Numerically induced stochasticity and long-time behavior of numerical trajectories - small ΔT analysis

In a one-dimensional anharmonic potential well, the period of an orbit is a function of its energy. The true motion in such a well is regular, since energy conservation constrains the velocity at each value of the coordinate. Nontheless, when the orbit is computed numerically, stochastic behavior can result. The phenomenon of numerically induced stochasticity has significance in several contexts. Firstly, a numerical investigation of the regions of phase space accessible to an orbit may lead to erroneous results, if the timestep is too large or the mover inappropriate. Furthermore, conclusions about orbital stability based on numerical integrations may be erroneous, since neighboring chaotic orbits diverge exponentially, even if the chaos is numerically induced. When studying the dynamics of a physical system, one should demonstrate that any chaos observed is not numerically induced. Also, linearized simulations of collective phenomena must avoid numerically induced stochasticity, since the zero-order and perturbed trajectories are 'neighboring'. Finally, trajectory crossings in PIC simulations can lead to enhanced noise and other errors. In addition to these investigations, an analysis is also made of the long-term behavior of numerical trajectories (small ΔT analysis). (Nogami, K.)