Hopf Bifurcation

Hopf bifurcation — normal form ẋ = μx − y − xr², ẏ = x + μy − yr². For μ ≤ 0 the origin is a stable focus. For μ > 0 the origin is unstable and a stable limit cycle of radius r* = √μ appears.

Starting exactly at the origin remains there mathematically. Any nonzero start is a transient, not an equilibrium, until it approaches the dashed limit cycle. Run integrates with RK4 using time steps refined to the elapsed frame time.