Re: OT: Wow, Vatilla Part 2
The hyperbolic mathematical model of a strange attractor was developed by the mathematicians Steve Smale, who we’ve discussed before, my mathematical grandfather, and Bob Williams. I met Bob once at a Chaos Conference around 1980.
Smale - Williams attractor
Mathematical theory of chaos based on a rigorous mathematical approach deals with strange attractors of hyperbolic type. In such attractor all belonging orbits in phase space are of saddle type, and the invariant sets of trajectories approaching the original one in forward or backward time (stable and unstable manifolds) intersect transversally, i.e. without touch.
Unfortunately, most of known physical systems like simple chaotic generators, nonlinear oscillators with periodic driving etc. do not belong to the class of systems with hyperbolic attractors. Chaos in them is associated as arule with the so-called quasi-attractor, which contains together with chaotic trajectories also a set of stable orbits of large periods (not observable in computations because of extremely narrow domains of attraction).
Hyperbolic strange attractors are robust (structurally stable). It means non-sensitivity of character of motions and mutual disposition of orbits in the phase space in respect to variation of the equations of the system. In contrast with the hyperbolic attractors, the quasi-attractors manifest sensitive dependence of details of the dynamics on parameters. Obviously, it is not so good in potential applications of chaos, like secure communications, signal masking etc. Hence, both from the point of view of fundamental studies and of applications it would be interesting to find physical examples of the hyperbolic chaos.
In text-books on nonlinear dynamics examples of the hyperbolic attractors are represented by abstract constructions. For instance, the Smale - Williams attractor is constructed for a three-dimensional map defined by the following procedure. Let us consider a domain of toral form, stretch it twice and fold and embed into the original object as shown in the picture. At each next iteration a number of coil is doubled. The object forming in a limit of infinite iterations is called the Smale - Williams solenoid. The transversal structure of it is a Cantor-like set.
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Similar to the construction of the Smale Horseshoe and Cantor Set, one takes the infinite intersection of life the forward iterates of the torus (mathematical donut). Frank his gives a Cantor Set of a new dimensional fibers which are expanding in time. Again, a hyperbolic dynamical system has a “moving frame” of eigenvectors to the total derivative Jacobian linear transformation matrix map. in this case, the direction of the one-dimensional fiber is the expanding direction and the “cross section” directions are contracting. Like the Horseshoe the Solenoid dynamical system can be coded as a Bernoulli system, a “fair coin toss”, so it is very chaotic and random, although it arises in a deterministic dynamical system.
Cheers,
Frank |
