A New Pathology in the Simulation of Chaotic Dynamical Systems on Digital Computers

Systematic distortions are uncovered in the statistical properties of chaotic dynamical systems when represented and simulated on digital computers using standard IEEE floating-point numbers. This is done by studying a model chaotic dynamical system with a single free parameter β, known as the generalized Bernoulli map, many of whose exact properties are known. Much of the structure of the dynamical system is lost in the floating-point representation. For even integer values of the parameter, the long time behaviour is completely wrong, subsuming the known anomalous behaviour for β = 2. For non-integer β, relative errors in observables can reach 14%. For odd integer values of β, floating-point results are more accurate, but still produce relative errors two orders of magnitude larger than those attributable to roundoff. The analysis indicates that the pathology described, which cannot be mitigated by increasing the precision of the floating point numbers, is a representative example of a deeper problem in the computation of expectation values for chaotic systems. The findings sound a warning about the uncritical application of numerical methods in studies of the statistical properties of chaotic dynamical systems, such as are routinely performed throughout computational science, including turbulence and molecular dynamics.