Henon map
2D Henon map is
|
|
x' = a + x2 + by,
y' = x.
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Unfortunately they use different parametrizations for the map.
E.g. we will get -x2 term after substitution (a,b,x,y)
→ (-a,b,-x,-y). It is evident that inverse Henon map is
x = y',
y = (x' - y' 2 - a)/b.
For small perturbation (δx, δy) of the point
(x, y) corresponding final deviation is
|
( |
|
)
=
( |
|
∂ x'/∂ x | ∂ x'/∂ y
| |
∂ y'/∂ x | ∂ y'/∂ y
|
|
)( |
|
)
=
( |
|
)( |
|
)
= J ( |
|
).
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Points of a small circle around (x, y) are mapped into an ellipse
around (x', y'). E.g. for real eigenvalues of the matrix J
λ1,2 = x ± (x2 +
b)1/2
principal axis of this ellipse coinside with eigenvectors of the matrix
and deformation of the initial circle is determined by the
λ1,2 values. Thus for a = 1.4 and b = 0.3
the fixed point x2 = y2 = -0.884 is unstable
with λ1 = 0.156 and λ2 = -1.92 .
Jacobian of the Henon map
|
Det(J) =
| |
|
| =
λ1 λ2 = -b.
|
The map is contracting for |b| < 1. All bounded attracting orbits
are located in this region and attractors have zero measure. Under iterations
of the map ellipses become narrower and elongated. For n-th iterations
|
(
|
|
) =
( |
|
)( |
|
) = [
∏i=0,n-1
( |
|
)]( |
|
).
|
As since one eigenvalue of the matrix product grows and the other decreases
therefore it is rather difficult to compute both values accurately (due to
roundoff errors). But one can replace the big eigenvalue by Sp(J) =
λ1 + λ2 ≈ λmax
with good accuracy. Then the biggest Lyapunov exponent is approximately
Lmax = 1/n ln|λmax|
≈ 1/n ln|Sp(J)|.
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.
Attractors
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).
Two more dynamic Henon fractals.
[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)
Contents
Previous: Baker's map
Next: Structure of the parameter space of
the Henon map
updated 14 Nov 06