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
- 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 ;
- If C ∈ (0, C 2), then f c(x) has no fixed points at (-e, e). 
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. 
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
- When C> 0, there are two equilibrium points: a saddle point another node (either an attractor or a repeller).
- When C = 0, there is a saddle-node point.
- When C <0, there are no equilibrium points.