x' = a + x.
^{2} + by,
y' = x | (1) |

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

| 2x b | J = | | = -b. | 1 0 |The map is contracting for

They are real for

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)

Contents Previous: 2D invertible maps Next: Strange attractors