5.2.2 Planar Andronov-Hopf bifurcation

5.2. ONE-PARAMETER LOCAL BIFURCATIONS IN ODES 147
One can also try to eliminate the w 2 ¯w-term by formally setting
h 21 = g 21
λ + ¯λ .
This is possible for small α ≠ 0, but the denominator vanishes at α = 0: λ(0) +
¯λ(0) = iω 0 − iω 0 = 0. To obtain a transformation that is smoothly dependent on α,
set h 21 = 0, which results in
c 1 = g 21
2 . ✷
Remarks:
(1) The remaining cubic w 2 ¯w-term is called a resonant term. Note that its
coefficient is the same as the coefficient of the cubic term z 2¯z in the original equation
in Lemma 5.3
(2) We leave to the reader a proof of Lemma 5.3 in real notations. ♦
Lemma 5.4 (Poincaré normalization) The equation
ż = λ(α)z +
∑
2≤k+l≤3
1
k!l! g kl(α)z k¯z l + O(|z| 4 ), (5.32)
where λ(α) = µ(α) + iω(α), µ(0) = 0, ω(0) = ω 0 > 0, can be transformed by an
invertible parameter-dependent change of complex coordinate, smoothly depending
on the parameter,
z = w + h 20(α)
w 2 + h 11 (α)w ¯w + h 02(α)
¯w 2
2
2
+ h 30(α)
w 3 + h 12(α)
w ¯w 2 + h 03(α)
¯w 3 + O(|w| 4 ),
6 2
6
for all sufficiently small |α|, into an equation with only one cubic term:
ẇ = λ(α)w + c 1 (α)w 2 ¯w + O(|w| 4 ), (5.33)
where
c 1 (0) =
i (
g 20 (0)g 11 (0) − 2|g 11 (0)| 2 − 1 )
2ω 0 3 |g 02(0)| 2
+ g 21(0)
. (5.34)
2
Proof: The superposition of the transformations defined by Lemmas 5.2 and 5.3
does the job. First, perform the transformation
z = w + h 20
2 w2 + h 11 w ¯w + h 02
2 ¯w2 , (5.35)
with
h 20 = g 20
λ , h 11 = g 11
¯λ , h 02 = g 02
2¯λ − λ ,
defined in Lemma 5.2. This annihilates all the quadratic terms but also changes the
coefficients of the cubic terms. The coefficient of w 2 ¯w will be 1 ˜g 2 21, say, instead of