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x
=
−
y
−
z
y
=
x
+
ay
z
=
b
+
z
(
x
−
c
)
.
Both the Lorenz and Rossler equations are involved with the study of
Navier-Stokes equations. Rossler is acclaimed to have used the parameter values
a
7, which we will also use. This system of equations
looks easier than the Lorenz system, with only one nonlinearity
xz
, but it is harder
to analyze. Figure 16.2b shows the Rossler attractor.
=
0
.
2,
b
=
0
.
2, and
c
=
5
.
16.4.3 Henon Map
The Henon map was devised by the theoretical astronomer Michel Henon to illu-
minate the microstructure of strange attractors in 1976. Previous scientists had en-
countered numerical difficulties when tackling the Lorenz system, so instead Henon
sought a mapping that captured its essential features but which also had an ad-
justable amount of dissipation. Henon chose to study mappings rather than dif-
ferential equations because maps are faster to simulate and their solutions can be
followed more accurately and for a longer time. The Henon map is given by
ax
n
x
n
+
1
=
y
n
+
1
−
y
n
+
1
=
bx
n
,
where
a
and
b
are adjustable parameters which are chosen as
a
=
1
.
4,
b
=
0
.
3.
16.4.4 The Henon-Heiles Equations
The Henon-Heiles model was introduced in 1964 by Michel Henon and Carl Heiles
as a model for the motion of a star inside a galaxy. With the Hamiltonian
2
p
1
+
q
2
1
1
3
q
2
,
q
1
+
p
2
+
q
1
q
2
=
+
−
H
p
y
, then the Hamiltonian can
be interpreted as a model for a single particle moving in two dimensions under the
action of a force described by a potential energy function
V
and if we let
q
1
=
x
,
q
2
=
y
,
p
1
=
p
x
, and
p
2
=
[9]. Hamilton's
equations for this system lead to the following equations for the dynamics of the
system:
(
x
,
y
)
=
∂
H
p
x
=
−
∂
H
x
p
x
=
p
x
x
=
−
x
−
2
xy
∂
∂
=
∂
H
p
y
=
−
∂
H
x
2
y
2
y
p
y
=
p
y
y
=
−
y
−
+
.
∂
∂
