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  • Question 30 - Attractors
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An attractor is a set to which a dynamical system evolves after a long enough time. That is, points that get close enough to the attractor remain close even if slightly disturbed. Geometrically, an attractor can be a point, a curve, a manifold, or even a complicated set with a fractal structure known as a strange attractor. Describing the attractors of chaotic dynamical systems has been one of the achievements of chaos theory.

Types of attractors

Attractors are parts of the phase space of the dynamical system.

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

Weakly attracting fixed point for complex quadratic polynomial

A fixed point is a point of a function that does not change under some transformation. If we regard the evolution of a dynamical system as a series of transformations, then there may or may not be a point which remains fixed under the whole series of transformation. In general there would not be such a point, but there may be one. The final state that a dynamical system evolves towards, such as the final states of a falling pebble, a damped pendulum, or the water in a glass corresponds to a fixed point of the evolution function, and will occur at the attractor, but the two concepts are not equivalent. A marble rolling around in a basin may have a fixed point in phase space even if it doesn't in physical space. Once it has lost momentum and settled into the bottom of the bowl it then has a fixed point in physical space, phase space, and is located at the attractor for that system.

Limit cycle

A limit cycle is a periodic orbit of the system that is isolated. Examples include the swings of a pendulum clock, the tuning circuit of a radio, and the heartbeat while resting. The ideal pendulum is not an example because its orbits are not isolated. In phase space of the ideal pendulum, near any point of a periodic orbit there is another point that belongs to a different periodic orbit.

Limit tori

There may be more than one frequency in the periodic trajectory of the system through the state of a limit cycle. If two of these frequencies form an irrational fraction (i.e. they are incommensurate), the trajectory is no longer closed, and the limit cycle becomes a limit torus. We call this kind of attractor Nt-torus if there are Nt incommensurate frequencies.

A time series corresponding to this attractor is a quasiperiodic series: A discretely sampled sum of Nt periodic functions (not necessarily sine waves) with incommensurate frequencies. Such a time series does not have a strict periodicity, but its power spectrum still consists only of sharp lines.

Strange attractor

An attractor is informally described as strange if it has non-integer dimension or if the dynamics on it 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. Strange attractors are often differentiable in a few directions, but some are like a Cantor dust, and therefore not differentiable.

Examples of strange attractors include the Hénon attractor, Rössler attractor, Lorenz attractor, Tamari attractor.

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