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Past exam of the mathematics course of the University of Cambridge
/
2024
/
ii
/
Paper 2
/
33A
/
b
/
iii
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Past exam of the mathematics course of the University of Cambridge
2024
ii
Paper 2
33A
b
Created
2026-09-24
Updated
2026-09-24
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Solution
iii
Solution
0
0
0
iii
With
x
=
v
+
w
and
y
=
v
−
w
, the
equations
at
r
=
1
become
v
˙
=
2
1
{
a
(
v
+
w
)
3
−
z
(
v
−
w
)}
,
(1)
w
˙
=
−
2
w
+
2
1
{
a
(
v
+
w
)
3
+
z
(
v
−
w
)}
,
z
˙
=
−
z
+
v
2
−
w
2
.
(2)
The
symmetry
(
v
,
w
,
z
)
↦
(
−
v
,
−
w
,
z
)
suggests
w
c
(
v
)
=
c
3
v
3
+
O
(
v
5
)
,
z
c
(
v
)
=
d
2
v
2
+
d
4
v
4
+
O
(
v
6
)
.
(3)
Substitute these into the invariance
equations
w
c
′
(
v
)
v
˙
=
w
˙
,
z
c
′
(
v
)
v
˙
=
z
˙
.
(4)
Comparison at orders
v
2
,
v
3
,
v
4
gives
d
2
=
1
,
c
3
=
4
a
+
1
,
d
4
=
1
−
a
.
(5)
Thus the
center manifold of the 2024 Cambridge cubic system
is
w
c
(
v
)
=
4
a
+
1
v
3
+
O
(
v
5
)
,
z
c
(
v
)
=
v
2
+
(
1
−
a
)
v
4
+
O
(
v
6
)
.
(6)
Ancestors
(11)
b
33A
Paper 2
ii
2024
Past exam of the mathematics course of the University of Cambridge
Mathematics course of the University of Cambridge
Course of the University of Cambridge
University of Cambridge
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