The Henon map

The Henon map is [unfortunately they use different parametrizations for the map, e.g. we will get -x2 after the substitution (a,b,x,y) -> (-a,b,-x,-y) in (1) ]
    x' = a + x2 + by,
y' = x
.
    (1)
The map is intricated enought [1-7] and we review only its simplest properties now.

Isoperiodic diagram for the (a,b) parameter plane of the Henon map is shown to the left below (computation algorithm and coloring scheme is explained in "swallows" and "shrimps"). To the right you see a strange attractor or periodic orbit (white points) on the dynamical (x,y) plane. Corresponding parameter values (-1.4, 0.3) are marked by the white cross to the left. 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.

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 can zoom both windows too.

Note that for b = 0 from (1) it follows x' = y'2 + a. And x = y2 + a yields a surprisingly good first-order approximation of the Henon attractor.

The parameter plane

The Jacobian of the map is
      | 2x  b |
  J = |       | = -b.
      | 1   0 |
The map is contracting for |b| < 1 and all attracting bounded orbits are located in this region. A "unit" square is ploted below for convenience. 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.

The fixed points

The Henon map has two fixed points
    x1 = y1 = (1 - b)/2 + [(1 - b)2/4 - a]1/2,     x2 = y2 = (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.

Attracting orbits

The fixed points x1,2 are marked to the right below by the "x" and "+" crosses corresondingly. The white orbit (started from "+") converges to two points of a period-2 orbit (click mouse in its vicinity to remove transient points).

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 can zoom both windows too.

You can test by mouse that there is always only one attracting orbit. Find the period-5 shrimp orbit. Test that outside the black basin points go to infinity and that strange attractor for all initial values is almost identical (exept for an initial transient).
Three more dynamical Henon fractals.

[1] M.Henon "A two-dimensional mapping with a strange attractor" Comm.Math.Phys. 50, 69 (1976).
[2] D.G. Sterling, H.R. Dullin, J.D. Meiss Homoclinic Bifurcations for the Henon Map arxiv.org/abs/chao-dyn/9904019
[3] H.R.Dullin, J.D.Meiss Generalized Henon Maps: the Cubic Difeomorphisms of the Plane
[4] Predrag Cvitanovic, Gemunu H. Gunaratne, Itamar Procaccia Topological and metric properties of Henon-type strange attractors Phys. Rev. A 38, 1503-1520 (1988) abstract
[5] Kai T. Hansenyx and Predrag Cvitanovic "Bifurcation structures in maps of Henon type" Nonlinearity 11 (1998) 1233-1261.
[6] Michael Benedicks, Marcelo Viana "Solution of the basin problem for Henon-like attractors" Invent. math. 143, 375-434 (2001)
[7] S.P.Kuznetsov Dynamical chaos, 2001 (in Russian)


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updated 27 June 04