We use the Toda chain model to demonstrate that numerical simulation of
integrable Hamiltonian dynamics using time discretization destroys
integrability and induces dynamical chaos. Specifically, we integrate this
model with various symplectic integrators parametrized by the time step Ï„
and measure the Lyapunov time TΛ​ (inverse of the largest Lyapunov
exponent Λ). A key observation is that TΛ​ is finite whenever
τ is finite but diverges when τ→0. We compare the Toda
chain results with the nonitegrable Fermi-Pasta-Ulam-Tsingou chain dynamics. In
addition, we observe a breakdown of the simulations at times TB​≫TΛ​ due to certain positions and momenta becoming extremely large
(``Not a Number''). This phenomenon originates from the periodic driving
introduced by symplectic integrators and we also identify the concrete
mechanism of the breakdown in the case of the Toda chain.Comment: 13 pages, 7 figure