文档介绍：12 H´enon attractor
The chaotic phenomena of the Lorenz equations may be exhibited by even
simpler systems.
We now consider a discrete-time, 2-D mapping of the plane into itself. The
points in R2 are considered to be the the Poincar´e section of a flow in higher
dimensions, say, R3 .
The restriction that d > 2 for a strange attractor does not apply, because
maps generate discrete points; thus the flow is not restricted by continuity
(., lines of points need not be parallel).
The H´enon map
The discrete time, 2-D mapping of H´enon is
2
X = Y + 1 X
k+1 k − k
Yk+1 = λXk
controls the nonlinearity.
•
λ controls the dissipation.
•
Pictorially, we may consider a set of initial conditions given by an ellipse:
Y
X
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Now bend the elllipse, but preserve the area inside it (we shall soon quantify
area preservation):
Y’
Map T1 : X∗= X
2 X’
Y ∗= 1 X + Y
−
Next, contract in the x-direction ( λ< 1)
| |
Y’’
Map T2 : X∗∗= λX∗
X
Y ∗∗= Y ∗’’
Finally, reorient along the x axis (. flip across the diagonal).
Y’’’
Map T3 : X∗∗∗= Y ∗∗
X
Y ∗∗∗= X∗∗’’’
posite of these maps is
T = T T T .
3 ∝ 2 ∝ 1
We readily find that T is the H´enon map:
2
X∗∗∗= 1 X + Y
−
Y ∗∗∗= λX
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Dissipation
The rate of dissipation may be quantified directly from the mapping via the
Jacobian.