Dynamic Properties of a Rational Landen Transformation

The properties of a two dimensional dynamical system are described. The structure of the system is explored through the study of t-lines and their images under the transformation operator, Φ6. The analysis of t-line, Lt, images leads to a closed form parametrization for the discriminant curve of the cubic. The parametrization, λ(s), is utilized to study the action of Φ6 on the discriminant curve. A parameterized form of Φ6, φ(s), for points that lie on the discriminant curve is obtained and the general properties of φ(s) are given. The qualitative behavior of regions in the a-b plane under Φ6 are studied through computer simulation. The t-Box is introduced, which facilitates determining image boundaries for transformed regions in the a-b plane. Analysis of t-Boxes leads to the establishment of global image boundaries.