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.

## Summary

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- 1 Definition
- 2 Shape
- 3 Example 1
- 4 Comparison
- 5 See also
- 6 References

## Definition

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).^{[2]}

## Shape

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:

- 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.