Abstract
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.
Highlights
Systematic distortions are uncovered in the statistical properties of chaotic real numbers can dramatically alter the dynamics of chaotic systems after a short dynamical systems when represented and simulated on digital computers amount of simulation time
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 to such systems, dynamicists retreated to the position that, while accuracy for individual orbits may not be possible, accuracy in an averaged sense may still be possicomputational science, including turbulence and molecular dynamics
While it has long been known that individual orbits of this map are chaotic in nature, and that even statistical averages are problematic for the particular case of the system parameter β = 2,[7,8,9] our present work demonstrates a more serious problem
Summary
A New Pathology in the Simulation of Chaotic Dynamical Systems on Digital Computers. Systematic distortions are uncovered in the statistical properties of chaotic real numbers can dramatically alter the dynamics of chaotic systems after a short dynamical systems when represented and simulated on digital computers amount of simulation time. This may involve combining probabilities of very different magnitudes, and so it is computed by first sorting the list of contributing probabilities and adding them from smallest to largest, using double-precision arithmetic, to avoid loss of significance We can work this out using single-precision floating-point numbers by enumerating all of the asymptotic limit cycles of the dynamics, as well as the fraction of initial states that lie in the basins of attraction of those limit cycles. The density of orbits as a function of period decays ex-
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