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Attractor

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Attractors are the pinnacle and origin of chaos theory.

An attractor is a 'set', 'curve', or 'space' that a system irreversibly evolves to if left undisturbed. It is other-wise known as a 'limit set'. There are three types of attractors; point attractors, periodic attractors and strange attractors, all of which are discussed below.

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Example

For instance, 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 phenomena 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:

<math>(x=k,\ {dx \over dt}=0,\ {d^2x \over dt^2}=0)</math>

Phase space

The trajectory representation of a single-variable system is:

<math> x = f(t)\mathbf{}</math>
That is, state(x) is a function of time(t). Similarly, for a multi-variable system, we express x as a vector:
<math>\mathbf{x} = \left[x_1, x_2, x_3, ... , x_n\right]</math>
And say that:
<math>\mathbf{x} = \mathbf{F}(t)</math>

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:
<math>{dx \over dt} = f(x)</math>
Or, in vector notation:
<math>{d\mathbf{x} \over dt} = \mathbf{F}(\mathbf{x})</math>

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

<math>F: X \rightarrow X^\prime</math>
As time approaches infinity(<math> t \rightarrow \infty</math>), the coordinate system contracts into a limit set, or attractor.

The three 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.

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 attractor. It is characterized by a set of coupled nonlinear partial 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 3 coupled nonlinear differential equations:

<math>{dx \over dt} = a (y - x)c</math>
<math>{dy \over dt} = x (b - z) - y</math>
<math>{dz \over dt} = xy - c z</math>
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.

Further Reading

  • "The Essence of Chaos" -Edward N. Lorenz
  • "Chaos: Making a New Science" -James Gleick



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