<abstract xmlns="http://www.w3.org/1999/xhtml"><p>We study a model problem describing vibrational resonance by means of a high-order averaging technique based on so-called word series. With the technique applied here, the tasks of constructing the averaged system and the associated change of variables are divided into two parts. It is first necessary to build recursively a set of so-called <italic>word basis</italic> functions and, after that, all the required manipulations involve only scalar coefficients that are computed by means of simple recursions. As distinct from the situation with other approaches, with word-series, high-order averaged systems may be derived without having to compute the associated change of variables. In the system considered here, the construction of high-order averaged systems makes it possible to obtain very precise approximations to the true dynamics.</p></abstract>

We study a model problem describing vibrational resonance by means of a high-order averaging technique based on so-called word series. With the technique applied here, the tasks of constructing the averaged system and the associated change of variables are divided into two parts. It is first necessary to build recursively a set of so-called word basis functions and, after that, all the required manipulations involve only scalar coefficients that are computed by means of simple recursions. As distinct from the situation with other approaches, with word-series, high-order averaged systems may be derived without having to compute the associated change of variables. In the system considered here, the construction of high-order averaged systems makes it possible to obtain very precise approximations to the true dynamics.

Vibrational resonance: a study with high-order word-series averaging

Universidad Carlos III de Madrid, Avenida de la Universidad 30

Applied Mathematics and Nonlinear Sciences

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"Vibrational resonance: a study with high-order word-series averaging"}

We study a model problem describing vibrational resonance by means of a high-order averaging technique based on so-called word series. With the technique...

n	# n-letter words with f_w ≠ 0	Error B = 0.52	Error B = 0.53
1	7	0.241	0.431
2	35	0.080	0.481
3	217	0.026	0.251
4	1,407	0.018	0.036
5	9,345	0.009	0.015
6	62,951	0.004	0.008
7	427,889	0.003	0.005

Vibrational resonance: a study with high-order word-series averaging

Published Online: Apr 06, 2016

Page range: 239 - 246

Received: Feb 07, 2016

Accepted: Apr 06, 2016

DOI: https://doi.org/10.21042/AMNS.2016.1.00018

Keywords
High-order averaging, vibrational resonance, formal series

© 2016 A. Murua, J.M. Sanz-Serna, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Fig. 1

Higher-order averaging

Vibrational resonance: a study with high-order word-series averaging

Published Online: Apr 06, 2016

Page range: 239 - 246

Received: Feb 07, 2016

Accepted: Apr 06, 2016

DOI: https://doi.org/10.21042/AMNS.2016.1.00018

KeywordsHigh-order averaging, vibrational resonance, formal series

© 2016 A. Murua, J.M. Sanz-Serna, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Fig. 1

Higher-order averaging

Keywords
High-order averaging, vibrational resonance, formal series