Pitchfork bifurcation

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 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. The normal form of the supercritical pitchfork bifurcation is For negative values of r {displaystyle r} , 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}}} . The normal form for the subcritical case is 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.