Bifurcation

Hopf bifurcation

The Hopf bifurcation is a 2D analogue to the Pitchfork bifurcation. It is most easily represented in Polar coördinates, in which case the radial equation is precisely that of a Pitchfork bifurcation. For the supercritical Hopf bifurcation case¹

which in cartesian coördinates becomes

Thus

• For 𝜇 <0 there exists a single stable focus at 𝐱∗ =(0,0)
• For 𝜇 =0 there exists a nonlinear centre at 𝐱∗ =(0,0)
• For 𝜇 >0 there exists a stable orbit at 𝑟 =√𝜇 enclosing an unstable focus 𝐱∗ =(0,0)

Visually, a stable focus ejects a stable orbit around itself and becomes stable, In the subcritical Hopf bifurcation, the roles are swapped: An unstable orbit contracts around a stable focus to produce an unstable focus.

Hopf bifurcation theorem

Let ˙𝑥 =𝑓(𝑥,𝑦,𝜇) and ˙𝑦 =𝑔(𝑥,𝑦,𝜇) where 𝑓,𝑔 are thrice-differentiable in 𝑥,𝑦 and once-differentiable in 𝜇. Let (𝑥∗(𝜇),𝑦∗(𝜇)) be a critical point with eigenvalues 𝛼(𝜇) ±𝑖𝜔(𝜇). If there exists a 𝜇∗ such that

• 𝛼(𝜇∗) =0
• 𝛼′(𝜇∗) ≠0
• 𝑄(𝑥∗,𝑦∗,𝜇∗) ≠0

where

then there is a Hopf bifurcation at 𝜇∗, which is supercitical for 𝑄 <0 and subcritical for 𝑄 >0. #m/thm/dynamics/flow

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