In bifurcation theory, a field within mathematics, a pitchfork bifurcation is a particular type of local bifurcation where the system transitions from one fixed point to three fixed points. Pitchfork bifurcations, like Hopf bifurcations, have two types – supercritical and subcritical. In continuous dynamical systems described by ODEs—i.e. flows—pitchfork bifurcations occur generically in systems with symmetry.
Supercritical case
The normal form of the supercritical pitchfork bifurcation is
d x d t = r x − x 3 . {\displaystyle {\frac {dx}{dt}}=rx-x^{3}.}
For r < 0 {\displaystyle r<0} , there is one stable equilibrium at x = 0 {\displaystyle x=0} . For r > 0 {\displaystyle r>0} there is an unstable equilibrium at x = 0 {\displaystyle x=0} , and two stable equilibria at x = ± r {\displaystyle x=\pm {\sqrt {r}}} .
Subcritical case
The normal form for the subcritical case is
d x d t = r x + x 3 . {\displaystyle {\frac {dx}{dt}}=rx+x^{3}.}
In this case, for r < 0 {\displaystyle r<0} the equilibrium at x = 0 {\displaystyle x=0} is stable, and there are two unstable equilibria at x = ± − r {\displaystyle x=\pm {\sqrt {-r}}} . For r > 0 {\displaystyle r>0} the equilibrium at x = 0 {\displaystyle x=0} is unstable.
Formal definition An ODE
x ˙ = f ( x , r ) {\displaystyle {\dot {x}}=f(x,r)\,}
described by a one parameter function f ( x , r ) {\displaystyle f(x,r)} with r ∈ R {\displaystyle r\in \mathbb {R} } satisfying:
− f ( x , r ) = f ( − x , r ) {\displaystyle -f(x,r)=f(-x,r)\,\,} (f is an odd function),
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