From Mathematical Dynamics to Scientific Evidence: Determinism, Chaos, Ergodicity, and the Epistemology of Computational Simulation

Source abstract

Computer simulations are often assessed through their reproducibility, numerical accuracy, or agreement with observed data, although these criteria support different kinds of scientific inference. This research note examines how determinism, chaotic sensitivity, ergodicity, stochasticity, and finite computational representation jointly constrain what may be inferred from simulated trajectories and statistics. Drawing on dynamical systems theory, numerical analysis, philosophy of simulation, and agent-based modeling, it distinguishes properties of a mathematical model from those of its implementation and from claims about an empirical target. A cross-disciplinary survey spans mechanics, engineering, computing, economics, and international relations; closer analysis centers on the double pendulum, logistic map, Lorenz-63 system, and a minimal stochastic agent-based model. Particular attention is given to invariant measures, finite-precision effects, pseudorandom generators and seeds, execution order, and cross-platform replication. Multiplicative wealth dynamics illustrate the difference between ensemble expectations and typical time evolution, while strategic interaction illustrates the limits of inferring intentions and mechanisms from simulated outcomes. The synthesis argues that trajectory agreement, statistical agreement, computational repeatability, and empirical adequacy are distinct evidential achievements. Accordingly, the credibility of a simulation-based claim must be evaluated relative to its target, model assumptions, implementation, observable, time horizon, and uncertainty specification. No new numerical experiments or general theorem are claimed; the contribution is a critical integration of established results and a set of bounded questions for comparative simulation studies.