From Encyclopedia of Mathematics
Revision as of 17:00, 7 February 2011 by (talk) (Importing text file)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

A point of self-intersection of a curve. For a parametrically given curve a node corresponds to two or more values of the parameter. E.g. for the curve the origin of coordinates is a node.

Figure: n066760a


A reference for "node of a curve" is [a1].


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

A node is a type of arrangement of the trajectories of an autonomous system of second-order ordinary differential equations


, a domain of uniqueness, in a neighbourhood of a stationary point . This type is characterized in the following way. There exists a neighbourhood of such that for all trajectories of the system beginning in the negative semi-trajectories leave in the course of time any compact set , while the positive semi-trajectories approach while not leaving and, moreover, being completed with , touch it in well-defined directions, or vice versa. The point 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) or is totally unstable (asymptotically stable for ). The Poincaré index of a node is 1 (cf. Singular point).

For a system (*) of class () with non-zero matrix , a stationary point is a node if the eigen values of are real and satisfy the conditions , ; it can also be a node in cases when , , . In case , will be a node if ; when this condition is not satisfied it may turn out to be focus. In any of the cases listed above the trajectories of the system converging to the node touch it in well-defined directions, defined by the eigen vectors of . If , there exist four such directions (if diametrically opposite ones are counted as distinct), and two trajectories of the system touch at in directions corresponding to the eigen value while two trajectories touch at in directions corresponding to the eigen value (Fig. a). These are ordinary nodes. If , then the eigen directions for at 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 every direction is tangent at to a unique trajectory of the system. Such a node is called dicritical (Fig. c). VOL 5 COL 475 !

Figure: n066760b

VOL 5 COL 475 !

Figure: n066760c

VOL 5 COL 475 !

Figure: n066760d

If the system (*) is linear (, where is a fixed matrix) then the point is a node only when the eigen values of are real and . Any ray ( an eigen vector of , 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: n066760e

Figure: n066760f

Figure: n066760g

In the case of an ordinary node all curvilinear trajectories are affine images of parabolas , .

The term "node" is also applied to a stationary point of a system of the form (*) of order with analogous behaviour of the trajectories in neighbourhoods of it.

For references see Singular point of a differential equation.



[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:
This article was adapted from an original article by A.F. Andreev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article