Difference between revisions of "Arc, contactless (free)"
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A smooth curve without self-intersections in the phase plane of a two-dimensional [[Autonomous system|autonomous system]] of differential equations | A smooth curve without self-intersections in the phase plane of a two-dimensional [[Autonomous system|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|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. | 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. | ||
Latest revision as of 18:48, 5 April 2020
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) |
Arc, contactless (free). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Arc,_contactless_(free)&oldid=18363