A Hopf bifurcation occurs when a complex-conjugate pair of eigenvalues crosses the imaginary axis and a periodic orbit is created or destroyed near the equilibrium.
The complex normal form near a simple Hopf bifurcation is . The sign of determines the critical cubic radial drift: negative gives supercritical saturation and decay at the critical parameter, while positive gives destabilizing drift.
In the radial normal forman unstable periodic orbit exists for and shrinks into the equilibrium as . This is a subcritical Hopf bifurcation.
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Hopf bifurcation is a critical phenomenon in dynamical systems that occurs when a system's stability changes, leading to the emergence of oscillatory behavior. It specifically involves the transition from a stable equilibrium point to a stable limit cycle or periodic orbit as certain system parameters are varied. To understand Hopf bifurcation in more detail, consider a dynamical system described by ordinary differential equations.