Regular and chaotic phase space fraction in the double pendulum

The double coplanar pendulum is an example of the coexistence of regular and chaotic dynamics for equal energy values but different initial conditions. Regular trajectories predominate for low energies; as the energy is increased, the system passes through values where chaotic trajectories are abundant, and then, increasing the energy further, it is again dominated by regular trajectories. Given that the energetically accessible states are bounded, a relevant question is about the fraction of phase space regular or chaotic trajectories as the energy varies. In this paper, we calculate the relative abundance of chaotic trajectories in phase space, characterizing the trajectories using the maximum Lyapunov exponent, and find that, for low energies, it grows exponentially.