The transcritical bifurcation occurs when in the combined space of phase space and controlparameter space two different manifolds of fixed points cross each other. At the crossing point the fixed points exchange there stability property. That is, the unstable fixed point becomes stable and vice versa. Note, that beyond the bifurcation point the number of fixed points has not changed contrary to saddle-node bifurcation where two fixed points appear or disappear. A transcritical bifurcation is not a generic bifurcation in a phase space with more than one dimension because it is unlikely that two lines cross each other in a space with more than two dimensions.
Pitchfork bifurcations are possible in dynamical systems with an inversion or reflection symmetry .
An example is the undriven pendulum . Pitchfork bifurcations are the generic bifurcations when such a symmetric solution changes its stability. The bifurcating solution does not have the full symmetry of the equation of motion. This phenomenon is called broken symmetry . A solution with a broken symmetry does not occur in isolation because the broken symmetry applied onto such a solution generates a new solution where the same symmetry is broken.