Namespaces
Variants
Actions

Difference between revisions of "Node"

From Encyclopedia of Mathematics
Jump to: navigation, search
Line 36: Line 36:
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.L. Coolidge,  "Algebraic plane curves" , Dover, reprint  (1959)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.L. Coolidge,  "Algebraic plane curves" , Dover, reprint  (1959)</TD></TR></table>
  
A node is a type of arrangement of the trajectories of an [[Autonomous system|autonomous system]] of second-order ordinary differential equations
 
 
<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/n/n066/n066760/n0667602.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n0667603.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n0667604.png" /> a domain of uniqueness, in a neighbourhood of a stationary point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n0667605.png" />. This type is characterized in the following way. There exists a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n0667606.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n0667607.png" /> such that for all trajectories of the system beginning in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n0667608.png" /> the negative semi-trajectories leave in the course of time any compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n0667609.png" />, while the positive semi-trajectories approach <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n06676010.png" /> while not leaving <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n06676011.png" /> and, moreover, being completed with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n06676012.png" />, touch it in well-defined directions, or vice versa. The point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n06676013.png" /> itself is also called a node, a nodal point or a basis point.
 
 
A node is either asymptotically stable in the sense of Lyapunov (cf. [[Lyapunov stability|Lyapunov stability]]) or is totally unstable (asymptotically stable for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n06676014.png" />). The Poincaré index of a node is 1 (cf. [[Singular point|Singular point]]).
 
 
For a system (*) of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n06676015.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n06676016.png" />) with non-zero matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n06676017.png" />, a stationary point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n06676018.png" /> is a node if the eigen values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n06676019.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n06676020.png" /> are real and satisfy the conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n06676021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n06676022.png" />; it can also be a node in cases when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n06676023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n06676024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n06676025.png" />. In case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n06676026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n06676027.png" /> will be a node if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n06676028.png" />; when this condition is not satisfied it may turn out to be [[Focus|focus]]. In any of the cases listed above the trajectories of the system converging to the node <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n06676029.png" /> touch it in well-defined directions, defined by the eigen vectors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n06676030.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n06676031.png" />, there exist four such directions (if diametrically opposite ones are counted as distinct), and two trajectories of the system touch at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n06676032.png" /> in directions corresponding to the eigen value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n06676033.png" /> while two trajectories touch at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n06676034.png" /> in directions corresponding to the eigen value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n06676035.png" /> (Fig. a). These are ordinary nodes. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n06676036.png" />, then the eigen directions for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n06676037.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n06676038.png" /> are either just two opposite directions (in this case the node is degenerate, cf. Fig. b) or all directions. In this last case, under the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n06676039.png" /> every direction is tangent at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n06676040.png" /> to a unique trajectory of the system. Such a node is called dicritical (Fig. c). VOL 5 COL 475 !
 
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/n066760b.gif" />
 
 
Figure: n066760b
 
 
VOL 5 COL 475 !
 
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/n066760c.gif" />
 
 
Figure: n066760c
 
 
VOL 5 COL 475 !
 
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/n066760d.gif" />
 
 
Figure: n066760d
 
 
If the system (*) is linear (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n06676041.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n06676042.png" /> is a fixed matrix) then the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n06676043.png" /> is a node only when the eigen values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n06676044.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n06676045.png" /> are real and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n06676046.png" />. Any ray <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n06676047.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n06676048.png" /> an eigen vector of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n06676049.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n06676050.png" /> a parameter) is a trajectory for it. Ordinary, degenerate and dicritical nodes for a linear system are depicted in Fig. d, Fig. eand Fig. f.
 
 
 
Figure: n066760g
 
 
In the case of an ordinary node all curvilinear trajectories are affine images of parabolas <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n06676051.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n06676052.png" />.
 
 
The term  "node"  is also applied to a stationary point of a system of the form (*) of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066760/n06676053.png" /> with analogous behaviour of the trajectories in neighbourhoods of it.
 
 
For references see [[Singular point|Singular point]] of a differential equation.
 
 
====Comments====
 
  
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Lefshetz,  "Differential equations: geometric theory" , Dover, reprint  (1977)  pp. Sect. IX.2</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Birkhoff,  G.-C. Rota,  "Ordinary differential equations" , Ginn  (1962)  pp. Sect. VI.8</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Lefshetz,  "Differential equations: geometric theory" , Dover, reprint  (1977)  pp. Sect. IX.2</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Birkhoff,  G.-C. Rota,  "Ordinary differential equations" , Ginn  (1962)  pp. Sect. VI.8</TD></TR></table>

Revision as of 06:27, 6 May 2012

The term which may refer to one of the following:

  • A singular (non-smooth) point of an algebraic curve which is a transversal intersection of smooth branches;
  • A vertex of the boundary of a nodal domain in the Harmonic analysis (related to the above and also to the [of the node in Physics]);
  • A singular point of vector field of a special type (with all eigenvalues of the linear part being to one side of the imaginary axis);
  • A point at which the function is evaluated for purposes of interpolation or numeric integration.

Node of a curve

For an algebraic (or analytic) planar curve, the node (or nodal point, also nodal singularity) is a point of self-intersection of smooth branches of the curve.

Example. For the curve $r =s\sin3\varphi$ (in the polar coordinates) the origin of coordinates is a node.

Warning. Often it is additionally required that the curve has only transversal self-intersections. In this case at most two smooth branches of the curve can cross each other at any given point, thus the above example will be not a node in this restricted sense.

Node of a vector field

A singular point of vector field $v(x)=Ax+\cdots$ (mostly on the plane, $x\in(\R^2,0)$, but higher-dimensional nodes in $(\R^n,0)$ are also considered) distinguished by the condition that the real part of all eigenvalues $\lambda_1,\dots,\lambda_n$ of the linear operator $A$ have the same sign. Thus the node can be stable, if $\operatorname{Re}\lambda_i<0$ for all $i=1,\dots,n$, and unstable if $\operatorname{Re}\lambda_i>0$ for all $i=1,\dots,n$. The stability is both in the Lyapunov sense and asymptotic (the unstable node is stable in the reverse time $\tau=-t$).

A stable (unstable) node is topologically equivalent to the standard node $\dot x=x$ (resp., $\dot x=-x$).

The node always has a polynomial Poincare-Dulac formal normal form, see Local normal forms for dynamical systems, linear for non-resonant nodes and integrable in quadratures for resonant nodes. The transformation bringing an analytic node into its normal form always converges.

Differential and analytic type of nodes

Although all nodes sharing the same stability type are topologically equivalent, their $C^k$-smooth classification (for $k\ge 1$) is nontrivial. For simplicity, we will talk only about the "genuine" (planar) nodes of real analytic vector fields. The following types can be distinguished.

  1. Dicritical node with equal eigenvalues $(\lambda,\lambda)$ and diagonal linear part. The corresponding vector field has a continuum of separatrices and can be analytically linearized to its topological normal form times a positive constant $|\lambda|>0$.
  2. Degenerate node with equal eigenvalues and nontrivial Jordan normal form of the linear part. This node has a single analytic separatrix but still can be analytically linearized. All other trajectories are characteristic with the common limit direction.
  3. "Ordinary" resonant node with the ratio of eigenvalues $(\lambda:\mu)=(1:n)$, $n\in\N$, $n\ge 2$. This node also has only one analytic separatrix and is $C^{n-1}$-linearizable, but not $C^n$-linearizable in general. However, in the exceptional case the resonant node may have continuum of analytic separatrices and be analytically linearizable.
  4. "Ordinary" nonresonant node always has two analytic separatrices and is analytically linearizable. All trajectories except for two have the common limit direction which coincides with the eigenvector having the smallest (in the absolute value) eigenvalue. Two exceptional trajectories form the other separatrix tangent to the second eigenvector.



References

[a1] J.L. Coolidge, "Algebraic plane curves" , Dover, reprint (1959)


References

[a1] S. Lefshetz, "Differential equations: geometric theory" , Dover, reprint (1977) pp. Sect. IX.2
[a2] G. Birkhoff, G.-C. Rota, "Ordinary differential equations" , Ginn (1962) pp. Sect. VI.8
How to Cite This Entry:
Node. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Node&oldid=26084
This article was adapted from an original article by A.F. Andreev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article