Hopf Bifurcation

From decay to sustained oscillation

Choose μ and an initial x₀,y₀. Before Run the views show the initial point and equilibrium branches. Run traces the motion; Pause, One step (0.02 model units) and PNG inspect it. μ changes apply to the current point, including when paused: the trajectory and time remain. Changing x₀ or y₀, or Clear trace, returns to a new initial-point preview. The slider is convenient; the number field accepts finer values and decimal comma.

Read the three views

The phase portrait uses equal x/y scales and retains its scale as the orbit decays. The orange circle is the stable limit cycle for the current μ, not a simulated orbit. In the bifurcation diagram, solid branches are stable, the dashed zero branch is unstable for μ>0. Blue is the actual transient amplitude and is never clipped to √2. The radius history retains recent samples and the target belonging to each past μ; changing μ does not relabel old history.

Model and exact solution

Supercritical normal form: ẋ=μx−y−x(x²+y²), ẏ=x+μy−y(x²+y²). In polar coordinates ṙ=μr−r³ and θ̇=1. μ<0 gives a stable focus; μ=0 is a weak, nonhyperbolic origin with algebraic decay r=r₀/√(1+2r₀²t). μ>0 gives an unstable origin and a stable cycle r*=√μ. An exact zero start stays zero; a small nonzero start must not be called that equilibrium. For constant μ≠0, 1/r²=exp(−2μt)/r₀²+(1−exp(−2μt))/μ, and θ=θ₀+t.

The browser evaluates this closed solution between μ changes, using logarithms and expm1 to retain very small radii and values near μ=0. Time is dimensionless, with period 2π; it is not a measured oscillator or an arbitrary system’s bifurcation solver. Fixed 0.02 sampling avoids frame-rate dependent traces. Playback targets one model unit/s and can slow under load; hidden tools suspend it. The last 1200 samples are kept. PNG pauses and records the displayed state; current μ and historical targets are distinct. Numerical display can underflow before the stored log-radius does.

Andronov–Hopf bifurcation (Scholarpedia)

Keep an experiment

An invalid input pauses playback and keeps the last valid experiment. Imported sessions are bounded and checked; a failed import keeps the existing experiment. Sessions keep at most 1200 trajectory points and 1200 history samples, not the complete elapsed trajectory.

Only input values are remembered in this browser. Calculated results and experiment histories are not saved.