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.