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In bifurcation theory, a field within mathematics, a transcritical bifurcation is a particular kind of local bifurcation, meaning that it is characterized by an equilibrium having an eigenvalue whose real part passes through zero. The normal form of a transcritical bifurcation, where r ranges from −5 to 5.
If the eigenvalue is equal to one, the bifurcation is either a saddle-node (often called fold bifurcation in maps), transcritical or pitchfork bifurcation. If the eigenvalue is equal to −1, it is a period-doubling (or flip) bifurcation, and otherwise, it is a Hopf bifurcation. Examples of local bifurcations include: Saddle-node (fold) bifurcation
A common simple bifurcation is the transcritical bifurcation, given by d x d t = r x − x 2 {\displaystyle {dx \over dt}=rx-x^{2}} and the bifurcation diagram in Figure 4 (black curves). The phase diagrams are shown in Figure 5.
The bifurcation diagram shows the forking of the periods of stable orbits from 1 to 2 to 4 to 8 etc. Each of these bifurcation points is a period-doubling bifurcation. The ratio of the lengths of successive intervals between values of r for which bifurcation occurs converges to the first Feigenbaum constant.
All systems exhibiting a certain type of bifurcation are locally (around the equilibrium) topologically equivalent to the normal form of the bifurcation. For example, the normal form of a saddle-node bifurcation is = + where is the bifurcation parameter. The transcritical bifurcation
Bifurcation diagram; Biological applications of bifurcation theory; ... Transcritical bifurcation This page was last edited on 16 April 2018, at 10:24 (UTC). Text ...
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In the mathematical area of bifurcation theory a saddle-node bifurcation, tangential bifurcation or fold bifurcation is a local bifurcation in which two fixed points (or equilibria) of a dynamical system collide and annihilate each other. The term 'saddle-node bifurcation' is most often used in reference to continuous dynamical systems.