Linear Algebra, Theory And Applications, 2012a

C.7. THESTABLEMANIFOLD 435
Lemma C.7.2 Consider the initial value problem for the almost linear system
x ′ = Ax + g (x) , x (0) = x 0 ,
where g is C 1 and A is of the special form
( )
A− 0
A =
0 A +
in which A − is a k × k matrix which has eigenvalues for which the real parts are all negative
and A + is a (n − k) × (n − k) matrix for which the real parts of all the eigenvalues are
positive. Then 0 is not stable. More precisely, there exists a set of points (a − , ψ (a − )) for
a − small such that for x 0 on this set,
lim
t→∞ x (t, x 0)=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 for all positive t.
Proof: Consider the initial value problem for the almost linear equation,
( )
x ′ a−
= Ax + g (x) , x (0) = a = .
a +
Then by the variation of constants formula, a local solution has the form
( )( )
Φ− (t) 0 a−
x (t, a) =
0 Φ + (t) a +
∫ t
( )
Φ− (t − s) 0
+
g (x (s, a)) ds (3.38)
0 Φ + (t − s)
0
Write x (t) forx (t, a) for short. Let ε>0 be given and suppose δ is such that if |x|