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Consider an [[Autonomous system|autonomous system]] of ordinary differential equations depending on a parameter
 
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>
+
$$ \tag{a1 }
 +
{\dot{x} } = f ( x, \alpha ) , \quad x \in \mathbf R  ^ {n} ,  \alpha \in \mathbf R  ^ {1} ,
 +
$$
  
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.
+
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
 
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>
+
$$ \tag{a2 }
 +
{\dot{x} } = f ( x, \alpha ) , \quad x \in \mathbf R  ^ {1} ,  \alpha \in \mathbf R  ^ {1} ,
 +
$$
  
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:
+
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) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110010/s11001014.png" />;
+
1) $  a = ( {1 / 2 } ) f _ {xx }  ( 0,0 ) \neq 0 $;
  
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
+
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
  
<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>
+
$$ \tag{a3 }
 +
{\dot{y} } = \beta + \sigma y  ^ {2} , \quad y \in \mathbf R  ^ {1} ,  \beta \in \mathbf R  ^ {1} ,
 +
$$
  
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" />.
+
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 <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:
+
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:
  
<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>
+
$$ \tag{a4 }
 +
\left \{
 +
\begin{array}{l}
 +
{ {\dot{y} } = \beta + \sigma y  ^ {2} , \  y \in \mathbf R  ^ {1} ,  \beta \in \mathbf R  ^ {1} , } \\
 +
{ {\dot{y} } _ {s} = - y _ {s} , \  y _ {s} \in \mathbf R ^ {n _ {s} } , } \\
 +
{ {\dot{y} } _ {u} = + y _ {u} , \  y _ {u} \in \mathbf R ^ {n _ {u} } . }
 +
\end{array}
 +
\right .
 +
$$
  
Fig.a1 shows the phase portraits of the system (a4) in the planar case, when <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" />.
+
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" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s110010a.gif" />
Line 31: Line 85:
 
Saddle-node (fold) bifurcation on the plane
 
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:
+
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:
  
<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>
+
$$
 +
Av = A  ^ {T} w = 0, \quad \left \langle  {w,v } \right \rangle = 1,
 +
$$
  
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]]],
+
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]]],
  
<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>
+
$$
 +
a = {
 +
\frac{1}{2}
 +
} \left . {
 +
\frac{d  ^ {2} }{d \tau  ^ {2} }
 +
} \left \langle  {w,f ( \tau v,0 ) } \right \rangle \right | _ {\tau = 0 .
 +
$$
  
For discrete-time dynamical systems, similar results are valid concerning bifurcations of fixed points with a simple eigenvalue <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]]].
+
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====
 
====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>
 
<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>

Latest revision as of 08:12, 6 June 2020


Consider an 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) $ x = 0 $ with a simple eigenvalue $ \lambda _ {1} = 0 $( cf. also Eigen value) of its Jacobian matrix $ A = f _ {x} ( 0,0 ) $. Then, generically, two equilibria collide, form a saddle node singular point, and disappear when $ \alpha $ passes through $ \alpha = 0 $. This phenomenon is called the saddle-node (or fold) bifurcation [a1], [a2], [a4]. It is characterized by one bifurcation condition $ \lambda _ {1} = 0 $( has 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) 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 $, [a2], [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; [a5], [a3], [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, [a2], under the non-degeneracy conditions 1) and 2), the system (a1) is locally topologically equivalent (cf. 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 $.

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 in $ \mathbf R ^ {n} $. Namely [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 [a2], [a8], [a6].

References

[a1] 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)
[a2] V.I. Arnol'd, "Geometrical methods in the theory of ordinary differential equations" , Grundlehren math. Wiss. , 250 , Springer (1983) (In Russian)
[a3] J. Carr, "Applications of center manifold theory" , Springer (1981)
[a4] J. Guckenheimer, Ph. Holmes, "Nonlinear oscillations, dynamical systems and bifurcations of vector fields" , Springer (1983) MR0709768 Zbl 0515.34001
[a5] A. Kelley, "The stable, center stable, center, center unstable and unstable manifolds" J. Diff. Eq. , 3 (1967) pp. 546–570 MR0221044
[a6] Yu.A. Kuznetsov, "Elements of applied bifurcation theory" , Springer (1995) MR1344214 Zbl 0829.58029
[a7] A. Vanderbauwhede, "Centre manifolds, normal forms and elementary bifurcations" Dynamics Reported , 2 (1989) pp. 89–169 MR1000977 Zbl 0677.58001
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How to Cite This Entry:
Saddle-node bifurcation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Saddle-node_bifurcation&oldid=24558
This article was adapted from an original article by Yu.A. Kuznetsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article