Quasi-periodicity, global stability and scaling in a model of Hamiltonian round-off
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Computer simulation of a dynamical system often in- volves rounding off, i.e., replacing a continuous dynami- cal mapping arising from a set of differential equations by a discrete one which can be implemented using integer arithmetic. We consider here an especially simple ex- ample of a round-off scheme in which the discrete map acts on an infinite square lattice labeled by pairs of inte- gers. After a certain number of time steps, every orbit of the continuum map returns to its initial position on the plane, whereas this is not necessarily true of the discrete map. Over time, the latter can trace out on the integer lattice a complicated path resembling a snowflake. We show that the longer orbits can be constructed from the shorter ones by a simple substitution rule, with an asso- ciated fractal dimension. In principle, one could imagine infinitely long orbits, but we are able to prove that such orbits are not present. Our complete characterization of all possible orbits lays the groundwork for an exact de- termination of the propagation of round-off error for as- ymptotically long times. Understanding and controlling the propagation of round- off errors is fundamental to any satisfactory numerical de- scription of a dynamical system. For a sufficiently simple continuum system the discretization of phase space and the replacement of Hamiltonian dynamics by a discrete map can produce behavior which is far more complicated ~and inter- esting! than that of the unperturbed system, and quite differ- ent from the addition of white noise which is often used to model the effects of round-off. In the present work we study an extreme example, in which the continuum system is little more than a linear os- cillator, while the propagation of round-off error is highly complex, being endowed with quasi-periodicity on a square lattice and scaling of periodic orbits with a nontrivial fractal dimension. Our main interest is in the long-time asymptotics, and the issues include both deterministic and statistical as- pects. We shall postpone to a separate publication a discus- sion of the statistical considerations, concentrating here on the organization of the phase portrait and the question of global stability, i.e., whether there exist unbounded escape orbits.