where A_{−} and A_{+} are square matrices of size k × k and
(n − k)
×
(n − k)
respectively. Also
assume A_{−} has eigenvalues whose real parts are all less than −α while A_{+} has eigenvalues
whose real parts are all larger than α. Assume also that each of A_{−} and A_{+} is upper
triangular.
Also, I will use the following convention. For v ∈ F^{n},
( )
v−
v = v+
where v_{−} consists of the first k entries of v.
Then from Theorem D.5.3 and Lemma D.6.1 the following lemma is obtained.
Lemma D.7.1Let A be of the form given in 4.33as explained above and let Φ_{+}
(t)
and Φ_{−}
(t)
bethe fundamental matrices corresponding to A_{+}and A_{−}respectively. Then there exist positiveconstants, α and γ such that
|Φ+ (t)y| ≤ Ce αt for all t < 0 (4.34)
(4.34)
|Φ − (t)y| ≤ Ce− (α+γ)t for all t > 0. (4.35)
(4.35)
Also for any nonzero x ∈ ℂ^{n−k},
|Φ+ (t)x| is unbounded. (4.36)
(4.36)
Proof:The first two claims have been established already. It suffices to pick α and γ such
that −
(α +γ )
is larger than all eigenvalues of A_{−} and α is smaller than all eigenvalues of A_{+}. It
remains to verify 4.36. From the Putzer formula for Φ_{+}
(t)
,
n−1
Φ (t)x = ∑ r (t)P (A)x
+ k=0 k+1 k
where P_{0}
(A )
≡ I. Now each r_{k} is a polynomial (possibly a constant) times an exponential. This
follows easily from the definition of the r_{k} as solutions of the differential equations
r′k+1 = λk+1rk+1 + rk.
Now by assumption the eigenvalues have positive real parts so
It follows from Lemma 13.6.4, for each a_{−} such that
|a− |
<
δ-
2C
, there exists a unique solution to
4.38 in E_{γ}.
As pointed out earlier, if
∫
ψ (a) ≡ − ∞ Φ (− s)g (x(s,a))ds
0 + +
then for x
(t,x0)
the solution to the initial value problem
x′ = Ax + g (x),x(0) = x0
has the property that if x_{0} is not of the form
( )
a−
ψ (a− )
, then
|x(t,x0)|
cannot be less than δ
for all t > 0.
On the other hand, if x_{0} =
( a )
−
ψ (a− )
for
|a− |
<
2δC
, then x
(t,x0)
,the solution to 4.38 is
the unique solution to the initial value problem
′
x = Ax + g (x),x (0) = x0.
and it was shown that
||x(⋅,x0)||
_{γ}< δ and so in fact,
−γt
|x(t,x0)| ≤ δe
showing that
tli→m∞ x (t,x0) = 0.
■
The following theorem is the main result. It involves a use of linear algebra and the above
lemma.
Theorem D.7.3Consider the initial value problem for the almost linear system
′
x = Ax +g (x),x(0) = x0
in which g is C^{1}and where at there are k < n eigenvalues of A which have negative real parts andn − k eigenvalues of A which have positive real parts. Then 0 is not stable. More precisely, thereexists a set of points
(a,ψ (a))
for a small and in a k dimensional subspace such that for x_{0}onthis set,
lt→im∞ x(t,x0) = 0
and for x_{0}not on this set, there exists a δ > 0 such that
|x (t,x )|
0
cannot remain less than δ forall positive t.
Proof: This involves nothing more than a reduction to the situation of Lemma D.7.2. From
Theorem 9.5.2 on Page 9.5.2A is similar to a matrix of the form described in Lemma D.7.2. Thus
A = S^{−1}
( A 0 )
−
0 A+
S. Letting y = Sx, it follows
( )
′ A − 0 ( − 1 )
y = 0 A+ y + g S y
Now
|x |
=
|| −1 ||
S Sx
≤
|||| − 1||||
S
|y|
and
|y|
=
|| −1 ||
SS y
≤
||S ||
|x|
. Therefore,
--1- |||| − 1||||
||S || |y| ≤ |x| ≤ S |y|.
It follows all conclusions of Lemma D.7.2 are valid for this theorem. ■
The set of points
(a,ψ (a))
for a small is called the stable manifold. Much more can be said
about the stable manifold and you should look at a good differential equations book for
this.