Forward period analysis and the long term simulation of a periodic Hamiltonian system

The period of a Morse oscillator and mathematical pendulum system are obtained, accurate to 100 significant digits, by forward period analysis (FPA). From these results, the long-term [0, 10^60] (time unit) solutions, which overlap from the Planck time to the age of the universe, are computed reliably and quickly with a parallel multiple-precision Taylor series (PMT) scheme. The application of FPA to periodic systems can reduce the computation loops of long-term reliable simulation from O(t^(1+1/M)) to O(lnt+t/h0) where T is the period, M the order and h0 a constant step-size. This scheme provides a way to generate reference solutions to test other schemes' long-term simulations.