Melnikov's method applied to the double pendulum

Melnikov's method is applied to the planar double pendulum proving it to be a chaotic system. The parameter space of the double pendulum is discussed, and the integrable cases are identified. In the neighborhood of the integrable case of two uncoupled pendulums Melnikov's integral is evaluated using residue calculus. In the two limiting cases of one pendulum becoming a rotator or an oscillator, the parameter dependence of chaos, i.e. the width of the separatrix layer is analytically discussed. The results are compared with numerically computed Poincaré surfaces of section, and good agreement is found.