An Analytical Discussion on the Self-Oscillations in the Van der Pol Equation Far from a Hard Mode Instability Point

Shu EZAKI; Kaoru YAMAFUJI; Kiyoshi TOKO; Kouichi RYU

An Analytical Discussion on the Self-Oscillations in the Van der Pol Equation Far from a Hard Mode Instability Point

Shu EZAKI, Kaoru YAMAFUJI, Kiyoshi TOKO, Kouichi RYU

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A nonperturbative method is presented for describing approximately the behavior of a self-oscillation of electric voltage in the Van der Pol equation over a wide range of the value of external parameter µ. To express an appreciably distorted wave form for the steady self-oscillation at µ1, a phase F of the voltage x, defined by x2A cos F (ωt), is approximated by a combination of several straight lines as a function of ωt from 0 to 2π with several numerical coefficients determined mainly from asymptotic behaviors of x for µ1 and µ1. It is shown that the resultant expression for x can describe well the numerical result over the wide range of µ. A bursting phenomenon induced by an oscillation of µ with a long period is also discussed on the basis of the present method, and the analytical results are in good agreement with the numerical ones.

Publication: IEICE TRANSACTIONS on transactions Vol.E66-E No.2 pp.139-145

Publication Date: 1983/02/25

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Shu EZAKI
Kaoru YAMAFUJI
Kiyoshi TOKO
Kouichi RYU

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Shu EZAKI, Kaoru YAMAFUJI, Kiyoshi TOKO, Kouichi RYU, "An Analytical Discussion on the Self-Oscillations in the Van der Pol Equation Far from a Hard Mode Instability Point" in IEICE TRANSACTIONS on transactions, vol. E66-E, no. 2, pp. 139-145, February 1983, doi: .
Abstract: A nonperturbative method is presented for describing approximately the behavior of a self-oscillation of electric voltage in the Van der Pol equation over a wide range of the value of external parameter µ. To express an appreciably distorted wave form for the steady self-oscillation at µ1, a phase F of the voltage x, defined by x2A cos F (ωt), is approximated by a combination of several straight lines as a function of ωt from 0 to 2π with several numerical coefficients determined mainly from asymptotic behaviors of x for µ1 and µ1. It is shown that the resultant expression for x can describe well the numerical result over the wide range of µ. A bursting phenomenon induced by an oscillation of µ with a long period is also discussed on the basis of the present method, and the analytical results are in good agreement with the numerical ones.
URL: https://global.ieice.org/en_transactions/transactions/10.1587/e66-e_2_139/_p

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@ARTICLE{e66-e_2_139,
author={Shu EZAKI, Kaoru YAMAFUJI, Kiyoshi TOKO, Kouichi RYU, },
journal={IEICE TRANSACTIONS on transactions},
title={An Analytical Discussion on the Self-Oscillations in the Van der Pol Equation Far from a Hard Mode Instability Point},
year={1983},
volume={E66-E},
number={2},
pages={139-145},
abstract={A nonperturbative method is presented for describing approximately the behavior of a self-oscillation of electric voltage in the Van der Pol equation over a wide range of the value of external parameter µ. To express an appreciably distorted wave form for the steady self-oscillation at µ1, a phase F of the voltage x, defined by x2A cos F (ωt), is approximated by a combination of several straight lines as a function of ωt from 0 to 2π with several numerical coefficients determined mainly from asymptotic behaviors of x for µ1 and µ1. It is shown that the resultant expression for x can describe well the numerical result over the wide range of µ. A bursting phenomenon induced by an oscillation of µ with a long period is also discussed on the basis of the present method, and the analytical results are in good agreement with the numerical ones.},
keywords={},
doi={},
ISSN={},
month={February},}

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TY - JOUR
TI - An Analytical Discussion on the Self-Oscillations in the Van der Pol Equation Far from a Hard Mode Instability Point
T2 - IEICE TRANSACTIONS on transactions
SP - 139
EP - 145
AU - Shu EZAKI
AU - Kaoru YAMAFUJI
AU - Kiyoshi TOKO
AU - Kouichi RYU
PY - 1983
DO -
JO - IEICE TRANSACTIONS on transactions
SN -
VL - E66-E
IS - 2
JA - IEICE TRANSACTIONS on transactions
Y1 - February 1983
AB - A nonperturbative method is presented for describing approximately the behavior of a self-oscillation of electric voltage in the Van der Pol equation over a wide range of the value of external parameter µ. To express an appreciably distorted wave form for the steady self-oscillation at µ1, a phase F of the voltage x, defined by x2A cos F (ωt), is approximated by a combination of several straight lines as a function of ωt from 0 to 2π with several numerical coefficients determined mainly from asymptotic behaviors of x for µ1 and µ1. It is shown that the resultant expression for x can describe well the numerical result over the wide range of µ. A bursting phenomenon induced by an oscillation of µ with a long period is also discussed on the basis of the present method, and the analytical results are in good agreement with the numerical ones.
ER -