Information AboutStrange Attractor |
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A trajectory of the dynamical system in the attractor does not have to satisfy any special constraints except for remaining on the attractor. The trajectory may be periodic or chaotic or of any other type. MOTIVATION AND DEFINITION , Thermodynamic Losses , or loss of material, among many causes.) The dissipation and the driving force tend to combine to kill out initial transients and settle the system 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'''. Invariant sets and limit sets are similar to the attractor concept. An invariant set is a set that evolves to itself under the dynamics. Attractors may contain invariant sets. A limit set are the states a system goes to after an infinite amount of time. Attractors are limit sets, but 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 that knock it off to never return. As an example, the Damped Pendulum has two invariant points: the point ''x''0 of minimum height and the point ''x''1 of maximum height. The point ''x''0 is also a limit set, as trajectories converge to it; the point ''x''1 is not a limit set. Because of the dissipation, the point ''x''0 is also an attractor. If there were no dissipation, ''x''0 would not be an attractor. Mathematical definition In a dynamical system with dynamics ''f''(''t'', •), the attractor Λ is a Subset of the phase space such that:
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 , Lines , Surface s, Volume s. 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'', as described in the section below. 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 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 will no longer be closed, and the limit cycle becomes a limit torus. We call this kind of attractor -torus if there are incommensurate frequencies. For example it is a 2-torus: A time series corresponding to this attractor is a ''quasiperiodic'' series: A discretely sampled sum of periodic functions (not necessarily sine waves) with incommensurate frequencies. Such a time series does no longer 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 the attractor are Chaotic . The term was coined by David Ruelle and Floris Takens to describe the attractor that resulted from a series of Bifurcation s of a system describing fluid flow. ''Strange attractors'' are often differentiable in a few directions and Like a Cantor Dust (and therefore not differentiable) in others. 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 Equation s are all known to have global attractors of finite dimension. For the three-dimensional, incompressible Navier-Stokes equation with periodic Boundary Condition s, if it has a global attractor, then this attractor will be of finite dimension. FURTHER READING
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