differential equations
238 .
B ,
B ,
,
C C
∈ ∈
− −
=
∫∫
−
ϕ ψ
ξ ξ
ϕ θ
η θ
ξ ψ
ϕ ψ
ϕ ψ
θ α
r
d d
In [4]
a projection
{ }
α λ
π
2 ,
1
0C
B :
M →
, is defined by
[ ]
α ϕ
α α
α ϕ
π ,
Ψ +
Ψ Φ
= + X
. With this projection the space
0C
B is decomposed as
{ }
π
α λ
Ker B
2 ,
1
0C
⊕ = M
. Since the solution
t
x of 8 and 9
belongs to ],
, [
1
n
R r
C −
, it is decomposed as t
y t
u x
t
+ Φ
= α
, 10 with
t
x t
u ,
α Ψ
= and
t
x I
t y
π −
= , where
,
_
t z
t z
t u
= -column
vector,
C ∈
t z
. Let us define
=
1 1
α λ
α λ
α
B
. The projection of eq. 8 on
{ }
α λ
2 ,
1
M is
, α
α α
α α
α α
t y
t u
F u
B u
+ Φ
Ψ Φ
+ Φ
= Φ
•
, and since
α
Φ
is invertible, this is equivalent to ,
α α
α α
t y
t u
F u
B u
+ Φ
Ψ +
=
•
. 11 By projecting the initial condition we find
ϕ α,
Ψ =
u .
3. Existence of the invariant manifold and the restricted equation If
{ }
\ α
α U ∈
, and
Re
2 ,
1
α λ
then, since these are the only two eigenvalues with positive real part and they are simple, there is a local
invariant manifold for the problem, namely the local unstable manifold, tangent to the space
{ }
α λ
2 ,
1
M , [3], [7].
For α
α = , since
Re
2 ,
1
=
α λ
, there is a local invariant manifold for the problem, namely the local center manifold, tangent to the space
{ }
2 ,
1
α λ
M ,
[7]. Hence, for every
U ∈
α with
≥ α
µ , there is a neighborhood
α V
of
∈
B , and a local invariant manifold α
α V W
loc
⊂ , which is the
differential equations
239 graph of a
1
C
function. That is, the local invariant manifold may be expressed as
{ }
{ }
α ϕ
ϕ ϕ
α
α λ
α
V w
W
loc
∩ ∈
+ =
2 ,
1
; M
, where
{ }
π
α λ
α
Ker w
→
2 ,
1
: M is a
1
C
function, ,
=
α
w and it has zero
differential at 0. Since
{ }
α λ
ϕ
2 ,
1
M ∈
, we have ,
1 1
α ϕ
α ϕ
α ϕ
z z
+ =
with
C ∈
z
. This relation induces a dependence of
z z
,
to ϕ
α
w that justifies the
notation z
z w
w ,
α α
ϕ = .
Equation 11 implies α
α ϕ
α ϕ
α ψ
α λ
α
, ,
1 1
1 1
t z
t z
w t
z t
z F
t z
t z
+ +
+ =
•
, 12
ϕ α
ψ ,
1
= z
. 13 Let
φ
α
t S
be the solution of eq. 1 corresponding to the initial condition φ , at the moment t .
If α
φ
loc
W ∈
, then ,
1 1
t z
t z
w t
z t
z t
S
α α
α ϕ
α ϕ
φ +
+ =
. 14 By using again the function f, 12 becomes
, ,
1 1
α φ
φ α
ψ α
λ
α α
r t
S t
S f
t z
t z
− +
=
•
, 15
or, ,
, ,
1
α α
λ t
z t
z g
t z
t z
+ =
•
16 by denoting
, ,
, ,
1
α φ
φ α
ψ α
α α
r t
S t
S f
t z
t z
g −
= . 17
4. The equations for the invariant manifold
The following proposition is a natural consequence of the invariance of α
loc
W . A similar result is given in [7], on the center manifold. We give the
proof for the sake of completeness. Proposition 1. Let
α φ
loc
W ∈
be the initial value for the problem 1. Then the function
α
w satisfies the following equations
differential equations
240
[ ]
, ,
, ,
, ,
, ,
,
1 1
r s
s t
z t
z w
s s
t z
t z
g s
t z
t z
g s
t z
t z
w t
− ∈
∂ ∂
= =
+ +
∂ ∂
α α
α ϕ
α α
ϕ α
18
, ,
, ,
, ,
, ,
,
2 1
α φ
φ α
α α
ϕ α
α ϕ
α
α α
α α
α
r t
S t
S f
r t
z t
z w
B t
z t
z w
A t
z t
z g
t z
t z
g t
z t
z w
t −
+ −
+ =
= +
+ ∂
∂
19 with zt solution of the Cauchy problem
16,13 and g defined by 17.
Proof Since
α φ
loc
W ∈
and α
loc
W is invariant,
α φ
α loc
W t
S ∈
. Let us denote, for
≥ t
and
[ ]
, r
s −
∈ ,
s t
x s
t S
+ =
φ φ
α
. Obviously
s t
s x
s t
t x
+ ∂
∂ =
+ ∂
∂
φ φ
. This and 14 imply
, ,
1 1
1 1
s t
z s
t z
s t
z t
z w
s s
t z
s t
z s
t z
t z
w t
α ϕ
α ϕ
α ϕ
α ϕ
α α
• •
• •
+ +
∂ ∂
= =
+ +
∂ ∂
20
here
1 1
s ds
d s
α ϕ
α ϕ
=
•
and thus s
t z
t z
w s
s t
z s
t z
s t
z t
z w
t ,
,
_ 1
1 1
1 _
α α
α ϕ
α λ
α ϕ
α λ
∂ ∂
=
−
+
−
+ ∂
∂
• •
. 21 With 16 we obtain 18.
On another side, since φ
α
t S
is a solution of equation 1, we have
, ,
,
_ 1
1
s r
t S
s t
S f
s r
t S
B s
t S
A s
t z
t z
w t
s t
z s
t z
φ φ
φ α
φ α
α ϕ
α ϕ
α α
α α
α
− +
+ −
+ =
∂ ∂
+ +
• •
differential equations
241 and, by taking
= s
, we obtain 19. This proposition allows the determination of the coefficients of the series of
powers in z and
_
z of the function
α
w . Indeed, let us write
, 1
, ,
2 k
j k
j jk
z z
F k
j r
t S
t S
f
∑
≥ +
= −
α α
φ φ
α α
22
k j
k j
jk
z z
g k
j t
z t
z g
∑
≥ +
=
2
1 ,
,
α α
,
k j
k j
jk
z z
w k
j z
z w
∑
≥ +
=
2
, 1
, ,
α θ
θ
α
23 where
1
α ψ
α
jk jk
F g
=
. By replacing 22 and 23 in 18 and by matching the obtained series, we
get first order linear differential equations for
jk
w
. Thus, equation 18 implies
. ,
1 1
1 ,
1
1 1
2 1
2 1
2 2
+ +
+ +
=
• −
• −
≥ +
≥ +
≥ +
≥ +
∑ ∑
∑ ∑
z z
kz z
z jz
s w
k j
s z
z g
k j
s z
z g
k j
z z
s w
ds d
k j
k j
k j
k j
jk k
j k
j jk
k j
k j
jk k
j k
j jk
α α
ϕ α
α ϕ
α α
24 In this equality
•
z and
•
z will be replaced with the right hand side of 16 to
obtain
. 1
1 ,
1 ,
1 1
1 ,
1
2 _
1 2
1 2
1 1
2 1
2 1
2 2
+
+
+ +
+
+
+
=
∑ ∑
∑ ∑
∑ ∑
∑
≥ +
− ≥
+ −
≥ +
≥ +
≥ +
≥ +
≥ +
m l
m l
lm k
j m
l m
l lm
k j
k j
jk k
j k
j jk
k j
k j
jk k
j k
j jk
k j
k j
jk
z z
g m
l z
kz z
z g
m l
z jz
s w
k j
z z
k j
s w
k j
s z
z g
k j
s z
z g
k j
z z
s w
ds d
k j
α α
α α
λ α
λ α
α ϕ
α α
ϕ α
α
25 By matching the same order terms we obtain first order differential
equations for
, .
α
jk
w
. A relation similar to 25 is obtained by substituting the series 22, 23 in
19, and by using 16 :
differential equations
242 .
1 ,
1 ,
1 1
1 1
1 ,
1 ,
1
2 2
2 1
2 1
2 2
_ 1
2 1
2 1
1 2
k j
k j
jk k
j k
j jk
k j
k j
jk k
j k
j jk
k j
k j
jk m
l m
l lm
k j
m l
m l
lm k
j k
j jk
k j
k j
jk
z z
F k
j z
z r
w k
j B
z z
w k
j A
z z
g k
j z
z g
k j
z z
g m
l z
kz z
z g
m l
z jz
w k
j z
z k
j w
k j
∑ ∑
∑ ∑
∑ ∑
∑ ∑
∑
≥ +
≥ +
≥ +
≥ +
≥ +
≥ +
− ≥
+ −
≥ +
≥ +
+ −
+ =
=
+
+ +
+
+
+ +
α α
α α
α α
ϕ α
α ϕ
α α
α α
α λ
α λ
α
26 The relations obtained by equating the terms with similar powers of z,
_
z in
26 are used as conditions to determine the constants that appear in the general form of
jk
w
obtained above. In this theoretical form the calculus is very lengthy and we do not make it
explicitly here. We firstly remark that the coefficients of the second order terms in z and
_
z in the expansion of
, ,
α φ
φ
α α
r t
S t
S f
i
− ,
n i
,..., 1
= , are independent
on those of
α
w , they depend only on the coefficients of the Taylor series of
α ,
, y x
f
i
. The similar assertion holds for the coefficients of
, ,
α
t z
t z
g
. Hence
α α
α
02 11
20
, ,
g g
g are known, given
i
f and
α ψ
1
. The following algorithm to determine
α
jk
w
must be used. -
α α
α
02 11
20
, ,
w w
w are determined from the equations obtained by
identifying the terms containing
2 2
, ,
z z
z z
respectively, in 25, with initial conditions obtained by the same method from 26; they depend
on α
α α
02 11
20
, ,
g g
g .
- α
α α
02 11
20
, ,
w w
w are used to compute
α
jk
g
, j+k=3, from 14, 17, 23 .
- α
jk
w
, j+k=3, are determined from the equations 25 and
conditions 26; they depend on
3 ,
≤ + k
j g
jk
α .
differential equations
243 -
3 ,
≤ + k
j w
jk
α are used to compute
α
jk
g
, j+k=4 from 14, 17, 23 .
- α
jk
w
, j+k=4, are determined from the equations 25 and
conditions 26; they depend on
4 ,
≤ + k
j g
jk
α .
-
4 ,
≤ + k
j w
jk
α are used to compute
α
jk
g
, j+k=5. We do not need so many terms to have an accurate form of the invariant
manifold, but we need them in order to discuss the behaviour of the solution z of 16 that determines the solution of 1 on the invariant manifold.
5. The Bautin bifurcation
