Trident fork

In mathematics a pitchfork bifurcation (in English “pitchfork”) is a type of branch premises of a differential equation of a dynamic system . Trident bifurcations can be supercritical or subcritical . [one]

In continuous dynamic systems described by an ordinary differential equation , trident bifurcations occur in systems with symmetry. In other words, this bifurcation is related to system symmetries. For example, in systems that have spatial symmetry between fixed points, left and right, they tend to appear and disappear in symmetric pairs (example: analysis of buckling of a straight beam tablet). [2]


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  • 1 Definition
  • 2 Stability
  • 3 Shape
  • 4 Example 1
  • 5 Super and subcritical
  • 6 See also
  • 7 References


If the ordinary differential equation dx / dt = f (x, C), described by a single parameter C of the function f (x, C), with C being a member or element of the real numbers (C ∈ R), yf (x, C) is an odd function to which:

f (-x, C) = – f (x, C), and also:

df / dx (0, C o ) = 0; d 2 f / dx 2 (0, C o ) = 0;

3 f / dx 3 (0, Co) is not zero;

df / dc (0, Co) = 0; d 2 f / dcdx (0, Co) is not zero

then the differential equation has a trident bifurcation at the fixed point (0, Co) [3]


3 f / dx 3 (0, C o )> 0 → subcritical

3 f / dx 3 (0, C o ) <0 → supercritical [4]


The normal way is:

dx / dt = Cx ∓ x 3

The normal form of a fork is a simple dynamic system that is equivalent to all systems showing this fork. [5]

Example 1

Trident bifurcations are common in physical systems that possess symmetry. An example is the Euler strut .

An increasing load is applied to the vertical strut on the Euler strut, until it finally buckles. Right and left buckling are equivalent and odd function symmetry is applied. Analysis shows that the system has a supercritical trident fork at the buckling point.

Super and subcritical

The important difference between supercritical and subcritical bifurcations is: “danger”.


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