Precise asymptotic behavior of solutions to damped simple pendulum equations

Shibata, Tetsutaro

Electronic Journal of Differential Equations, Tome 2009 (2009).

Résumé

Summary: 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 $f(u) = -u$.

Détail

Classification : 34B15
Keywords: damped simple pendulum, asymptotic formula