ME201 Advanced Dynamics (Fall 2007)
HW5 Solution (
100 pts
)
1.
30 pts
Consider the standard map studied in class.
I
0
=
I
+
δ
sin(2
πφ
)
φ
0
=
φ
+
I
+
δsin
(2
πφ
)
Study the change of the phase space of the standard map as
δ
changes from 0 to 2
π
. Speciﬁcally,
plot the cases of
δ
=
{
0
.
2
/
2
π,
0
.
6
/
2
π,
1
/
2
π,
2
π
}
Describe your investigation.
In the following ﬁgure we have iterations of the standard map with varying
δ
. We ﬁnd that as
δ
increases, the existence of new resonant orbits arise. Orbits of diﬀerent period arise which are
surrounded by islands and KAM curves. In between these curves and the islands exist zones of chaos
or instability. Note that the ﬁgures look diﬀerent based on initial conditions chosen (i.e. a rational
or irrational grid of points). Here we have chosen random initial conditions.
Figure 1: Standard map with increasing
δ
2.
40 pts
Find the criterion for existence of 1/2 period orbit of the standard map (hint: study the
second iterate of the map). Compare your analytical criterion with what you see in the numerical
phase portrait for small
δ <
1
/
2
π
. What happens close to that trajectory as
δ
is increased?
ME201 Advanced Dynamics
1
HW5 Solution
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View Full DocumentYou may consider the standard map as a function of the form:
I
(
k
+ 1) =
f
(
I
(
k
)
,φ
(
k
))
φ
(
k
+ 1) =
g
(
I
(
k
)
,φ
(
k
))
We are looking for conditions in which
I
(
k
+ 2) =
I
(
k
),
φ
(
k
+ 2) =
φ
(
k
), this is the 1/2 period
orbit. Note also that this will occur at all points which are the ﬁrst period orbits:
I
(
k
+ 1) =
I
(
k
),
φ
(
k
+1) =
φ
(
k
) so we need to identify these ﬁrst (it is assumed in the problem that we are interested
in the orbits that are strictly 1/2 periodic). The conditions for a single period orbit are:
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 Fall '08
 Mezic,I
 Trajectory, Existence, Celestial mechanics, Iterated function, period orbit

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