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

1 = −2a1 + a2 + "F1(t, 1, ˙1, 2, ˙2)¨

2 = 2a1 − 2a2 + "F2(t, 1, ˙1, 2, ˙2)where a and " 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 " is small and the smooth functions F1 and F2 define the perturbation which are periodic functions in it and in resonance p:q with some of the periodic solutions of the unperturbed double pendulum, being p and q positive integers relatively prime.

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