Tuesday, 11 March 2008

Building intuitions. Informed approach.

On page 44, of "Chaos, Dynamics and Fractals, an algorithmic approach to deterministic chaos", by J. L. McCauley, I read:

"Geometrically, strange attractors are made up of a continuum of points in a way that is qualitatively and nontrivially different from smooth curves and tori: they have a fragmented structure that can be like that of a Cantor set(Chapter 4), and the motion on the Cantor-like set is either completely unstable in the sense that nearby initial conditions yield orbits that develop very unsimilar spatial patterns as a function of time (Chapter 4), or else it is only marginally stable at a boundary of chaos (Chapters 5 and 6)."

Strange attractors, a continuum of points substantially different from smooth curves and tori, the classical attractors. They are more like Cantor sets, dusts, points scattered seemingly haphazardly, which makes the motion unstable as nearby initial conditions produce orbits, tracked by the motion of the phase point developing unsimilar spatial patterns, (the strange attractors?) as the time progresses to infinity. It can be seen as well as being stable marginally on the verge of chaos.

Stability is the essence? There is stability, despite being marginal even when the system develops strange attractors?

In page 41, of "Chaos, Dynamics and Fractals, an algorithmic approach to deterministic chaos", by J. L. McCauley, I read:

"We begin with the way that deterministic chaos was discovered numerically in a system of three coupled nonlinear differential equations by the meteorologist E. Lorenz, in an attempt to integrate the model on a computer. In addition, Lorenz showed analytically that orbits are attracted to enter a certain phase space volume from which escape is impossible, but which contains no stable classical attractors (equilibria, limit cycles, and tori). Consequently, he discovered numerically that nearby initial conditions yielded trajectories with entirely different spatial patterns."

Chaotic orbits, defined by the system of the three nonlinear equations, are attracted to enter a certain phase space volume from which escape is impossible. Phase space volume from which escape is impossible, trapped as time goes by, even to infinity. How that abstract phase space can be visualised in its real form? The model built by the three nonlinear equations, is a replica, it simulates events taking place in the real world. No matter how stripped down, the model is, still depicts a real situation. Weather development and orbits forever trapped in, within the boundaries of a particular phase space. Since it can not escape, confers by it, stability. Within the limits of the phase space, the sum or set of particular states, happen again and again. As the same states are repeated over and over, therefore stability , stable states.

States that, their role is to represent or are directly related to states of the world. The points plotted, swirling around within the boundaries of the phase space, the leading phase point represent the numerical values of variables of the real world. Be that temperature, moisture, wind velocity all closely intertwined, influenced by one another, constantly changing affecting world states and as their numerical counterpart reveals, being trapped within the confines of attractors, the world states that bring about are determined by the range of values permitted by the confining attractors. So there would either be rain, snow or even sunshine. The attractors provide the measure of states of the world they model about. And as the particular attractors, or subsets of attractors, or phase space patterns developed, sensitively depend on the initial conditions of the system, the small fluctuations exponentially amplified, brings about the constantly changing face of the world.


No comments:

Post a Comment

Note: only a member of this blog may post a comment.