On The Periodic Solutions of a Perturbed Double Pendulum

We provide sufficient conditions for the existence of periodic solutions of the planar perturbed double pendulum with small oscillations having equations of motion\[\begin{array}{l}\ddot{\T}_{1}=-2a\T_{1}+a\T_{2}+\e F_1(t,\T_1,\dot \T_1,\T_2, \dot \T_2),\\\ddot{\T}_{2}=2a\T_{1}-2a\T_{2}+\e F_2(t,\T_1,\dot \T_1,\T_2, \dot\T_2),\end{array}\]where $a$ and $\e$ are real parameters. The two masses of the unperturbed double pendulum are equal, and its two stems have the same length $l$. In fact $a=g/l$ where $g$ is the acceleration ofthe gravity. Here the parameter $\e$ is small and the smooth functions $F_1$ and $F_2$ define the perturbation which are periodic functions in $t$ and in resonance $p$:$q$ with some of the periodicsolutions of the unperturbed double pendulum, being $p$ and $q$ positive integers relatively prime.