Long Term Simulations of the Double Pendulum by Keeping the Value of Hamiltonian Constant

Abstract

One of the simplest dynamical models which preserve the energy and have properties of chaos may be double pendulum. Many relevant studies have been performed by conventional integral schemes which can't control the error of Hamiltonian. In this paper, I will show the simulation results of double pendulum with the Hamiltonian conserved integral scheme. Non-dimensional Hamilton's equations of the double pendulum are derived. Initial values of the simulations are decided by mode analysis of the linearized system. Parametric survey is performed so that the system property may change from linear via non-linear to chaos. The results of simulations are as follows. The double pendulum which starts synchronously may change into the reverse steady mode by way of chaos-like behavior, even if it starts with small amplitude, which has been supposed to keep the same coupled harmonic oscillation. On the contrary, asynchronous mode keeps the same mode more moderately. The quantity Ln(d/T), where d is a deviation of one round trip, and T is a one round time, defined here shows a good indicator of chaos-like behavior.