Hopf bifurcations and self-sustained oscillations

In a planar normal form, use a nonnegative radius r and phase θ. The angular equation makes the state rotate, while the radial equation determines whether its amplitude grows, decays, or settles. For $\dot r=\mu r-r^3$, the origin attracts below threshold; above threshold it repels and a circle at radius $\sqrt\mu$ attracts nearby radii. A trajectory on that circle continues moving. A limit cycle is a periodic orbit isolated from nearby periodic orbits, not a stationary state at every point on a ring.

This distinction separates self-sustained oscillation from the ideal harmonic oscillator. The latter has a different closed orbit for every amplitude and no radial attraction, so a small kick permanently changes the amplitude. An attracting limit cycle restores its selected amplitude after a small transverse kick. Displacing along the cycle changes phase instead, and that phase displacement need not decay in an autonomous system. A stable cycle thus has a neutral direction associated with time translation as well as contracting transverse directions.

A generic Hopf point requires a conjugate eigenvalue pair with nonzero imaginary part and a real part crossing zero at nonzero speed as the control parameter varies. Remaining eigenvalues must stay off the imaginary axis in the standard local setting. Radial stability here concerns the planar center dynamics; attraction in the full system additionally requires every remaining eigenvalue to have negative real part. The leading cubic amplitude coefficient, expressed by the first Lyapunov coefficient after reduction, must also be nonzero. Merely finding imaginary eigenvalues at one parameter is insufficient: the eigenvalues might touch without crossing, or the nonlinear amplitude coefficient might vanish.

The opposite cubic sign gives $\dot r=\mu r+r^3$. Here an unstable small cycle exists for negative μ around a locally stable origin, then shrinks into it as the origin loses stability. This is the local subcritical case. The formula predicts neither a stable large-amplitude oscillation nor its endpoint. Adding a stabilizing higher-order term can produce coexistence and hysteresis, but those outcomes require the extra global structure. Always state the radial sign convention when attaching the words supercritical and subcritical.

Near a supercritical onset the leading amplitude scales with the square root of distance from threshold, while the period remains finite because the crossing frequency is nonzero. That contrasts with some global bifurcations whose periods diverge. An amplitude-frequency plot can therefore help distinguish mechanisms, provided transients have settled. Close to threshold radial relaxation is slow, so a short record can make an ordinary transient look like a stable low-amplitude cycle or an abrupt jump.

For a physical oscillator, identify how power enters and leaves over one period. A periodic orbit requires zero net energy change over a cycle, but that balance alone does not prove stability: a displaced amplitude must experience a restoring imbalance. Averaging can derive an approximate amplitude equation when a slow envelope is separated from rapid oscillation. Its validity is local in amplitude and parameter and must be checked against the original dynamics. A phase portrait, amplitude equation, and energy account should tell the same qualified story.