u(t,(epsilon)) of the free van der Pol equation, using the Taylor series expansion about (epsilon) = 0 calculated by Andersen and Geer {SIAM J. Appl. Math., 1982}. Using Pade approximants, three pairs of complex-conjugate branch point singularities in the (epsilon)-plane are recognized. Two pairs apparently remain fixed, while the third pair moves towards the origin and the imaginary axis as t increases from 0 to (pi)/2. For (pi)/4 (LESSTHEQ) t (LESSTHEQ) (pi)/2, this third singularity becomes closest to the origin in complex (epsilon)-plane. We then convert the series into one involving a different expansion parameter. The perturbation solution computed from the new revised series converges for much larger values of (epsilon) than the original series, which agrees well with some results of previous investigations.The second problem we study is the forced non-linear motion of a simple pendulum. We study three cases corresponding to the driving frequency being either greater than, equal to, or less than the natural frequency. There appears to be only one type of singularity which lies close to the imaginary axis and remains fixed for all values of independent variable t. The limit cycles and ratio and root tests indicate that the perturbation series converges only for small values of (epsilon). A new revised series is used to approximate the periodic solution for all values of (epsilon).Finally, we study nonresonant excitations of van der Pol's equation subjected to a periodic external force. Three pairs of singularities in the complex (epsilon)-plane are found for this type of motion. As the forcing amplitude increases, two pairs move away from the origin while approaching the real axis and eventually disappear. The third pair of singularities appear in the complex (epsilon)-plane as the amplitude increases and then moves towards the origin along the imaginary axis. The relationship between these singularities and the quenching of free oscillations is discussed." @default.
- W2497892639 created "2016-08-23" @default.
- W2497892639 creator A5041225000 @default.
- W2497892639 date "1982-01-01" @default.
- W2497892639 modified "2023-09-24" @default.
- W2497892639 title "Perturbation analysis of nonlinear oscillations using symbolic and numerical computations" @default.
- W2497892639 hasPublicationYear "1982" @default.
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