5.2.2 Planar Andronov-Hopf bifurcation

146 CHAPTER 5. LOCAL BIFURCATION THEORY
The matrix in the left-hand-side is nonsingular near Hopf bifurcation, since its determinant
is equal to
(9ω 2 + µ 2 )(ω 2 + µ 2 ) 2 ,
which reduces to 9ω0 6 > 0 at α = 0. Thus, for any given quadratic coefficients
a ij , b ij in (5.29), one can find from (5.31) g ij , h ij with i, j = 0, 1, 2, i + j = 2, such
that system (5.30) will have no quadratic terms for all sufficiently small |α|. This is
equivalent to Lemma 5.2. Clearly, the complex notation is simpler. ♦
Lemma 5.3 The equation
ż = λz + g 30
6 z3 + g 21
2 z2¯z + g 12
2 z¯z2 + g 03
6 ¯z3 + O(|z| 4 ),
where λ = λ(α) = µ(α) + iω(α), µ(0) = 0, ω(0) = ω 0 > 0, and g ij = g ij (α), can be
transformed by an invertible parameter-dependent change of complex coordinate
z = w + h 30
6 w3 + h 21
2 w2 ¯w + h 12
2 w ¯w2 + h 03
6 ¯w3 ,
for all sufficiently small |α|, into an equation with only one cubic term:
where c 1 = c 1 (α).
Proof:
The inverse transformation is
ẇ = λw + c 1 w 2 ¯w + O(|w| 4 ),
Therefore,
w = z − h 30
6 z3 − h 21
2 z2¯z − h 12
2 z¯z2 − h 03
6 ¯z3 + O(|z| 4 ).
ẇ = ż − h 30
2 z2 ż − h 21
2 (2z¯zż + z2 ˙¯z) − h 12
2 (ż¯z2 + 2z¯z ˙¯z) − h 03
2 ¯z2 ˙¯z + · · ·
(
g30
= λz +
6 − λh ) (
30
z 3 g21
+
2 2 − λh 21 − ¯λh )
21
z 2¯z 2
+
(
g12
2 − λh )
12
−
2
¯λh 12 z¯z 2 +
(
g03
6 − ¯λh 03
2
)
¯z 3 + · · ·
= λw + 1 6 (g 30 − 2λh 30 )w 3 + 1 2 (g 21 − (λ + ¯λ)h 21 )w 2 ¯w
+ 1 2 (g 12 − 2¯λh 12 )w ¯w 2 + 1 6 (g 03 + (λ − 3¯λ)h 03 ) ¯w 3 + O(|w| 4 ).
Thus, by setting
h 30 = g 30
2λ , h 12 = g 12
2¯λ , h 03 = g 03
3¯λ − λ ,
we can annihilate all cubic terms in the resulting equation except the w 2 ¯w -term,
which we have to treat separately. The substitutions are valid since all the involved
denominators are nonzero for all sufficiently small |α|.