Let us summarize the geometrical features of a turning point. Locally there are no solutions on one side of a turning point and two solutions on the other side. At a turning point two solutions are born or two solutions annihilate each other. Viewing the two half-branches that meet in a turning point as parts of one smooth branch, we realize that only one branch is involved in a turning point
In the early days of bifurcation, turning points were not regarded as members of the family of bifurcation points. Rather, the name “bifurcation point” was reserved to situations where two (or more) branches with distinct tangents intersect. Two such types of bifurcation in a narrower sense are
- transcritical bifurcation
- pitchfork bifurcation
The points of bifurcation are emphasized by dots. These bifurcation points have a two-sided character. Pitchfork bifurcations are points at which one of the two branches is one-sided
1Seydel, R. (2010). Practical Bifurcation and Stability Analysis. Interdisciplinary Applied Mathematics, vol 5. Springer, New York, NY.