Attractor

In a mathematical field of dynamical systems, an attractor is a style of states toward which a system tends to evolve, for a wide set of starting conditions of the system. System values that receive close enough to the attractor values proceed close even whether slightly disturbed.

In finite-dimensional systems, the evolving variable may be represented ]

If the evolving variable is two- or three-dimensional, the attractor of the dynamic process can be represented strange attractor below. whether the variable is a scalar, the attractor is a subset of the real number line. Describing the attractors of chaotic dynamical systems has been one of the achievements of chaos theory.

A trajectory of the dynamical system in the attractor does not gain to satisfy any special constraints apart from for remaining on the attractor, forward in time. The trajectory may be periodic or chaotic. If a set of points is periodic or chaotic, but the flow in the neighborhood is away from the set, the set is not an attractor, but instead is called a repeller or repellor.

Mathematical definition

Let t represent time and let ft, • be a function which specifies the dynamics of the system. That is, if a is a module in an n-dimensional phase space, representing the initial state of the system, then f0, a = a and, for a positive good of t, ft, a is the a thing that is caused or produced by something else 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 R^{2 with coordinates x,v, where x is the position of the particle, v is its velocity, a = x,v, in addition to the evolution is assumption by}

An attractor is a subset A of the phase space characterized by the coming after or as a result of. three conditions:

Since the basin of attraction contains an open set containing A, every an fundamental or characteristic part of something abstract. that is sufficientlyto 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 R^{n, the Euclidean norm is typically used.}

Many other definitions of attractor arise in the literature. For example, some authors require that an attractor pretend positive measure preventing a point from being an attractor, others relax the requirement that BA be a neighborhood.