Perturbed Damped Pendulum: Finding Periodic Solutions

Using the equation of motion of the damped pendulum, we introduce the averaging method on the study of periodic solutions of dynamical systems with small perturbation. We provide sufficient conditions for the existence of periodic solutions of the perturbed damped pendulum with small oscillations having equations of motion\[\ddot{\T}=-a\T-b\dot{\T}+\e f(t,\T,\dot{\T}),\]where $a>0,\,b>0$ and $\e$ are real parameters, with $a=g/l$, $g$ the acceleration of the gravity, $l$ the length of the rod and $b$ the damping coefficient. Here the parameters $b$ and $\e$ are small and the smooth function $f$ is $T$--periodic in $t$. The averaging theory provides a useful means to study dynamical systems, accessible to Master and PhD students.