On the independence of the slow and fast scales in multiple-scale expansions, with application to Van der Pol's equation
Abstract
When implementing the method of multiple scales, one is traditionally instructed to treat the slow and fast time scales as if they were independent. Despite the intuitive motivation and the effectiveness of this perturbation method, one cannot failt to notice that these two scales relate to the same unique variable, so independence can only be formal. How sensible is it, then, to split a variable asymptotically into two (or more) independent ones? In this paper, we elucidate this issue with Van der Pol's equation, one of the simplest weakly nonlinear oscillators, as well as a simple example of a Hopf bifurcation. The discussion involves carrying the multiple-scale analysis up to arbitrarily large order and dealing with the divergent character of the resulting asymptotic series. Using the technique of optimal truncation, we re-connect the two scales. Specifically, we show that an initial translation of the fast coordinate leads to a non-trivial, exponentially small, phase shift that depends on the slow coordinate. This phase shift breaks the independence of the slow and fast scales and is found to result from the nonlinearity. Numerical simulations confirm its existence, as well as the predicted scaling. The calculation is carried out in sufficient detail to provide confidence in the generality of our result, both in its essence and in its form. In particular, we find strong indications that a Hopf bifurcation with a quadratic nonlinearity would lead to the same phenomenon, but with a larger magnitude.
Versions
Related
- On Some Karhunen–Loève Expansions Related to SU(2) with Application to Tests of Symmetry
- A determinant formula for multiple harmonic sums and their linear independence
- Laurent series expansions for the Barnes multiple zeta function
- Faber-Walsh field expansions for the planar conductivity problem with multiple inclusions
- The threshold for linear independence of multiple zeta values in positive characteristic
BibTeX
@misc{kozyreff2026independence,
title = {{On the independence of the slow and fast scales in multiple-scale expansions, with application to Van der Pol's equation}},
author = {Gregory Kozyreff and John R. King},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.27038},
eprint = {2606.27038},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.27038},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the independence of the slow and fast scales in multiple-scale expansions, with application to Van der Pol's equation
AU - Gregory Kozyreff
AU - John R. King
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2606.27038
UR - https://arxiv.org/abs/2606.27038
AB - When implementing the method of multiple scales, one is traditionally instructed to treat the slow and fast time scales as if they were independent. Despite the intuitive motivation and the effectiveness of this perturbation method, one cannot failt to notice that these two scales relate to the same unique variable, so independence can only be formal. How sensible is it, then, to split a variable asymptotically into two (or more) independent ones? In this paper, we elucidate this issue with Van der Pol's equation, one of the simplest weakly nonlinear oscillators, as well as a simple example of a Hopf bifurcation. The discussion involves carrying the multiple-scale analysis up to arbitrarily large order and dealing with the divergent character of the resulting asymptotic series. Using the technique of optimal truncation, we re-connect the two scales. Specifically, we show that an initial translation of the fast coordinate leads to a non-trivial, exponentially small, phase shift that depends on the slow coordinate. This phase shift breaks the independence of the slow and fast scales and is found to result from the nonlinearity. Numerical simulations confirm its existence, as well as the predicted scaling. The calculation is carried out in sufficient detail to provide confidence in the generality of our result, both in its essence and in its form. In particular, we find strong indications that a Hopf bifurcation with a quadratic nonlinearity would lead to the same phenomenon, but with a larger magnitude.
ER -