This Demonstration plots the points (, ), where and for , starting with and . With different parameters and you will get similar pictures. Start with only 1000 steps and then choose an appropriate number of steps and an adequate plot range. You will get other interesting pictures, for example, with the parameters (0.31, 0.884), (0.456, 0.83), (0.49, 0.717), (0.564, 0.594), and (-0.612, 0.872).
The recurrence equations constitute a two-parameter canonical form of a nonlinear coupled oscillator.