Precise asymptotic behavior of solutions to damped simple pendulum equations

• Main Author: Tetsutaro Shibata
• Format: Article
• Language: English
• Published: Texas State University 2009-11-01
• Series: Electronic Journal of Differential Equations
• Subjects: Damped simple pendulum; asymptotic formula
• Online Access: http://ejde.math.txstate.edu/Volumes/2009/142/abstr.html

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 $f(u) = -u$.