On the periodic solutions of a perturbed double pendulum

Abstract. We provide sufficient conditions for the existence of periodic solutions of the planar perturbed double pendulum with small oscillations having equations of motion θ̈1 = −2aθ1 + aθ2 + εF1(t, θ1, θ̇1, θ2, θ̇2), θ̈2 = 2aθ1 − 2aθ2 + εF2(t, θ1, θ̇1, θ2, θ̇2), where a and ε 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 of the gravity. Here the parameter ε is small and the smooth functions F1 and F2 define the perturbation which are periodic functions in t and in resonance p:q with some of the periodic solutions of the unperturbed double pendulum, being p and q positive integers relatively prime.