s_{0}=0.4, s_{i+1}=-0.5*s_{i}^{2}+2.8

Center:-0+0i; Zoom = 1

Now the sequence is generated by a*s_{i}^{2}+b, a and b real numbers. You may be familiar with sequences like this, the Logistic Map and the real axis of the Mandelbrot set are two examples.

The parameters can be tweaked to produce a variety of behaviors in the sequence. For a < 0 and b > 0 and small the sequence will converges to a point. As b increases it goes through a period doubling phase where is converges to a 2 cycle, then 4, 8 as so on. Eventually after hitting every power of two, it becomes chaotic. The Wikipedia page linked above describes this phenomena in more detail.