The Van der Pol oscillator is a nonlinear damping and non-conservative oscillator. Energy is generated at low amplitude and dissipated at high amplitude. This nonlinear oscillator was first introduced by Dutch electrical engineer and physicist B. Van der Pol and it was originally used to investigate vacuum tubes. Nowadays, it is used in both physical and biological sciences. It is also used in sociology and even in economics. It has a limit cycle and in earlier it was determined by the classical perturbation methods when the nonlinear term is small. Then the harmonic balance method was used to determine the limit cycle for stronger nonlinear case. Moreover, many researchers have been analyzed this oscillator by various numerical approaches. In this article, a new analytical approach based on harmonic balance method is presented to determine the limit cycle as well as approximate solutions of this nonlinear oscillator. The frequency as well as the limit cycle obtained by new approach has been compared with those obtained by other existing methods. The present method gives better result than other existing results and also close to the corresponding numerical result (considered to the exact result). Moreover, the present method is simpler than the existing harmonic balance method.
| Published in | Science Journal of Applied Mathematics and Statistics (Volume 7, Issue 4) |
| DOI | 10.11648/j.sjams.20190704.11 |
| Page(s) | 51-55 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2024. Published by Science Publishing Group |
Van der Pol Oscillator, Limit Cycle, Harmonic Balance Method, Frequency
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APA Style
Md. Abul Kashem Mondal, Md. Helal Uddin Molla, Md. Shamsul Alam. (2019). A New Analytical Approach for Solving Van der Pol Oscillator. Science Journal of Applied Mathematics and Statistics, 7(4), 51-55. https://doi.org/10.11648/j.sjams.20190704.11
ACS Style
Md. Abul Kashem Mondal; Md. Helal Uddin Molla; Md. Shamsul Alam. A New Analytical Approach for Solving Van der Pol Oscillator. Sci. J. Appl. Math. Stat. 2019, 7(4), 51-55. doi: 10.11648/j.sjams.20190704.11
AMA Style
Md. Abul Kashem Mondal, Md. Helal Uddin Molla, Md. Shamsul Alam. A New Analytical Approach for Solving Van der Pol Oscillator. Sci J Appl Math Stat. 2019;7(4):51-55. doi: 10.11648/j.sjams.20190704.11
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Download@article{10.11648/j.sjams.20190704.11,
author = {Md. Abul Kashem Mondal and Md. Helal Uddin Molla and Md. Shamsul Alam},
title = {A New Analytical Approach for Solving Van der Pol Oscillator},
journal = {Science Journal of Applied Mathematics and Statistics},
volume = {7},
number = {4},
pages = {51-55},
doi = {10.11648/j.sjams.20190704.11},
url = {https://doi.org/10.11648/j.sjams.20190704.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjams.20190704.11},
abstract = {The Van der Pol oscillator is a nonlinear damping and non-conservative oscillator. Energy is generated at low amplitude and dissipated at high amplitude. This nonlinear oscillator was first introduced by Dutch electrical engineer and physicist B. Van der Pol and it was originally used to investigate vacuum tubes. Nowadays, it is used in both physical and biological sciences. It is also used in sociology and even in economics. It has a limit cycle and in earlier it was determined by the classical perturbation methods when the nonlinear term is small. Then the harmonic balance method was used to determine the limit cycle for stronger nonlinear case. Moreover, many researchers have been analyzed this oscillator by various numerical approaches. In this article, a new analytical approach based on harmonic balance method is presented to determine the limit cycle as well as approximate solutions of this nonlinear oscillator. The frequency as well as the limit cycle obtained by new approach has been compared with those obtained by other existing methods. The present method gives better result than other existing results and also close to the corresponding numerical result (considered to the exact result). Moreover, the present method is simpler than the existing harmonic balance method.},
year = {2019}
}
TY - JOUR T1 - A New Analytical Approach for Solving Van der Pol Oscillator AU - Md. Abul Kashem Mondal AU - Md. Helal Uddin Molla AU - Md. Shamsul Alam Y1 - 2019/10/09 PY - 2019 N1 - https://doi.org/10.11648/j.sjams.20190704.11 DO - 10.11648/j.sjams.20190704.11 T2 - Science Journal of Applied Mathematics and Statistics JF - Science Journal of Applied Mathematics and Statistics JO - Science Journal of Applied Mathematics and Statistics SP - 51 EP - 55 PB - Science Publishing Group SN - 2376-9513 UR - https://doi.org/10.11648/j.sjams.20190704.11 AB - The Van der Pol oscillator is a nonlinear damping and non-conservative oscillator. Energy is generated at low amplitude and dissipated at high amplitude. This nonlinear oscillator was first introduced by Dutch electrical engineer and physicist B. Van der Pol and it was originally used to investigate vacuum tubes. Nowadays, it is used in both physical and biological sciences. It is also used in sociology and even in economics. It has a limit cycle and in earlier it was determined by the classical perturbation methods when the nonlinear term is small. Then the harmonic balance method was used to determine the limit cycle for stronger nonlinear case. Moreover, many researchers have been analyzed this oscillator by various numerical approaches. In this article, a new analytical approach based on harmonic balance method is presented to determine the limit cycle as well as approximate solutions of this nonlinear oscillator. The frequency as well as the limit cycle obtained by new approach has been compared with those obtained by other existing methods. The present method gives better result than other existing results and also close to the corresponding numerical result (considered to the exact result). Moreover, the present method is simpler than the existing harmonic balance method. VL - 7 IS - 4 ER -
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