Hopf Bifurcation — Brusselator

The Brusselator exhibits a Hopf bifurcation as B crosses 1 + A²: a stable fixed point loses stability and a limit cycle emerges.

Top notes — Hopf: complex pair crosses imaginary axis; cycle appears.

Side notes — nullclines y=B/x and y=((B+1)x − A)/x².

A: B: dt: Speed: Seeds:

Brusselator: x' = A − (B+1)x + x²y, y' = Bx − x²y · Fixed point (x*,y*)=(A, B/A). Hopf threshold B_H = 1 + A².

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Time Series

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Hopf Bifurcation: Amplitude vs B

A: B min: B max: Samples: Transient iters: Measure iters:

For each B in [Bmin, Bmax] at fixed A, integrate, discard transients, then plot amplitude of x(t) (max−min)/2. Expect emergence near B = 1 + A².

Side notes — mark B = 1 + A² as Hopf threshold.

Bottom notes — add references or derivation.