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Saddle node bifurcation

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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 skies bifurcation in reference to the sudden creation of two fixed points.

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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 .

Here x is the state variable and r is the bifurcation parameter.

  • If r < 0 there are two equilibrium points, a stable equilibrium point at − − r and an unstable one at + − r .
  • At 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 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 ) which has a fixed point at x = 0 for r = 0 with ∂ f ∂ x ( 0 , 0 ) = 0 is locally topologically equivalent to d x d t = r ± x 2 , provided it satisfies ∂ 2 f ∂ x 2 ( 0 , 0 ) ≠ 0 and ∂ f ∂ r ( 0 , 0 ) ≠ 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 d y d t = − y .

    As can be seen by the animation obtained by plotting phase portraits by varying the parameter α ,

  • When α is negative, there are no equilibrium points.
  • When α = 0 , there is a saddle-node point.
  • When α is positive, there are two equilibrium points: that is, one saddle point and one node (either an attractor or a repellor).

    • A saddle-node bifurcation also occurs in the consumer equation (see transcritical bifurcation) if the consumption term is changed from p x to p , that is, the consumption rate is constant and not in proportion to resource x .

    Other examples are in modelling biological switches (see a tutorial for the computational techniques in modelling biological switches with an easy to understand synthetic toggle switch that demonstrates the bistability and hysteresis behavior showing the saddle-nodes or tipping points).

    References

    Saddle-node bifurcation Wikipedia
    (Text) CC BY-SA


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