x' = a + x2 + by,
y' = x.
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.
| 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).
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.|
 M.Henon "A two-dimensional mapping with a strange attractor"
Comm.Math.Phys. 50, 69 (1976).
 D.G. Sterling, H.R. Dullin, J.D. Meiss Homoclinic Bifurcations for the Henon Map arxiv.org/abs/chao-dyn/9904019
 H.R.Dullin, J.D.Meiss Generalized Henon Maps: the Cubic Difeomorphisms of the Plane
 Predrag Cvitanovic, Gemunu H. Gunaratne, Itamar Procaccia Topological and metric properties of Henon-type strange attractors Phys. Rev. A 38, 1503-1520 (1988) abstract
 Kai T. Hansenyx and Predrag Cvitanovic "Bifurcation structures in maps of Henon type" Nonlinearity 11 (1998) 1233-1261.
 Michael Benedicks, Marcelo Viana "Solution of the basin problem for Henon-like attractors" Invent. math. 143, 375-434 (2001)
 S.P.Kuznetsov Dynamical chaos, 2001 (in Russian)