View source for Arc, contactless (free)

A smooth curve without self-intersections in the phase plane of a two-dimensional [[Autonomous system|autonomous system]] of 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/a/a013/a013070/a0130701.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

at each point of which the phase velocity vector of the system (cf. [[Phase velocity vector|Phase velocity vector]]) is defined, is non-zero and is not a vector tangent to the curve. The concept was introduced by H. Poincaré [[#References|[1]]], and is extensively employed in the qualitative theory of differential equations [[#References|[2]]]. Thus, it is possible to draw a contactless segment of sufficiently small length through an arbitrary ordinary point of a trajectory of the system (*). A contactless arc is characterized by the fact that all the trajectories of system (*) intersecting the curve intersect it in the same direction. If the derivative along the flow of the system (*) (cf. [[Differentiation along the flow of a dynamical system|Differentiation along the flow of a dynamical system]]) at each point of the smooth curve does not vanish, this curve is a contactless arc. A closed contactless arc is said to be a contactless cycle.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Poincaré,   "Mémoire sur les courbes définies par une équation différentielle I - IV" , ''Oeuvres de H. Poincaré'' , '''1''' , Gauthier-Villars  (1916)  pp. 3–222</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Lefshetz,   "Differential equations: geometric theory" , Interscience  (1962)</TD></TR></table>