Computation of true chaotic orbits using cubic irrationals

• We propose a method for computing true orbits of 1D piecewise linear fractional maps.
• The method uses cubic irrationals as numbers and involves only integer arithmetic.
• True orbits generated by the method display the same properties as typical orbits.
• We can simulate maps whose simulation has been difficult, such as the Bernoulli map.

We introduce a method that enables us to generate long true orbits of discrete-time dynamical systems defined by one-dimensional piecewise linear fractional maps with integer coefficients. The method uses cubic irrationals to represent numbers and involves only integer arithmetic to compute true orbits. By applying the method to the Bernoulli map and a modified Bernoulli map, we show that it successfully generates true chaotic and intermittent orbits, respectively, in contrast with conventional simulation methods. We demonstrate through simulations concerning invariant measures and the power spectrum that the statistical properties of the true orbits generated agree well with those of typical orbits of the two maps.