Abstract
For problems that can be described as small perturbations of linear oscillators, the usual regular perturbation expansions can be shown to have limited validity. Difficulties arise from resonant forcing terms interacting with the expansions to produce misleading secular growth in the solutions at moderate to large times. We describe two methods, called the Poincare–Lindstedt method and the method of multiple timescales, based on suppressing resonant terms in the perturbed system. Both methods capture the slowly varying cumulative effects of the perturbations, yielding solutions that are valid for longer times.
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Notes
- 1.
Other problems, and additional timescales needed for higher order expansions could involve higher-order (slower) timescales such as \(\tau _k=\varepsilon ^k t\) for \(k=2,3,\ldots \).
- 2.
This is the simplified version for symmetric matrices and self-adjoint differential equations. The general version of the Fredholm alternative is similar:
The solution of the non-homogeneous problem \(\mathbf{{A}}_0 \mathbf{{x}}=\mathbf{{b}}\) will be unique if and only if the adjoint problem \(\mathbf{{A}}^\dagger _0 \mathbf{{y}}=\mathbf{{0}}\) has only the trivial solution [39, 93].
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Witelski, T., Bowen, M. (2015). Weakly-Nonlinear Oscillators. In: Methods of Mathematical Modelling. Springer Undergraduate Mathematics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-23042-9_9
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DOI: https://doi.org/10.1007/978-3-319-23042-9_9
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Publisher Name: Springer, Cham
Print ISBN: 978-3-319-23041-2
Online ISBN: 978-3-319-23042-9
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