Hopf bifurcation and limit cycles

For a sufficiently smooth one-parameter planar vector field, follow a smooth equilibrium branch p(mu). At the candidate value mu=0, suppose the Jacobian has eigenvalues plus and minus i omega zero with omega zero strictly positive. The real part alpha(mu) must cross zero with nonzero parameter derivative. Finally, the first Lyapunov coefficient, the leading cubic amplitude coefficient after normal-form reduction, must be nonzero. These are the standard local nondegeneracy conditions, stated here for a smooth family so the required derivatives exist.

The theorem then supplies a small branch of periodic orbits on one side of the crossing, with amplitude proportional to the square root of the appropriately oriented parameter to leading order. The coefficient sign determines radial stability and which side carries the branch. The theorem is local in state and parameter, and normal-form coordinates need not be physical Euclidean amplitude. In higher dimensions, other eigenvalues must stay off the imaginary axis; attraction of the cycle in the full space additionally requires the remaining transverse directions to contract.

Consider the exact polar model r dot = mu r−c r cubed and theta dot = omega zero, with c positive and omega zero positive. The corresponding Cartesian field is smooth at the origin. For positive mu, the nonzero stationary radius is the square root of mu divided by c. The radial derivative there is minus twice mu, so radial perturbations decay. The origin has changed from a stable focus for negative mu to an unstable focus for positive mu: this is the supercritical orientation of Hopf.

At mu=0, the cubic damping still attracts nonzero nearby radii algebraically, despite the purely imaginary linear pair. Reversing the cubic sign gives r dot = mu r+c r cubed. For negative mu it has an unstable cycle at square root of minus mu divided by c, surrounding a stable origin. That is the subcritical local branch. The cubic model alone does not create a distant stable cycle or prove hysteresis; an additional saturation term and a global branch analysis would be needed for those claims.

A damped linear oscillator can oscillate while its amplitude tends to zero; it has no nontrivial periodic attractor. An undamped linear oscillator has a whole family of closed ellipses, so none is isolated. Letting linear damping cross zero changes stability, but without a nonlinear amplitude coefficient the generic Hopf theorem's nondegeneracy hypothesis fails. This counterexample explains why the linear spectrum identifies a candidate transition rather than supplying the periodic-orbit conclusion on its own.

A numerical continuation study can track equilibria, locate eigenvalue crossings, and continue periodic orbits, but it must resolve small amplitudes near the transition and distinguish transient oscillation from recurrence. The measured radius may depend on the chosen coordinates, while the existence and stability of an isolated orbit are coordinate-invariant properties. Close to a supercritical transition, the radial relaxation rate tends to zero, so a short simulation can look settled before its amplitude has reached the cycle. Predict the slow rate from the normal form before choosing an observation time.