repellors

Repellors are a term used primarily in dynamical systems to describe an invariant set that pushes nearby trajectories away as time advances. In broader usage, the word may refer to devices or substances that deter animals or pests, but in scientific writing the pest-control sense is usually called a repeller.

Formally, a set S in a dynamical system is a repellor if it is invariant and there exists a neighborhood U of S such that for every x in U, the forward-time orbit of x moves away from S and leaves every smaller neighborhood of S as t grows. Equivalently, repellors are the time-reversal of attractors: reversing time turns a repellor into an attractor. In the linearization near a fixed point, a fixed point is a repellor if all eigenvalues of the Jacobian have positive real parts, indicating local instability.

Examples include the origin for the ordinary differential equation dx/dt = x, where x = 0 is unstable and trajectories diverge from 0 as t increases. In chaotic dynamics, repellors can be fractal invariant sets that control the forward evolution of nearby orbits and are studied using their unstable manifolds.

Repellors are distinguished from attractors by their forward-time behavior; they are often contrasted in discussions of stability and bifurcations. The term is also encountered as a variant spelling; the more common everyday usage is repeller or repellant, referring to biological or chemical deterrents.