Chair-node bifurcation

In mathematics , a saddle-node fork (or tangent, or in English “saddle-node”, “tangent, or” fold “) is a local or global fork of an ordinary differential equation in which two fixed points (or equilibria, or critics) of a dynamic system collide and annihilate each other. The phrase “chair-node bifurcation” is often used in reference to continuous dynamic systems . In discrete systems, the same fork has the name (“fold fork”). Chair-node bifurcations are the generic way that the number of equilibrium solutions of a dynamic system changes when any parameter is varied.

When the fork is global (not local), it goes by the name blue sky fork in reference to creating two fixed points. [one]

Chair-node bifurcations are associated with hysteresis cycles and catastrophes. If the phase space is one-dimensional, one of the equilibrium points is unstable (the chair), while the other is stable (the node).

The name “chair-node” comes from the two-dimensional bifurcation on the phase plane, in which a saddle point and a node come together and disappear, but the other dimension is not significant and this bifurcation is naturally one-dimensional.


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  • 1 Definition
  • 2 Shape
  • 3 Example 1
  • 4 Comparison
  • 5 See also
  • 6 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 a function to which:

f (0, C o ) = 0, and also:

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

df / dc (0, C o )> 0

then there is an interval between (C 1 , 0) and (0, C 2 ) with e> 0 where

  1. If C ∈ (C 1, 0), then f c(x) has 2 fixed points at (-e, e) with the positive being non-stable and the negative being stable ;
  2. If C ∈ (0, C 2), then f c(x) has no fixed points at (-e, e). [2]


The normal way is:

dx / dt = C ∓ x 2 , where C is the parameter.

The normal form of a fork is a simple dynamic system that is equivalent to all the systems that show this fork. [3]

Example 1

An example of a two-dimensional chair-node fork occurs in the dynamic system with the differential equations:

dx / dt = C – x 2

dy / dt = -y

see that:

  1. When C> 0, there are two equilibrium points: a saddle point another node (either an attractor or a repeller).
  2. When C = 0, there is a saddle-node point.
  3. When C <0, there are no equilibrium points.


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