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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.
Created Date: 8/30/2012 4:52:52 PM
To see how this number arises, consider the real one-parameter map =.Here a is the bifurcation parameter, x is the variable. The values of a for which the period doubles (e.g. the largest value for a with no period-2 orbit, or the largest a with no period-4 orbit), are a 1, a 2 etc.
Other examples are in modelling biological switches. [4] Recently, it was shown that under certain conditions, the Einstein field equations of General Relativity have the same form as a fold bifurcation. [5] A non-autonomous version of the saddle-node bifurcation (i.e. the parameter is time-dependent) has also been studied. [6]
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.
Download as PDF; Printable version; ... Biological applications of bifurcation theory; ... Period-doubling bifurcation; Pitchfork bifurcation; S.