The images on this page were produced through the repeated iteration of various quadratic equations, such as f(x)=x²+c. The x-axis of the images show the value of the constant c from the minimum on the left to the maximum on the right, while the y-axis of the image show the results for using the function's result as it's input in the next iteration for several cycles.   The iteration follows the form of :

[ f(x)₁=x²+c, f(x)₂=f(x)₁²+c, ..., f(x)_n=f(x)_(n-1)²+c ]
You will notice that for each value of c, the values of f(x) fall within a restricted range of y-coordinates. This range of y-coordinates illustrate the attractors for the equation at each value of c. The attractors are simply the set of values which the iteration converges to.   Areas where the values of f(x) fail to converge upon a limited set of values show the chaotic behavior that can result from a deterministic nonlinear system. These diagrams are sometimes called bifurcation diagrams because of the way that the attractors apparently branch or bifurcate over a range of c values.

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The function used to form these images is f(x)=x2+c. They show a series of zooms into the diagram, going from image one to image six. The first 50 iterations are not shown in these images to more clearly illustrate the attractors of the system.

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The following images are the result of various quadratic equations with all iterations displayed.
f(x)=cx2+c c:[-0.5 ... 0.5]	f(x)=x2+x+c c:[0 ... -2]
f(x)=x2+2x+c c:[-2 ... 0]	f(x)=x2+3x+c c:[-1 ... 0]
f(x)=x2-x+c c:[-1 ... 1]	f(x)=2x2+c c:[-1 ... 0.125]

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This image resulted from an error
made in the display of f(x)=x²+c.