Stable Attractor - Mathematical Definition

Mathematical Definition

Let t represent time and let f(t, •) be a function which specifies the dynamics of the system. That is, if a is an n-dimensional point in the phase space, representing the initial state of the system, then f(0, a) = a and, for a positive value of t, f(t, a) is the result of the evolution of this state after t units of time. For example, if the system describes the evolution of a free particle in one dimension then the phase space is the plane R2 with coordinates (x,v), where x is the position of the particle, v is its velocity, a=(x,v), and the evolution is given by

An attractor is a subset A of the phase space characterized by the following three conditions:

• A is forward invariant under f: if a is an element of A then so is f(t,a), for all t > 0.

• There exists a neighborhood of A, called the basin of attraction for A and denoted B(A), which consists of all points b that "enter A in the limit t → ∞". More formally, B(A) is the set of all points b in the phase space with the following property:

For any open neighborhood N of A, there is a positive constant T such that f(t,b) ∈ N for all real t > T.

• There is no proper subset of A having the first two properties.

Since the basin of attraction contains an open set containing A, every point that is sufficiently close to A is attracted to A. The definition of an attractor uses a metric on the phase space, but the resulting notion usually depends only on the topology of the phase space. In the case of Rn, the Euclidean norm is typically used.

Many other definitions of attractor occur in the literature. For example, some authors require that an attractor have positive measure (preventing a point from being an attractor), others relax the requirement that B(A) be a neighborhood.