Quadratic map bifurcations

Qualitative change in iteration dynamics when parameter c is changed is called bifurcation.

Tangent bifurcations

For c > 1/4 quadratic map iterations diverge to infinity. At c = 1/4 the blue parabola touches the green diagonal and attracting x1 and repelling x2 fixed points appear. This phenomenon is called the tangent bifurcation.

Drag the blue parabola by mouse.
To the left you see attracting period-3 orbit. In the center of the right picture f o3(x) is similar to parabola therefore the cycle is born in the tangent bifurcation at c = -1.75 and then repeats all bifurcations of quadratic map.

Period doubling bifurcations

For c < -3/4 derivative in the left fixed point f '(x) = m < -1, i.e. the point becomes repelling. In the vicinity of this fixed point after the first iteration perturbation δx1 = mδx ~ -δx changes its sign but after the second iteration δx2 = m2δx ~ δx again. Therefore an attracting period-2 orbit   xo → x1 → xo ... appears. This phenomenon is called the period doubling bifurcation.
The second iteration of the map f o2(x) (to the right above) has two attracting points (corresponding to the stable cycle) and one unstable fixed point between them. Note that near the center of the picture f o2(x) is similar to parabola and under decreasing c will have period doubling bifurcation again after which period-4 stable orbit appears and so on. Unstable cycles never disappear more.

Bifurcation diagram for quadratic map

There is a good way to trace bifurcations of quadratic map on the (x, c) plane by the bifurcation map (similar to the "logistic bifurcation map"). Let us plot iterations fc: x1 = 0 → x2 → x3 → ...→ xMaxIt for all real c on the (x, c) plane. Colors (from blue to red) show how often orbit visits the pixel (colors are changed under zooming). Usually initial points are omitted but transient process shows how fast iterations converge to an attractor. You can watch iterations of fc(x) for corresponding c values to the right.
Controls: Click mouse with Alt/Ctrl to zoom image. Max number of iterations = 2000. See coordinates of the image center and Δx, Δc in the text field. The vertical line goes through x = 0.
The top part of the picture begins with tangent bifurcation at c = 1/4. For c > 1/4 points go away to infinity. For -3/4 < c < 1/4 there is single attracting fixed point. Filaments broadening show how the critical orbit points are attracted to the fixed point. For c = 0 the fixed point coincides with the critical one (the blue vertical line) and becomes superstable. At c ~ -3/4 we see a branching point due to period doubling bifurcation. After that the fixed point loses its stability and attracting period-2 cycle appears. At c = -1 the cycle goes through the critical point and becomes superstable and so on. Cascade of period doubling bifurcations ends at c = -1.401155. Further you see region of chaotic dynamics. It ends by crisis at c = -2. For c < -2 there is an invariant Cantor set of unstable orbits with zero measure and iterations of the critical point diverge. In the chaotic band there are white narrow holes of windows of periodic dynamics. The biggest one corresponds to period-3 window. There are 3 junction regions in it and under iterations critical orbit jumps sequentially between these regions.

The bifurcation map patterns

pattern Fig.1 shows that caustics in distribution of points of chaotic orbits are generated by an extremum of a map. Therefore singularities (painted in the red) on the bifurcation diagram appear at images of the critical point fc on(0). Let us denote gn(c) = fcon(0), then
    go(c) = 0,   g1(c) = c,   g2(c) = c2 + c, ...
The curves g0,1,...,6(c) are shown in Fig.2.
Note that attractor of the map is located between g1(c) and g2(c) curves. If a caustic crosses the vertical line x = 0 then
    gk(c) = fcok(0) = 0.
I.e. there is a superstable period-k cycle. In the neighbourhood of this c value there is corresponding window of regular dynamics.
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updated 12 July 2006