Electronic Journal of Differential Equations,
Vol. 2009(2009), No. 142, pp. 1-15.

Title: Precise asymptotic behavior of solutions to
       damped simple pendulum equations

Author: Tetsutaro Shibata (Hiroshima Univ., Japan)

Abstract:
 We consider the simple pendulum equation
 $$\displaylines{
 -u''(t) + \epsilon f(u'(t)) = \lambda\sin u(t), \quad t \in I:=(-1, 1),\cr
 u(t) > 0, \quad t \in I, \quad u(\pm 1) = 0,
 }$$
 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 $f(u) = -u$.

Submitted April 21, 2009. Published November 07, 2009.
Math Subject Classifications: 34B15.
Key Words: Damped simple pendulum; asymptotic formula.