5.2.2 Planar Andronov-Hopf bifurcation

5.2. ONE-PARAMETER LOCAL BIFURCATIONS IN ODES 141
ρ 1
α < 0
α > 0
α = 0
(0)
ρ (α)
0
ρ 0
Figure 5.9: Fixed points of the return map.
limit cycle of the system, we can conclude that system (5.20) (or equation (5.19))
with any O(|z| 4 ) terms has a unique (stable) limit cycle bifurcating from the origin
and existing for α > 0 as in equation (5.19). Therefore, in other words, higher-order
terms do not affect the limit cycle bifurcation in some neighborhood of z = 0 for |α|
sufficiently small.
Step 2 (Construction of a homeomorphism). The established existence and uniqueness
of the limit cycle is enough for all applications. Nevertheless, extra work must
be done to prove the topological equivalence of the phase portraits.
( x 1 , x 2) ( x
~
1 ,
~
x 2)
τ 0 τ 0
ρ 0 ρ 0
Figure 5.10: Construction of the homeomorphism near the Hopf bifurcation.
Fix α small but positive. The real systems corresponding to (5.17) and (5.19)
both have a unique limit cycle in some neighborhood of the origin. Assume that
the time reparametrization resulting in the constant return time 2π is performed
in (5.17) (see the previous step). Also, apply a linear scaling of the coordinates in
(5.17) such that the point of intersection of the cycle and the horizontal half-axis is
at x 1 = √ α.
Define a map z ↦→ ˜z by the following construction. Take a point z = x 1 + ix 2
and find values (ρ 0 , τ 0 ), where τ 0 is the minimal time required for an orbit of system
(5.19) to approach the point z starting from the horizontal half-axis with ρ = ρ 0 .
Now, take the point on this axis with ρ = ρ 0 and construct an orbit of (5.17) on the