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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.
Symmetry breaking in pitchfork bifurcation as the parameter ε is varied. ε = 0 is the case of symmetric pitchfork bifurcation.. In a dynamical system such as ¨ + (;) + =, which is structurally stable when , if a bifurcation diagram is plotted, treating as the bifurcation parameter, but for different values of , the case = is the symmetric pitchfork bifurcation.
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
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
Diagram showing pitchfork bifurcation geometry given by a slice through cusp catastrophe. Date: 1 December 2005: Source: Created in OpenOffice Draw, exported as SVG, size explicitly added in text editor. Author: Jheald: SVG development
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Center manifolds play an important role in bifurcation theory because interesting behavior takes place on the center manifold and in multiscale mathematics because the long time dynamics of the micro-scale often are attracted to a relatively simple center manifold involving the coarse scale variables.
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.