Reference
Bifurcation atlas
The six codimension-1 bifurcations every nonlinear-dynamics student sees, with their normal forms and (μ, x*) diagrams. Orange dots mark stable fixed points; blue boxes mark unstable ones.
Saddle-node
ẋ = μ + x²
Two fixed points collide and annihilate as μ crosses 0 from negative to positive.
Transcritical
ẋ = μ x − x²
Two fixed points exchange stability when they cross at μ = 0.
Pitchfork (supercritical)
ẋ = μ x − x³
At μ = 0 the central fixed point loses stability; two stable branches emerge.
Pitchfork (subcritical)
ẋ = μ x + x³
Two unstable branches collide with the central fixed point at μ = 0; dangerous bifurcation.
Hopf (supercritical)
ż = (μ + i ω) z − |z|² z (radial: ṙ = μ r − r³)
A fixed point loses stability; a stable limit cycle of small radius √μ emerges.
Period-doubling (flip)
x_{n+1} = −(1 + μ) x_n + x_n³
A period-T orbit becomes unstable; a stable period-2T orbit appears nearby.
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