View source for Saddle-node bifurcation

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

$$ \tag{a1 }
{\dot{x} } = f ( x, \alpha ) , \quad x \in \mathbf R  ^ {n} ,  \alpha \in \mathbf R  ^ {1} ,
$$

where  $  f $
is a smooth function. Suppose that at  $  \alpha = 0 $
the system (a1) has an equilibrium (cf. also [[Equilibrium position|Equilibrium position]])  $  x = 0 $
with a simple eigenvalue  $  \lambda _ {1} = 0 $(
cf. also [[Eigen value|Eigen value]]) of its [[Jacobian|Jacobian]] matrix  $  A = f _ {x} ( 0,0 ) $. 
Then, generically, two equilibria collide, form a [[Saddle node|saddle node]] singular point, and disappear when  $  \alpha $
passes through  $  \alpha = 0 $. 
This phenomenon is called the saddle-node (or fold) bifurcation [[#References|[a1]]], [[#References|[a2]]], [[#References|[a4]]]. It is characterized by one bifurcation condition  $  \lambda _ {1} = 0 $(
has [[Codimension|codimension]] one) and appears generically in one-parameter families.

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

$$ \tag{a2 }
{\dot{x} } = f ( x, \alpha ) , \quad x \in \mathbf R  ^ {1} ,  \alpha \in \mathbf R  ^ {1} ,
$$

that has at  $  \alpha = 0 $
the equilibrium  $  x = 0 $
with  $  \lambda _ {1} = f _ {x} ( 0,0 ) = 0 $. 
If the following non-degeneracy (genericity) conditions hold:

1)  $  a = ( {1 / 2 } ) f _ {xx }  ( 0,0 ) \neq 0 $;

2)  $  f _  \alpha  ( 0,0 ) \neq 0 $, 
then (a2) is locally topologically equivalent (cf. [[Equivalence of dynamical systems|Equivalence of dynamical systems]]) near the origin to the normal form

$$ \tag{a3 }
{\dot{y} } = \beta + \sigma y  ^ {2} , \quad y \in \mathbf R  ^ {1} ,  \beta \in \mathbf R  ^ {1} ,
$$

where  $  \sigma = { \mathop{\rm sign} } a = \pm  1 $, 
[[#References|[a2]]], [[#References|[a6]]]. The system (a3) has two equilibria (one stable and one unstable)  $  y _ {1,2 }  = \pm  \sqrt {- \sigma \beta } $
for  $  \sigma \beta < 0 $
and no equilibria for  $  \sigma \beta > 0 $.

In the  $  n $-
dimensional case, the Jacobian matrix  $  A $
evaluated at the equilibrium  $  x = 0 $
has a simple eigenvalue  $  \lambda _ {1} = 0 $, 
as well as  $  n _ {s} $
eigenvalues with  $  { \mathop{\rm Re} } \lambda _ {j} < 0 $, 
and  $  n _ {u} $
eigenvalues with  $  { \mathop{\rm Re} } \lambda _ {j} > 0 $(
$  n _ {s} + n _ {u} + 1 = n $). 
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  $  {\mathcal M} _  \alpha  $
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:

$$ \tag{a4 }
\left \{
\begin{array}{l}
{ {\dot{y} } = \beta + \sigma y  ^ {2} , \  y \in \mathbf R  ^ {1} ,  \beta \in \mathbf R  ^ {1} , } \\
 { {\dot{y} } _ {s} = - y _ {s} , \  y _ {s} \in \mathbf R ^ {n _ {s} } , } \\
 { {\dot{y} } _ {u} = + y _ {u} , \  y _ {u} \in \mathbf R ^ {n _ {u} } . } 
\end{array}
 \right .
$$

Fig.a1 shows the phase portraits of the system (a4) in the planar case, when  $  n = 2 $, 
$  n _ {s} = 1 $, 
$  n _ {u} = 0 $, 
and  $  \sigma = 1 $.

<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  $  a $
can be computed (to within a scalar multiple) in terms of the right-hand sides of (a1), given two eigenvectors  $  v,w \in \mathbf R  ^ {n} $
corresponding to the zero eigenvalue of  $  A $
and of its transpose  $  A  ^ {T} $, 
respectively:

$$ 
Av = A  ^ {T} w = 0, \quad \left \langle  {w,v } \right \rangle = 1,
$$

where  $  \langle  {w,v } \rangle = \sum _ {i = 1 }   ^ {n} w _ {i} v _ {i} $
is the [[Inner product|inner product]] in  $  \mathbf R  ^ {n} $. 
Namely [[#References|[a6]]],

$$ 
a = {
\frac{1}{2}
 } \left . {
\frac{d  ^ {2} }{d \tau  ^ {2} }
 } \left \langle  {w,f ( \tau v,0 ) } \right \rangle \right | _ {\tau = 0 }  .
$$

For discrete-time dynamical systems, similar results are valid concerning bifurcations of fixed points with a simple eigenvalue  $  \mu _ {1} = 1 $
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>