Quadratic map

More complicated analytic quadratic map is
    xn+1 = fc( xn ) = xn2 + c.
It appears e.g. in dynamics of biological populations. On complex plane it generates the famous Mandelbrot and Julia fractal sets. In spite of apparent simplicity it has very rich dynamics. For this map regions of regular and chaotic dynamics are entangled in an intricate manner and scenarios of transition to chaos are common for many other dynamical systems.

Iteration diagram

Dynamics of 1D real maps is useful to trace on iteration diagram shown below. The blue curve is the N-th iteration f oN(x) = f(f(...f(x))). Diagonal y = x is the green line. -2 ≤ x,y ≤ 2. As since f(0) = C then for N = 1 the C value coincides with y(0). Dependence xn on n is plotted in the right window.

Controls: Drag the blue curve by mouse to change C value. Hold <Shift> to change the starting point xo. Press <Enter> to set new parameters from the text fields.
To plot the first iteration we draw vertical red line from the starting point xo = 0 toward the blue curve y = f(x) = x2 + c, where yo = f(xo). To get the second iteration we draw red horizontal line to the green diagonal y = x, where x1 = yo = f(xo). Then draw again vertical line to the blue curve to get y1 = f(x1) and so on.
Points fc: xo → x1 → x2 → ... for some c and xo values make orbit of the point xo (it is plotted in the right part of this applet).

Critical points

For an analytic map points where f '(xc ) = 0 are called critical points. Every stable cycle attracts at least one critical point. Quadratic map has the only critical point xc = 0. Therefore it can have only one attracting cycle and xc is used usually as the starting point to find the cycle.

Fixed points

For C = -1/2 iterations go quickly to an attracting fixed point   x• = f(x• ) of the map. Fixed points correspond to intersections of y = x and y = f(x) (green and blue) curves. There are always two fixed points for a quadratic map because of two roots of quadratic equation
    f(x•) - x• = x•2 + c - x• = 0,     x1,2 = 1/2 -+ (1/4 - c)1/2.
The first derivative of a map at a fixed point
    m = f '(x•) = 2x•
is called multiplier (or the eigenvalue) of the point. For small enough δx
    f(x• + δx) = f(x•) + mδx + O(δx2) ≈ x• + mδx.
So a fixed point is stable (attracting), superstable, repelling, indifferent (neutral) according as its multiplier satisfies |m| < 1, |m| = 0, |m| > 1 or |m| = 1.
The second fixed point (the right intersection) is always repelling. For |x| > x2 iterations go to infinity. For |x| < x2 they go to the attracting fixed point x1. This interval is the basin of attraction of the point.

Bifurcations. Tangent bifurcation

Qualitative change in iteration dynamics when parameter c is changed is called bifurcation. For c = 1/4 the blue parabola touches with the green diagonal and two fixed points merge together. For c > 1/4 they become complex and iterations diverge. This phenomenon is called the tangent bifurcation.

Period doubling bifurcation

For c < -3/4 derivative in the left fixed point f '(x•) = m < -1, i.e. the point becomes repelling (change xo to test that). In the vicinity of this point after the first iteration perturbation δx1 = mδx ~ -δx changes its sign and after the second iteration δx2 = m2δx ~ δx again. Therefore an attractive period-2 orbit   xo → x1 → xo ... appears here. This phenomenon is called the period doubling bifurcation.
The second iteration of the map f o2(x) (to the right above) get 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.

At last superstable period-3 orbit and f o3(x) corresponding to the period-3 window of regular dynamics is shown. In the center of the right picture f o3(x) is similar to parabola again and it repeats all bifurcations of quadratic map.

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updated 12 July 06