Saddle-node bifurcation explained

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:

(dx)/(dt) = 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 - \sqrt{ - r} and an unstable one at + \sqrt{ - 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

\tfrac{dx}{dt} = f(r,x) which has a fixed point at x = 0 for r = 0 with \tfrac{\partial f}{\partial x}( 0 , 0 ) = 0 is locally topologically equivalent to (dx)/(dt) = r \pm x ^{2} , provided it satisfies \tfrac{\partial ^{2} f}{\partial x ^{2}(0,0)} \ne 0 and \tfrac{\partial f}{\partial r}( 0 , 0 ) \ne 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:

(dx)/(dt) = \alpha - x ^{2} (dy)/(dt) = - y.

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

\alpha ,

• When

\alpha is negative, there are no equilibrium points.

• When

\alpha = 0 , there is a saddle-node point.

• When

\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.^[1] Recently, it was shown that under certain conditions, the Einstein field equations of General Relativity have the same form as a fold bifurcation.^[2] A non-autonomous version of the saddle-node bifurcation (i.e. the parameter is time-dependent) has also been studied.^[3]

See also

• Pitchfork bifurcation
• Transcritical bifurcation
• Hopf bifurcation
• Saddle point

References

• Book: Kuznetsov . Yuri A. . Elements of Applied Bifurcation Theory . 1998 . Springer . Second . 0-387-98382-1 .
• Book: Strogatz, Steven H. . Nonlinear Dynamics and Chaos . Addison Wesley . 1994 . 0-201-54344-3.
  • Book: Chong . K. H. . Samarasinghe . S. . Kulasiri . D. . Zheng . J. . Computational Techniques in Mathematical Modelling of Biological Switches . In Weber, T., McPhee, M.J. and Anderssen, R.S. (eds) MODSIM2015, 21st International Congress on Modelling and Simulation (MODSIM 2015). Modelling and Simulation Society of Australia and New Zealand, December 2015, pp. 578-584 . 2015 . 978-0-9872143-5-5.
• Book: Kohli . Ikjyot Singh . Haslam . Michael C. . Einstein Field Equations as a Fold Bifurcation . Journal of Geometry and Physics Volume 123, January 2018, Pages 434-437 . 2018.

Notes and References

1. Chong . Ket Hing . Samarasinghe . Sandhya . Kulasiri . Don . Zheng . Jie . 2015 . Computational techniques in mathematical modelling of biological switches . 21st International Congress on Modelling and Simulation . 10220/42793 .
2. 10.1016/j.geomphys.2017.10.001 . Einstein's field equations as a fold bifurcation . Journal of Geometry and Physics . 123 . 434–7 . 2018 . Kohli . Ikjyot Singh . Haslam . Michael C . 1607.05300 . 2018JGP...123..434K . 119196982 .
3. Li. Jeremiah H.. Ye. Felix X. -F.. Qian. Hong. Huang. Sui. 2019-08-01. Time-dependent saddle–node bifurcation: Breaking time and the point of no return in a non-autonomous model of critical transitions. Physica D: Nonlinear Phenomena. 395. 7–14. 10.1016/j.physd.2019.02.005. 31700198 . 6836434 . 0167-2789. 1611.09542. 2019PhyD..395....7L .