The Symmetric Level Index System with an Application to Chaos

In this paper we present a brief summary of the Symmetric Level Index, SLI, system of number representation and computer arithmetic with only enough detail to allow the reader to understand the remaining discussion. This discussion is focused on an elementary example of chaos, the logistic map or shift map. This map, which was discussed by Ulam and von Neumann in the context of random number generators, is one of the classical one-dimensional examples of a chaotic map. However, in the conventional realm of floating-point computing, the chaotic behavior is completely lost. To a great extent the correct behavior of this map is recovered if the arithmetic is performed in the SLI system. This statement is valid for all three of the distinguished cases, where the process is seeded with an irrational, a rational, or a rational with a power of 2 denominator.