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 there exists a single stable focus at
• For there exists a nonlinear centre at
• For there exists a stable orbit at enclosing an unstable focus

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 and where are thrice-differentiable in and once-differentiable in . Let be a critical point with eigenvalues . If there exists a such that

where

then there is a Hopf bifurcation at , which is supercitical for and subcritical for . #m/thm/dynamics/flow

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