Bifurcation diagram of the Henon map

Isoperiodic diagram of the Henon map on the (a,b) parameter plane is shown below ("unit" square is ploted for convenience). Algorithm of computations and coloring scheme are explained in "swallows" and "shrimps". We see the familiar quadratic map dynamics along the b = 0 line (period doubling bifurcation cascade and chaotic sea ending at the "nose" tip). It is amazing that the period-3 Mandelbrot midget originates from the strip 3 but the period-5 midget originates from the period-5 shrimps (see also Structure of the parameter space of the Henon map).
Controls: Click mouse in the window to find period p of a point. Click mouse + <Alt>/<Ctrl> to zoom In/Out.

Attractors

The Henon map has two fixed points
    x1,2 = y1,2 = (1 - b)/2 ± [(1 - b)2/4 - a]1/2.
They are real for a < ao = (1 - b)2/4 . The first fixed point is always unstable (apparently it is located at the border of the basin of attraction of bounded orbits). The second one is stable for a > a1 = -3(1 - b)2/4 in the (red) region marked by 1 (here a1 is the period doubling bifurcation curve). The point (x2 , y2 ) is usually used as the starting point in all applets.
To the right below you see a strange attractor or periodic orbit (white points) on the dynamical (x,y) plane. Corresponding parameter values (initially a=-1.4 and b=0.3) are marked by the white cross to the left. Two fixed points are marked by "×" and "+". The black region is the basin of attraction of the bounded orbit. Colors show how fast corresponding point go to infinity. The outer grey region (to the left) corresponds to parameter values when there are not (apparently :) attracting bounded orbits. Note that for b = 0 it follows x' = y'2 + a. And x = y2 + a yields a surprisingly good first-order approximation for the Henon attractor. Lyapunov exponents are computed for 1000 iterations. For a=-1.4 and b=0.3 computed value agrees well with known Λ = 0.42 value. L are negative for attracting periodic orbits.

Controls: click mouse into the left (parameter) window to see corresponding dynamical plot. Click mouse to the right to change starting point for the white orbit. You see parameters a, b or xo, yo,L in the browser status bar. You can zoom both windows too.

Swallows

Milnor motivated appearence of swallows in the following way. For small |b| the Henon map is similar to the quadratic map for which attracting periodic orbit has at least one point close to the critical point x = 0 . This point will stay close to x = 0 along a curve on the plane (a,b) (in the tangent direction there are bifurcation cascade of the quadratic family). If one more point of this orbit is close to x = 0 along a second curve then at the crossing of these two curves dynamics is similar to dynamics of composition of these quadratic maps and Milnor's swallow appears. If there is one more such curve then a pod of close swallows appears (see 3 Milnor's swallows).

[1] D.G. Sterling, H.R. Dullin, J.D. Meiss Homoclinic Bifurcations for the Henon Map arxiv.org/abs/chao-dyn/9904019
[2] H.R.Dullin, J.D.Meiss Generalized Henon Maps: the Cubic Difeomorphisms of the Plane
[3] Predrag Cvitanovic, Gemunu H. Gunaratne, Itamar Procaccia Topological and metric properties of Henon-type strange attractors Phys. Rev. A 38, 1503-1520 (1988) abstract
[4] Kai T. Hansenyx and Predrag Cvitanovic "Bifurcation structures in maps of Henon type" Nonlinearity 11 (1998) 1233-1261.
[5] Michael Benedicks, Marcelo Viana "Solution of the basin problem for Henon-like attractors" Invent. math. 143, 375-434 (2001)


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updated 14 Nov 06