Linear Algebra, Theory And Applications, 2012a

C.5. GEOMETRIC THEORY OF AUTONOMOUS SYSTEMS 431
where y 0 = x 0 − a.
Let A = Df (a) . Then from the definition of the derivative of a function,
y ′ = Ay + g (y) , y (0) = y 0 (3.32)
where
g (y)
lim = 0.
y→0 |y|
Thus there is never any loss of generality in considering only the equilibrium point 0 for an
almost linear system. 1 Therefore, from now on I will only consider the case of almost linear
systems and the equilibrium point 0.
Theorem C.5.5 Consider the almost linear system of equations,
x ′ = Ax + g (x) (3.33)
where
g (x)
lim = 0
x→0 |x|
and g is a C 1 function. Suppose that for all λ an eigenvalue of A, Re λ0andK such that for Φ (t) the
fundamental matrix for A,
|Φ(t) x| ≤Ke −δt |x| .
Let ε>0 be given and let r be small enough that Kr < ε and for |x| < (K +1)r, |g (x)| <
η |x| where η is so small that Kη < δ,andlet|y 0 |