Precise asymptotic behavior of solutions to damped simple pendulum equations

We consider the simple pendulum equation $$displaylines{ -u''(t) + epsilon f(u'(t)) = lambdasin u(t), quad t in I:=(-1, 1),cr u(t) > 0, quad t in I, quad u(pm 1) = 0, }$$ where $0 < epsilon le 1$, $lambda > 0$, and the friction term is either $f(y) = pm|y|$ or $f(y)...

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Bibliographic Details
Main Author: Tetsutaro Shibata
Format: Article
Language:English
Published: Texas State University 2009-11-01
Series:Electronic Journal of Differential Equations
Subjects:
Online Access:http://ejde.math.txstate.edu/Volumes/2009/142/abstr.html
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Summary:We consider the simple pendulum equation $$displaylines{ -u''(t) + epsilon f(u'(t)) = lambdasin u(t), quad t in I:=(-1, 1),cr u(t) > 0, quad t in I, quad u(pm 1) = 0, }$$ where $0 < epsilon le 1$, $lambda > 0$, and the friction term is either $f(y) = pm|y|$ or $f(y) = -y$. Note that when $f(y) = -y$ and $epsilon = 1$, we have well known original damped simple pendulum equation. To understand the dependance of solutions, to the damped simple pendulum equation with $lambda gg 1$, upon the term $f(u'(t))$, we present asymptotic formulas for the maximum norm of the solutions. Also we present an asymptotic formula for the time at which maximum occurs, for the case $f(u) = -u$.
ISSN:1072-6691