Let’s discuss attractors. Let’s discuss what they are and then let’s specifically point out the most important ones: strange attractors. We are discussing “rounder”-types now.
An attractor is a set of states (points in the phase space), invariant under the dynamics, towards which neighboring states in a given basin of attraction asymptotically approach in the course of dynamic evolution. An attractor is defined as the smallest unit which cannot be itself decomposed into two or more attractors with distinct basins of attraction. This restriction is necessary since a dynamical system may have multiple attractors, each with its own basin of attraction.Dynamic evolution is the procession of chaotic variation. Chaos is not truly disordered; it is controlled by respective attractors. As stated, attractors have restrictions placed on them in order for them to function individually.
Our most typically defined attractor is the Lorenz Attractor.
The Lorenz attractor is an attractor that arises in a simplified system of equations describing the two-dimensional flow of fluid of uniform depth
, with an imposed temperature difference
, under gravity
, with buoyancy
, thermal diffusivity
, and kinematic viscosity
. The full equations are:
(1)
(2)In the early 1960s, Lorenz accidentally discovered the chaotic behavior of this system when he found that, for a simplified system, periodic solutions of the form
(3)
(4)grew for Rayleigh numbers larger than the critical value, Ra>Ra_c. Furthermore, vastly different results were obtained for very small changes in the initial values, representing one of the earliest discoveries of the so-called butterfly effect.
The Lorenz attractor has a correlation exponent of
and capacity dimension
. As one of his list of challenging problems for mathematics, Smale posed the open question of whether the Lorenz attractor is a strange attractor. This question was answered in the affirmative by Tucker (2002), whose technical proof makes use of a combination of normal form theory and validated interval arithmetic.
The conclusion?
Yes. It is a strange attractor.
An attracting set that has zero measure in the embedding phase space and has fractal dimension. Trajectories within a strange attractor appear to skip around randomly.
Key words have been made bold. They are “non-linear with non-integral dimensions.” To be precise, they are fractal. Their dimensional analysis cannot even be perceived.
“But isn’t this just abstract mathematics?”
An example Lorenz attractor. This rounder-type strange attractor was created in reality through the perturbations of tiny glass sediments within a fluid glass container. All sediments were set to diffuse “chaotically.” Notice these sediments begin to take a pattern around two defined points, which themselves remain as simple “empty spaces.” The attractor that is set in those spaces is not visible; it cannot be defined by our dimensional reality. Its effect on our world only resonates without showing its true form.
