Namespaces
Variants
Actions

Difference between revisions of "Saddle point"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (MR/ZBL numbers added)
(TeX done)
 
Line 1: Line 1:
A point on a smooth surface such that the surface near the point lies on different sides of the tangent plane. If a point on a twice continuously-differentiable surface is a saddle point, then the [[Gaussian curvature|Gaussian curvature]] of the surface at the point is non-positive. A saddle point is a generalization of a [[Hyperbolic point|hyperbolic point]].
+
A point on a smooth surface such that the surface near the point lies on different sides of the tangent plane. If a point on a twice continuously-differentiable surface is a saddle point, then the [[Gaussian curvature]] of the surface at the point is non-positive. A saddle point is a generalization of a [[hyperbolic point]].
  
  
  
 
====Comments====
 
====Comments====
A surface all of whose points are saddle points is a [[Saddle surface|saddle surface]].
+
A surface all of whose points are saddle points is a [[saddle surface]].
  
A saddle point of a differentiable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083050/s0830501.png" /> is a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083050/s0830502.png" /> of the differentiable manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083050/s0830503.png" /> which is critical, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083050/s0830504.png" />, non-degenerate, i.e. the Hessian matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083050/s0830505.png" /> is non-singular, and such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083050/s0830506.png" /> is not a local maximum or a local minimum, i.e. the Hessian matrix is indefinite. Thus, a non-degenerate critical point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083050/s0830507.png" /> is a saddle point if its index (the number of negative eigenvalues of the Hessian matrix at that point) is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083050/s0830508.png" />. (The index does not depend on the local coordinates chosen.) The graph of a real-valued function of two variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083050/s0830509.png" /> near a saddle point looks like a saddle. See also [[Saddle point in game theory|Saddle point in game theory]].
+
A saddle point of a differentiable function $f : M \to \mathbf{R}$ is a point $x$ of the differentiable manifold $M$ which is critical, i.e. $\mathrm{d} f (x) = 0$, non-degenerate, i.e. the Hessian matrix $\left({ \partial^2 f / \partial x^i \partial x^j }\right)$ is non-singular, and such that $x$ is not a local maximum or a local minimum, i.e. the Hessian matrix is indefinite. Thus, a non-degenerate critical point of $f : M \to \mathbf{R}$ is a saddle point if its index (the number of negative eigenvalues of the Hessian matrix at that point) is $\ne 0,\,\dim M$. (The index does not depend on the local coordinates chosen.) The graph of a real-valued function of two variables $f : \mathbf{R}^2 \to \mathbf{R}$ near a saddle point looks like a saddle. See also [[Saddle point in game theory]].
  
A [[Saddle|saddle]] of a differential equation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083050/s08305010.png" /> is also often called a saddle point of that differential equation. More generally, given a dynamical system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083050/s08305011.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083050/s08305012.png" /> (or on a differentiable manifold) one considers the eigenvalues of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083050/s08305013.png" /> at an equilibrium point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083050/s08305014.png" />. If both positive and negative real parts occur, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083050/s08305015.png" /> is called a saddle, a saddle point or, sometimes, a Poincaré saddle point.
+
A [[saddle]] of a differential equation on $\mathbf{R}^2$ is also often called a saddle point of that differential equation. More generally, given a dynamical system $\dot x = f(x)$ on $\mathbf{R}^n$ (or on a differentiable manifold) one considers the eigenvalues of $D F(x_0)$ at an equilibrium point $x_0$. If both positive and negative real parts occur, $x_0$ is called a saddle, a saddle point or, sometimes, a Poincaré saddle point.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.W. Hirsch,  S. Smale,  "Differential equations, dynamical systems, and linear algebra" , Acad. Press  (1974)  pp. 190ff  {{MR|0486784}} {{ZBL|0309.34001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D.R.J. Chillingworth,  "Differential topology with a view to applications" , Pitman  (1976)  pp. 150ff  {{MR|0646088}} {{ZBL|0336.58001}} </TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  M.W. Hirsch,  S. Smale,  "Differential equations, dynamical systems, and linear algebra" , Acad. Press  (1974)  pp. 190ff  {{MR|0486784}} {{ZBL|0309.34001}} </TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top">  D.R.J. Chillingworth,  "Differential topology with a view to applications" , Pitman  (1976)  pp. 150ff  {{MR|0646088}} {{ZBL|0336.58001}} </TD></TR>
 +
</table>
 +
 
 +
{{TEX|done}}

Latest revision as of 18:52, 26 May 2017

A point on a smooth surface such that the surface near the point lies on different sides of the tangent plane. If a point on a twice continuously-differentiable surface is a saddle point, then the Gaussian curvature of the surface at the point is non-positive. A saddle point is a generalization of a hyperbolic point.


Comments

A surface all of whose points are saddle points is a saddle surface.

A saddle point of a differentiable function $f : M \to \mathbf{R}$ is a point $x$ of the differentiable manifold $M$ which is critical, i.e. $\mathrm{d} f (x) = 0$, non-degenerate, i.e. the Hessian matrix $\left({ \partial^2 f / \partial x^i \partial x^j }\right)$ is non-singular, and such that $x$ is not a local maximum or a local minimum, i.e. the Hessian matrix is indefinite. Thus, a non-degenerate critical point of $f : M \to \mathbf{R}$ is a saddle point if its index (the number of negative eigenvalues of the Hessian matrix at that point) is $\ne 0,\,\dim M$. (The index does not depend on the local coordinates chosen.) The graph of a real-valued function of two variables $f : \mathbf{R}^2 \to \mathbf{R}$ near a saddle point looks like a saddle. See also Saddle point in game theory.

A saddle of a differential equation on $\mathbf{R}^2$ is also often called a saddle point of that differential equation. More generally, given a dynamical system $\dot x = f(x)$ on $\mathbf{R}^n$ (or on a differentiable manifold) one considers the eigenvalues of $D F(x_0)$ at an equilibrium point $x_0$. If both positive and negative real parts occur, $x_0$ is called a saddle, a saddle point or, sometimes, a Poincaré saddle point.

References

[a1] M.W. Hirsch, S. Smale, "Differential equations, dynamical systems, and linear algebra" , Acad. Press (1974) pp. 190ff MR0486784 Zbl 0309.34001
[a2] D.R.J. Chillingworth, "Differential topology with a view to applications" , Pitman (1976) pp. 150ff MR0646088 Zbl 0336.58001
How to Cite This Entry:
Saddle point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Saddle_point&oldid=41585
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article