saddle-node bifurcation

A typical example of a differential equation with a saddle-node bifurcation is:

:$\backslash frac\{dx\}\{dt\}=r+x^2.$

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

• If there are two equilibrium points, a stable equilibrium point at $-\backslash sqrt\{-r\}$ and an unstable one at $+\backslash sqrt\{-r\}$.
• 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.{{sfn|Kuznetsov|1998|pp=80–81}}

File:Fold Bifurcation.webm

In fact, this is a normal form of a saddle-node bifurcation. A scalar differential equation $\backslash tfrac\{dx\}\{dt\}\; =\; f(r,x)$ which has a fixed point at $x\; =\; 0$ for $r\; =\; 0$ with $\backslash tfrac\{\backslash partial\; f\}\{\backslash partial\; x\}(0,0)\; =\; 0$ is locally topologically equivalent to $\backslash frac\{dx\}\{dt\}\; =\; r\; \backslash pm\; x^2$, provided it satisfies $\backslash tfrac\{\backslash partial^2\backslash !\; f\}\{\backslash partial\; x^2\}(0,0)\; \backslash ne\; 0$ and $\backslash tfrac\{\backslash partial\; f\}\{\backslash partial\; r\}(0,0)\; \backslash ne\; 0$. The first condition is the nondegeneracy condition and the second condition is the transversality condition.{{sfn|Kuznetsov|1998|loc=Theorems 3.1 and 3.2}}

Image:Saddlenode.gif

An example of a saddle-node bifurcation in two dimensions occurs in the two-dimensional dynamical system:

:$\backslash frac\; \{dx\}\; \{dt\}\; =\; \backslash alpha\; -\; x^2$

:$\backslash frac\; \{dy\}\; \{dt\}\; =\; -\; y.$

As can be seen by the animation obtained by plotting phase portraits by varying the parameter $\backslash alpha$,

• When $\backslash alpha$ is negative, there are no equilibrium points.
• When $\backslash alpha\; =\; 0$, there is a saddle-node point.
• When $\backslash alpha$ 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.{{cite conference |last1=Chong |first1=Ket Hing |last2=Samarasinghe |first2=Sandhya |last3=Kulasiri |first3=Don |last4=Zheng |first4=Jie |year=2015 |title=Computational techniques in mathematical modelling of biological switches |conference=21st International Congress on Modelling and Simulation |hdl=10220/42793 }} Recently, it was shown that under certain conditions, the Einstein field equations of General Relativity have the same form as a fold bifurcation.{{cite journal |doi=10.1016/j.geomphys.2017.10.001 |title=Einstein's field equations as a fold bifurcation |journal=Journal of Geometry and Physics |volume=123 |pages=434–7 |year=2018 |last1=Kohli |first1=Ikjyot Singh |last2=Haslam |first2=Michael C |arxiv=1607.05300 |bibcode=2018JGP...123..434K |s2cid=119196982 }} A non-autonomous version of the saddle-node bifurcation (i.e. the parameter is time-dependent) has also been studied.{{Cite journal|last1=Li|first1=Jeremiah H.|last2=Ye|first2=Felix X. -F.|last3=Qian|first3=Hong|last4=Huang|first4=Sui|date=2019-08-01|title=Time-dependent saddle–node bifurcation: Breaking time and the point of no return in a non-autonomous model of critical transitions|journal=Physica D: Nonlinear Phenomena|volume=395|pages=7–14|doi=10.1016/j.physd.2019.02.005|pmid=31700198 |pmc=6836434 |issn=0167-2789|arxiv=1611.09542|bibcode=2019PhyD..395....7L }}

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

{{reflist}}

• {{cite book | last1 = Kuznetsov | first1 = Yuri A. | title = Elements of Applied Bifurcation Theory | year = 1998 | publisher = Springer | edition = Second | isbn = 0-387-98382-1 }}
• {{cite book | last = Strogatz | first = Steven H. | title = Nonlinear Dynamics and Chaos | publisher = Addison Wesley | year = 1994 | isbn = 0-201-54344-3}}
• {{MathWorld|title=Fold Bifurcation|urlname=FoldBifurcation}}
• {{cite book | last1 = Chong | first1 = K. H. | last2 = Samarasinghe | first2 = S. | last3 = Kulasiri | first3 = D. | last4 = Zheng | first4 = J. | title = Computational Techniques in Mathematical Modelling of Biological Switches | publisher = 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 | year = 2015 | isbn = 978-0-9872143-5-5}}
• {{cite book | last1 = Kohli | first1 = Ikjyot Singh | last2 = Haslam | first2 = Michael C. | title = Einstein Field Equations as a Fold Bifurcation | publisher = Journal of Geometry and Physics Volume 123, January 2018, Pages 434-437 | year = 2018}}

{{DEFAULTSORT:Saddle-Node Bifurcation}}

Category:Bifurcation theory

Category:Articles containing video clips