In the mathematical area of bifurcation theory a saddle-node bifurcation, tangential bifurcation or fold bifurcation is a local bifurcation in which two fixed points (or equilibria) of a dynamical system collide and annihilate each other. The term 'saddle-node bifurcation' is most often used in reference to continuous dynamical systems. In discrete dynamical systems, the same bifurcation is often instead called a fold bifurcation. Another name is blue sky bifurcation in reference to the sudden creation of two fixed points. If the phase space is one-dimensional, one of the equilibrium points is unstable (the saddle), while the other is stable (the node). Saddle-node bifurcations may be associated with hysteresis loops and catastrophes.
Normal form A typical example of a differential equation with a saddle-node bifurcation is:
d x d t = r + x 2 . {\displaystyle {\frac {dx}{dt}}=r+x^{2}.}
Here x {\displaystyle x} is the state variable and r {\displaystyle r} is the bifurcation parameter.
If r < 0 {\displaystyle r<0} there are two equilibrium points, a stable equilibrium point at − − r {\displaystyle -{\sqrt {-r}}} and an unstable one at + − r {\displaystyle +{\sqrt {-r}}} . At r = 0 {\displaystyle r=0} (the bifurcation point) there is exactly one equilibrium point. At this point the fixed point is no longer hyperbolic. In this case the fixed point is called a saddle-node fixed point. If r > 0 {\displaystyle r>0} there are no equilibrium points.
In fact, this is a normal form of a saddle-node bifurcation. A scalar differential equation d x d t = f ( r , x ) {\displaystyle {\tfrac {dx}{dt}}=f(r,x)} which has a fixed point at x = 0 {\displaystyle x=0} for r = 0 {\displaystyle r=0} with ∂ f ∂ x ( 0 , 0 ) = 0 {\displaystyle {\tfrac {\partial f}{\partial x}}(0,0)=0} is locally topologically equivalent to d x d t = r ± x 2 {\displaystyle {\frac {dx}{dt}}=r\pm x^{2}} , provided it satisfies ∂ 2 f ∂ x 2 ( 0 , 0 ) ≠ 0 {\displaystyle {\tfrac {\partial ^{2}\!f}{\partial x^{2}}}(0,0)\neq 0} and ∂ f ∂ r ( 0 , 0 ) ≠ 0 {\displaystyle {\tfrac {\partial f}{\partial r}}(0,0)\neq 0} . The first condition is the nondegeneracy condition and the second condition is the transversality condition.
Example in two dimensions
An example of a saddle-node bifurcation in two dimensions occurs in the two-dimensional dynamical system:
d x d t = α − x 2 {\displaystyle {\frac {dx}{dt}}=\alpha -x^{2}}
d y d t = − y . {\displaystyle {\frac {dy}{dt}}=-y.}
As can be seen by the animation obtained by plotting phase portraits by varying the parameter α {\displaystyle \alpha } ,
When α {\displaystyle \alpha } is negative, there are no equilibrium points. When α = 0 {\displaystyle \alpha =0} , there is a saddle-node point. When α {\displaystyle \alpha } is positive, there are two equilibrium points: that is, one saddle point and one node (either an attractor or a repellor). Other examples are in modelling biological switches. Recently, it was shown that under certain conditions, the Einstein field equations of General Relativity have the same form as a fold bifurcation. A non-autonomous version of the saddle-node bifurcation (i.e. the parameter is time-dependent) has also been studied.
See also Pitchfork bifurcation Transcritical bifurcation Hopf bifurcation Saddle point
Notes
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