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In the study of dynamical systems, an attractor is a 'set', 'curve', or 'space' to which a system irreversibly evolves, if left undisturbed. It is otherwise known as a 'limit set'. There are five known types of attractors; point attractors, periodic point attractors, periodic attractors, strange attractors, and spatial attractors, all of which are discussed below. Attractors are the pinnacle and origin of chaos theory.

Contents

1 Example

2 Phase space

3 Five types of attractors

3.1 Point attractor
3.2 Periodic point attractor
3.3 Periodic attractor (a.k.a. limit-cycle)
3.4 Strange attractor
3.5 Spatial attractor

4 Further reading

5 External links

Example

If you drop a book, it will land on the floor, and stop moving. This final state is the attractor of the system of "the book dropping". The book has now lost its potential energy, and is in a state of equilibrium. The type of attractor exhibited by this phenomenon is known as a 'point attractor', because the limit set consists of a single point: position = constant, velocity = zero, acceleration = zero. Mathematically stated (see differential equations), we say:

$(x=k,\ {dx \over dt}=0,\ {d^2x \over dt^2}=0).$

Phase space

The trajectory representation of a single-variable system is:

$x = x(t).\,$

That is, state (x) is a function of time (t). Similarly, for a multi-variable system, we express x as a vector:

$\mathbf{x} = \left[x_1, x_2, x_3, ... , x_n\right].\,$

And say that:

$\mathbf{x} = \mathbf{x}(t).\,$

The phase space representation of a single-variable system, however, expresses the change of state of the system with respect to time (dx/dt) as a function of the current state of the system:

${dx \over dt} = f(x).$

Or, in vector notation:

${d\mathbf{x} \over dt} = \mathbf{F}(\mathbf{x}).$

Where F is a transformation matrix (see control systems) or tensor describing a nonlinear transformation , mapping x onto a new coordinate system:

$F: X \rightarrow X^\prime.$

As time approaches infinity (t → ∞), the coordinate system contracts into a limit set, or attractor.

Five types of attractors

Point attractor

A point attractor is a fixed point that a system evolves towards, such as a falling book, a damped pendulum , or the halting state of a computer. Compare this to a fixed point of a function.

Periodic point attractor

A periodic point attractor is a finite-length repeating loop of discrete states, i.e. a repeating succession of 'quasi'-point attractors (quasi in that they are only point attractors in a (temporally) local sense). Examples include the time on a digital clock or an infinite loop of a computer.

Periodic attractor (a.k.a. limit-cycle)

A periodic attractor is a repeating loop of states. A planet orbiting around a star is an example of a periodic attractor. Also, an undamped pendulum and an infinite loop on a digital computer are examples of periodic attractors.

Strange attractor

A strange attractor is a non-periodic attractor. This is the most common type of (not spatially-extended) attractor. It is characterized by a set of coupled nonlinear ordinary differential equations. The first strange attractor discovered was the Lorenz attractor, discovered by the meteorologist Edward Lorenz, while simulating weather on a computer.

The Lorenz attractor is defined by a set of three coupled nonlinear differential equations:

${dx \over dt} = a (y - x)$

${dy \over dt} = x (b - z) - y$

${dz \over dt} = xy - c z$

where a = 10, b = 28, c = 8 / 3. Strange attractors have fractal structure.

These last two types of attractors are exhibited by what are called dissipative systems. Dissipative systems are systems not in thermodynamic equilibrium, but constantly "evolving towards" equilibrium. That is, they are characterized by a flow of entropy, and mutually, a flow of energy.

Spatial attractor

Spatial attractors are unique from the other types of attractors in that they are spatially extended. Examples of spatial attractors include Turing structures and pseudo-examples include periodic point attractors in cellular automata. See also excitable medium.

Above: Pseudo-example of a spatial attractor - a glider in Conway's Game of Life.

External links

A gallery of polynomial strange attractors

Categories: Dynamical systems

01-04-2007 01:18:14
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