Algebraic-numerical method for the slightly perturbed harmonic oscillator

Authors:
A. Nadeau, J. Guyard and M. R. Feix

Journal:
Math. Comp. **28** (1974), 1057-1066

MSC:
Primary 65L05

MathSciNet review:
0349020

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Abstract: The solution of slightly perturbed harmonic oscillators can easily be obtained in the form of a series given by Poisson's method. However, this perturbation method leads to secular terms unbounded for large time (the time unit being the fundamental period of the harmonic oscillator), which prevent the use of finite series. The analytical elimination of such terms was first solved by Poincaré and, more recently, generalized by Krylov and Bogoliubov. Unfortunately, these methods are very difficult to handle and are not easily carried out for high orders.

A numerical reinitialization method is combined here with the Poisson perturbation treatment to avoid the growth of secular terms and therefore to get the solution at any time. The advantages of such a method is that the analytical work can be carried to high orders keeping the step of numerical integration to a relatively large value (compared to a purely numerical method). This algorithm has been tested on the Mathieu equation. A method for the computation of the eigenvalues of this equation is given. By properly selecting the order of the perturbation and the time step of reinitialization, we can recover, at any order, all the effects of the slight perturbation (including *all* the unstable zones).

Consequently, such a method is a useful intermediate between purely analytical and purely numerical algorithms.

**[1]**N. N. Bogoliubov and Y. A. Mitropolsky,*Asymptotic methods in the theory of non-linear oscillations*, Translated from the second revised Russian edition. International Monographs on Advanced Mathematics and Physics, Hindustan Publishing Corp., Delhi, Gordon and Breach Science Publishers, New York, 1961. MR**0141845****[2]***The plasma in a magnetic field: A symposium on magnetohydrodynamics.*, Stanford University Press, Stanford, Calif., 1958. MR**0115576****[3]**H. R. LEWIS, Jr., "Class of exact invariants for classical and quantum time-dependent harmonic oscillators,"*J. Mathematical Phys.*, v. 9, 1968, p. 1976.**[4]**Jacques Guyard, André Nadeau, Germain Baumann, and Marc R. Feix,*Eigenvalues of the Hill equation to any order in the adiabatic limit.*, J. Mathematical Phys.**12**(1971), 488–492. MR**0280104**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1974-0349020-9

Keywords:
Nonlinear harmonic oscillator,
Poisson perturbation method,
secular terms,
Krylov-Bogoliubov method,
Mathieu equation

Article copyright:
© Copyright 1974
American Mathematical Society