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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.
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
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.
But when the mass reaches a certain point – the bifurcation point – the beam will suddenly buckle, in a direction dependent on minor imperfections in the setup. This is an example of a pitchfork bifurcation. Changes in the control parameter eventually changed the qualitative behavior of the system.
Diagram showing pitchfork bifurcation geometry given by a slice through cusp catastrophe. Created in OpenOffice Draw, exported as SVG, size explicitly added in text editor. File usage
The exact value of the bifurcation point for a window of period 3 is known, and if the value of this bifurcation point r is , then = + =. The outline of this bifurcation can be understood by considering the graph of f 3 ( x ) {\displaystyle f^{3}(x)} (vertical axis x n + 3 {\displaystyle x_{n+3}} , horizontal axis x n {\displaystyle x_{n}} ).
An example would be plotting the , value every time it passes through the = plane where is changing from negative to positive, commonly done when studying the Lorenz attractor. In the case of the Rössler attractor, the x = 0 {\displaystyle x=0} plane is uninteresting, as the map always crosses the x = 0 {\displaystyle x=0} plane at z = 0 ...
Download as PDF; Printable version; ... Biological applications of bifurcation theory; ... Period-doubling bifurcation; Pitchfork bifurcation; S.