Attractor

In dynamical systems, an attractor is a set to which the system evolves after long enough time. Once trajectories get close enough to an attractor, they remain close even if slightly disturbed. An attractor can be a point, a periodic trajectory, or even complicated sets with fractal structures—strange attractors. Describing the attractors of chaotic dynamical system has been one of the achievements of chaos theory.

1 Motivation and definition
- 1.1 Mathematical definition
2 Types of attractors
3 Partial differential equations
4 Further reading
5 External links
6 References

Motivation and definition

Dynamical systems are often described in terms of differential equations. These equations describe the behavior of the system for a short period of time. To determine the behavior for longer periods it is necessary to integrate the equations, either through analytical means or through iteration, often with the aid of computer. Dynamical systems that come from applications tend to be dissipative: if it were not for driving force the motion would cease. (The dissipation may come from internal friction, thermodynamic losses, or loss of material, among many causes.) Because of dissipation there tends to be an initial transient motion before the system settles into its typical behavior. The part of the phase space of the dynamical system corresponding to the typical behavior is the attracting set or attractor.

There are three concepts in dynamical systems that are similar: invariant sets, limit sets, and attractors. An invariant set is a set that evolves to itself under the dynamics. A limit set represent the states a system goes to after an infinite amount of time. For example, in a damped pendulum, the system eventually comes to rest at its lowest point—a limit point and an invariant point. A limit set is invariant, but invariant points need not be limit points. For example, the equilibrium point when the pendulum is at its maximum height. If carefully placed in that position it will remain there forever. So it is an invariant point. But it is not a limit point, for once the system leaves that position it never returns. Not all limit sets are attractors. It is possible to have a system converge to a limit set, but if placed in the limit set, have small perturbations knock it off to never return.

Mathematical definition

In a dynamical system with dynamics f(t, •), the attractor Λ is a subset of the phase space such that:

there is a neighborhood of Λ, the attraction basin, that will converge to any open containing Λ, and
f(t, Λ) ⊃ Λ for large t.

The open subset condition assures that phase space points in the neighborhood of the attractor converge to it.

Types of attractors

Attractors are parts of the phase space of the dynamical system. Until the 1960s, as evidenced by contemporaneous textbooks, attractors were thought of as being geometrical subsets of the phase space: points, lines, surfaces, volumes. The (topologically) wild sets that had been observed were thought to be fragile anomalies. Stephen Smale was able to show that his horseshoe map was robust and that its attractor had the structure of a Cantor set.

Two simple attractors are the fixed point and the limit cycle. There can be many other geometrical sets that are attractors. When these sets (or the motions on them) are hard to describe, then the attractor is a strange attractor.

Fixed point

A fixed point is a point that a system evolves towards, such as the final states of a falling pebble, a damped pendulum, or the water in a glass. It corresponds to a fixed point of the evolution function that is also attracting.

Limit cycle

A limit cycle is a repeating sequence of states, a periodic orbit of the dynamics. Examples include the swings of a pendulum clock, the tuning circuit of a radio, and the heartbeat while resting. If the dynamical system is discrete in time, then the limit cycle could be a finite-length repeating loop of discrete states.

Strange attractor

An attractor is informally described as strange if it has non-integer dimension or if the dynamics on the attractor are chaotic. The term was coined by David Ruelle and Floris Takens to describe the attractor that resulted from a series of bifurcations of a system describing fluid flow.

The Hénon attractor and the Lorenz attractor are examples of strange attractors.

Partial differential equations

Parabolic partial differential equations may have finite-dimensional attractors. The diffusive part of the equation damps higher frequencies and in some cases leads to a global attractor. The Ginzburg-Landau, the Kuramoto-Sivashinsky, and the two-dimensional, forced Navier-Stokes equations are all known to have global attractors of finite dimension.

For the three-dimensional, incompressible Navier-Stokes equation with periodic boundary conditions, if it has a global attractor, then this attractor will be of finite dimension.

External links

A gallery of polynomial strange attractors

References

D. Ruelle and F. Takens (1971). "On the nature of turbulence". Communications of Mathematical Physics 20: 167–192.

David Ruelle (1989). Elements of Differentiable Dynamics and Bifurcation Theory. Academic Press. ISBN 0126017107.

R. Temam (1997). Infinite dimensional dynamical systems in mechanics and physics, 2nd edition. Springer-Verlag. ISBN 038794866X.

Categories: Dynamical systems