Direction and Stability of the Hopf Bifurcation In Theorem 2, we obtained a condition under which system (2) undergoes Hopf bifurcation at τ₀; that is, a family of periodic solutions bifurcate from the positive steady state point (K_∗, L_∗) at the critical value τ₀.

In a differential equation a Hopf bifurcation typically occurs when a complex conjugate pair of eigenvalues of the linearised flow at a fixed point becomes purely imaginary. This implies that a Hopf bifurcation can only occur in systems of dimension two or higher.

A Hopf Bifurcation occurs when a periodic solution or limit cycle, surrounding an equilibrium point, arises or goes away as a parameter µ varies. When a stable limit cycle surrounds an unstable equilibrium point, the bifurcation is called a supercritical Hopf bifurcation.

You can check whether the bifurcation is Hopf or not by looking at the imaginary components of the dominant eigenvalues whose real parts are at a critical value (zero); if there are non-zero imaginary components, it must be a Hopf bifurcation. Re(λ)={r±√r2−42if r2≥4,orr2if r2<4.