# Arc, contactless (free)

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.

#### References

[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: http://encyclopediaofmath.org/index.php?title=Arc,_contactless_(free)&oldid=45210