View source for Saddle-node bifurcation

Consider an [[Autonomous system|autonomous system]] of ordinary differential equations depending on a parameter

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110010/s1100101.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110010/s1100102.png" /> is a smooth function. Suppose that at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110010/s1100103.png" /> the system (a1) has an equilibrium (cf. also [[Equilibrium position|Equilibrium position]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110010/s1100104.png" /> with a simple eigenvalue <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110010/s1100105.png" /> (cf. also [[Eigen value|Eigen value]]) of its [[Jacobian|Jacobian]] matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110010/s1100106.png" />. Then, generically, two equilibria collide, form a [[Saddle node|saddle node]] singular point, and disappear when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110010/s1100107.png" /> passes through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110010/s1100108.png" />. This phenomenon is called the saddle-node (or fold) bifurcation [[#References|[a1]]], [[#References|[a2]]], [[#References|[a4]]]. It is characterized by one bifurcation condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110010/s1100109.png" /> (has [[Codimension|codimension]] one) and appears generically in one-parameter families.

To formulate relevant facts more precisely, first consider a smooth differential equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110010/s11001010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

that has at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110010/s11001011.png" /> the equilibrium <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110010/s11001012.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110010/s11001013.png" />. If the following non-degeneracy (genericity) conditions hold:

1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110010/s11001014.png" />;

2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110010/s11001015.png" />, then (a2) is locally topologically equivalent (cf. [[Equivalence of dynamical systems|Equivalence of dynamical systems]]) near the origin to the normal form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110010/s11001016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110010/s11001017.png" />, [[#References|[a2]]], [[#References|[a6]]]. The system (a3) has two equilibria (one stable and one unstable) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110010/s11001018.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110010/s11001019.png" /> and no equilibria for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110010/s11001020.png" />.

In the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110010/s11001021.png" />-dimensional case, the Jacobian matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110010/s11001022.png" /> evaluated at the equilibrium <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110010/s11001023.png" /> has a simple eigenvalue <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110010/s11001024.png" />, as well as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110010/s11001025.png" /> eigenvalues with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110010/s11001026.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110010/s11001027.png" /> eigenvalues with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110010/s11001028.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110010/s11001029.png" />). According to the centre manifold theorem (cf. [[Centre manifold|Centre manifold]]; [[#References|[a5]]], [[#References|[a3]]], [[#References|[a7]]]), there is an invariant one-dimensional centre manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110010/s11001030.png" /> near the origin, the restriction of (a1) to which has the form (a2). Moreover, [[#References|[a2]]], under the non-degeneracy conditions 1) and 2), the system (a1) is locally topologically equivalent (cf. [[Equivalence of dynamical systems|Equivalence of dynamical systems]]) near the origin to the suspension of the normal form (a3) by the standard saddle:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110010/s11001031.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>

Fig.a1 shows the phase portraits of the system (a4) in the planar case, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110010/s11001032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110010/s11001033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110010/s11001034.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110010/s11001035.png" />.

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s110010a.gif" />

Figure: s110010a

Saddle-node (fold) bifurcation on the plane

The coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110010/s11001036.png" /> can be computed (to within a scalar multiple) in terms of the right-hand sides of (a1), given two eigenvectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110010/s11001037.png" /> corresponding to the zero eigenvalue of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110010/s11001038.png" /> and of its transpose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110010/s11001039.png" />, respectively:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110010/s11001040.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110010/s11001041.png" /> is the [[Inner product|inner product]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110010/s11001042.png" />. Namely [[#References|[a6]]],

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110010/s11001043.png" /></td> </tr></table>

For discrete-time dynamical systems, similar results are valid concerning bifurcations of fixed points with a simple eigenvalue <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110010/s11001044.png" /> of the Jacobian matrix [[#References|[a2]]], [[#References|[a8]]], [[#References|[a6]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A.A. Andronov, E.A. Leontovich, I.I. Gordon, A.G. Maier, "Theory of bifurcations of dynamical systems on a plane" , Israel Program Sci. Transl. (1971) (In Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> V.I. Arnol'd, "Geometrical methods in the theory of ordinary differential equations" , ''Grundlehren math. Wiss.'' , '''250''' , Springer (1983) (In Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> J. Carr, "Applications of center manifold theory" , Springer (1981)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> J. Guckenheimer, Ph. Holmes, "Nonlinear oscillations, dynamical systems and bifurcations of vector fields" , Springer (1983) {{MR|0709768}} {{ZBL|0515.34001}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> A. Kelley, "The stable, center stable, center, center unstable and unstable manifolds" ''J. Diff. Eq.'' , '''3''' (1967) pp. 546–570 {{MR|0221044}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> Yu.A. Kuznetsov, "Elements of applied bifurcation theory" , Springer (1995) {{MR|1344214}} {{ZBL|0829.58029}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> A. Vanderbauwhede, "Centre manifolds, normal forms and elementary bifurcations" ''Dynamics Reported'' , '''2''' (1989) pp. 89–169 {{MR|1000977}} {{ZBL|0677.58001}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> D.C. Whitley, "Discrete dynamical systems in dimensions one and two" ''Bull. London Math. Soc.'' , '''15''' (1983) pp. 177–217 {{MR|0697119}} {{ZBL|0513.58033}} </TD></TR></table>