Arc, contactless (free)

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A smooth curve without self-intersections in the phase plane of a two-dimensional autonomous system of differential equations

$$ \tag{* } \dot{x} = P ( x , y ) ,\ \dot{y} = Q ( x , y ) , $$

at each point of which the phase velocity vector of the system (cf. Phase velocity vector) is defined, is non-zero and is not a vector tangent to the curve. The concept was introduced by H. Poincaré [1], and is extensively employed in the qualitative theory of differential equations [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) 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.


[1] H. Poincaré, "Mémoire sur les courbes définies par une équation différentielle I - IV" , Oeuvres de H. Poincaré , 1 , Gauthier-Villars (1916) pp. 3–222
[2] S. Lefshetz, "Differential equations: geometric theory" , Interscience (1962)
How to Cite This Entry:
Arc, contactless (free). Encyclopedia of Mathematics. URL:,_contactless_(free)&oldid=45210
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article