# Repelling set

*repellor, in a dynamical system $\{S_t\}$.*

A subset of the phase space of the system that is an attractor for the reverse system $\{S_{-t}\}$. In general, an attractor of a dynamical system (cf. also Routes to chaos) is a non-empty subset $D$ of the phase space such that all trajectories from a neighbourhood of $D$ tend to $D$ when time increases. More precisely, let $A(D)$ be the domain of attraction (or basin) of $D$, i.e. the set of all points $y$ in the phase space for which $S_ty\to D$ as $t\to\infty$ (that is, for every neighbourhood $V$ of $D$ there is an $r>0$ such that $S_ty\in V$ for all $t\geq r$). If the phase space is locally compact and $D$ is compact, then $A(D)=\{y\colon\emptyset\neq\Omega_y\subset D\}$, where $\Omega_y$ is the $\omega$-limit set of $y$ (cf. Limit set of a trajectory) (certain authors take this as a definition of $A(D)$ in the general case). Now, a subset $D$ of the phase space is called an attractor whenever $D$ has an open neighbourhood $U$ such that $U\subset A(D)$; in that case $A(D)$ 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 [a1]. It should be noted that in other literature the definition of an attractor is what is called a stable attractor in [a1]. For discussions on the "correct" definition of an attractor see [a2], Sect. 5.4, and [a3]. See also Strange attractor.

#### References

[a1] | N.P. Bahtia, G.P. Szegö, "Stability theory of dynamical systems" , Springer (1970) |

[a2] | J. Guckenheimer, P. Holmes, "Non-linear oscillations, dynamical systems, and bifurcations of vector fields" , Springer (1983) |

[a3] | D. Ruelle, "Small random perturbations of dynamical systems and the definition of attractors" Comm. Math. Phys. , 82 (1981) pp. 137–151 |

**How to Cite This Entry:**

Repelling set.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Repelling_set&oldid=32987