Saddle-knot bifurcation

Bifurcation diagram of a saddle-node bifurcation. Stable fixed points are shown in red, unstable in blue.

The saddle-node bifurcation ( English saddle-node bifurcation ), fold bifurcation (English fold bifurcation ), tangent bifurcation (English tangent bifurcation ), limit point or turning point is a certain type of bifurcation of a nonlinear dynamic system .

The normal form of the saddle-node bifurcation is

${\displaystyle {\dot {x}}=\mu -x^{2},}$

where is the bifurcation parameter. ${\displaystyle \mu }$

This normal form has for fixed points : ${\displaystyle \mu \geq 0}$

${\displaystyle {x_{1/2}}^{*}=\pm {\sqrt {\mu }}.}$

That means, it does not exist for any fixed point, for exactly one fixed point and otherwise two. The first fixed point is stable ( knot ), the second unstable ( saddle ). The saddle and knot collide at the bifurcation point . Looking at a system with a higher order in ${\displaystyle \mu <0}$ ${\displaystyle \mu =0}$ ${\displaystyle \mu =0}$ ${\displaystyle x}$

${\displaystyle {\dot {x}}=\mu -x^{2}+O(x^{3}),}$

so these terms do not influence the behavior of the system in a sufficiently small area around the saddle-node point . That is, the system is locally topologically equivalent at the origin to the normal form. In general, the bifurcation is characterized in that an eigenvalue of the Jacobian matrix of the dynamic system becomes zero at a critical value of the bifurcation parameter. ${\displaystyle \mu =0}$ ${\displaystyle D_{x}f(x,\mu )}$ ${\displaystyle {\dot {x}}=f(x,\mu )}$

See also

• Pitchfork bifurcation
• Hopf bifurcation
• Transcritical bifurcation

literature

• Yuri A. Kuznetsov: Elements of Applied Bifurcation Theory (=  Applied Mathematica Sciences . Volume 112 ). 2nd Edition. Springer, 1995, ISBN 0-387-98382-1 .