Exact Simulation of One-Dimensional Chaotic Dynamical Systems Using Algebraic Numbers

摘要

We introduce a method of true orbit generation that allowed us to perform, with digital computers, exact simulations of discrete-time dynamical systems defined by one-dimensional piecewise linear and linear fractional maps with integer coefficients by generalizing the method proposed by Saito and Ito (Physica D 268, 100-105 (2014)). The salient features of the new method are that it can use algebraic numbers of an arbitrarily high odd degree to represent numbers, and that it only involves integer arithmetic to compute true orbits. We demonstrated that it succeeded in generating true chaotic and intermittent orbits, respectively, by applying the method to a tent map and a map associated with a mediant convergents algorithm, in contrast with conventional methods of simulation. We particularly demonstrated through simulations regarding invariant measures that the statistical properties of the generated true orbits agreed well with those of the typical orbits of the two maps.