View source for Repelling set

''repellor, in a dynamical system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081310/r0813101.png" />.''

A subset of the phase space of the system that is an attractor for the reverse system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081310/r0813102.png" />. In general, an attractor of a [[Dynamical system|dynamical system]] (cf. also [[Routes to chaos|Routes to chaos]]) is a non-empty subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081310/r0813103.png" /> of the phase space such that all trajectories from a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081310/r0813104.png" /> tend to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081310/r0813105.png" /> when time increases. More precisely, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081310/r0813106.png" /> be the domain of attraction (or basin) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081310/r0813107.png" />, i.e. the set of all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081310/r0813108.png" /> in the phase space for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081310/r0813109.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081310/r08131010.png" /> (that is, for every neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081310/r08131011.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081310/r08131012.png" /> there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081310/r08131013.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081310/r08131014.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081310/r08131015.png" />). If the phase space is locally compact and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081310/r08131016.png" /> is compact, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081310/r08131017.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081310/r08131018.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081310/r08131019.png" />-limit set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081310/r08131020.png" /> (cf. [[Limit set of a trajectory|Limit set of a trajectory]]) (certain authors take this as a definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081310/r08131021.png" /> in the general case). Now, a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081310/r08131022.png" /> of the phase space is called an attractor whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081310/r08131023.png" /> has an open neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081310/r08131024.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081310/r08131025.png" />; in that case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081310/r08131026.png" /> is an open invariant subset of the phase space. If an attractor, respectively repellor, consists of one point, then one speaks of an attracting, respectively repelling, point. For details (e.g., on stability of attractors) see [[#References|[a1]]]. It should be noted that in other literature the definition of an attractor is what is called a stable attractor in [[#References|[a1]]]. For discussions on the  "correct"  definition of an attractor see [[#References|[a2]]], Sect. 5.4, and [[#References|[a3]]]. See also [[Strange attractor|Strange attractor]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.P. Bahtia,   G.P. Szegö,   "Stability theory of dynamical systems" , Springer  (1970)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Guckenheimer,   P. Holmes,   "Non-linear oscillations, dynamical systems, and bifurcations of vector fields" , Springer  (1983)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D. Ruelle,   "Small random perturbations of dynamical systems and the definition of attractors"  ''Comm. Math. Phys.'' , '''82'''  (1981)  pp. 137–151</TD></TR></table>