Newton maps: Fractals from Newton's method for the circle map

To understand why two interacting oscillators synchronize with each other, or lock together and resonate at some rational frequency ratio, dynamical-systems theory shows that one should study circle maps and their periodic orbits. One can easily explore the structure of these periodic orbits using Newton maps, derived from Newton's method for finding the roots of an equation. I present here some interesting and beautiful examples of fractals encountered in Newton maps while investigating the periodic orbits of circle maps.