Abstract

In 1952, as a first approach to computational nonlinear problems, Enrico Fermi, John Pasta and Stanislaw Ulam simulated the loaded string model, perturbed with small, nonlinear interaction terms. Because Poincare's theorem guarantees the non-existence of a complete set of integrals for the three-body problem, they expected to see the diffusion of energy from its single-mode initial condition to all other modes of the string . But for every combination of initial conditions, the energy remained bounded within the lowest few modes. No theoretical explanation existed for this failure of the underlying hypothesis that ergodicity follows from the lack of a complete set of integrals of the motion in a Hamiltonian model. I trace the history of this problem from the FPU simulation up to the point that a consensus was reached about its solution twenty years later. During this period, the simulation of nonlinearly-perturbed integrable models became the methodology for a new era in dynamics. Through the use of simulation, dynamicists discovered both deterministic chaos, in which the exponential separation of pair orbits generate randomness in deterministic macroscopic systems, and a new kind of structure--related to the KAM theorem--that provides limited order in the absence of analytic integrals of the motions. Historically, I map the set of conceptually-related journal articles into a chronological inference topology that tracks the emergent understanding of this so-called "fundamental problem of dynamics." Simulating non-integrable models on a digital computer requires the discretization of time and space. In turn, these approximations affect what the simulation can reveal about the model, and the model about reality. As the central feature of this new methodology, simulations play the role of experiments on mathematical models. Although similar in function to physical experiments, simulations differ significantly because they explore a mathematical realm. I present a discussion of the issues that emerge with the use of simulation as a heuristic device, and I lay the groundwork for an epistemology of simulation to answer the questions "What can a simulation tell us about, and why should we believe what it tells us?"