Thursday, 20 August 2009

Low-dimensional systems embedded in macroscopic systems. Enhanced fluctuations?

I feel I have stumbled once again onto something which will augment my knowledge, my perspective. It will drag me away from my current endeavours to bring me back enhanced and ready for new challenges.

Writing unhindered by any considerations deemed irrelevant for my pursuits. Enough introduction, back to the point.

Šil'nikov phenomena, are referred to, in this paper titled 'Poincaré versus Boltzmann in Šil'nikov phenomena'. It was the climax of my google searching, following terms like, subordinate Šil'nikov bifurcations, Sil'nikov-saddle-node interaction, Šil'nikov phenomena, Sil'nikov chaos.

Analyse the terms and deepen my knowledge by examining the effects, the mentioned concepts had on the knowledge already held in my reverberated neural nets and in the process ascend whatever fragments are there, and emerge in new knowledge, new ways of looking at things.

In the abstract of the paper, 'Poincaré versus Boltzmann in Šil'nikov phenomena', it mentions

"By suitable adjustment of the control parameters in a CO2 laser with feedback we show experimental evidence of Sil'nikov chaos, ..."

... control parameters, as in parameters and variables. Parameters, of which their values are considered if not unchanged, changing within a very narrow limiting and limited range, as in any fractal-making program, what you tweak ever so slightly, to get all kinds, versions of the initial fractal. Suitably adjusting the parameters, deemed as controlling in the CO2 laser. Feedback providing the iteration part, the recursiveness in function, for chaos coming.

Experimental evidence of Sil'nikov chaos? What kind of chaos is this?
It further mentions,

".. characterized by intensity pulses almost equal in shape, but irregularly separated in time."

Intensity pulses irregularly separated in time, their experimental setup signature of chaos. Irregularity signifying chaos.

"The times of return to a Poincare section are statistically spread,.."

Poincare sections traversing its flow in time, incisions right through its fractal space, revealed statistically spread times of return, the order hidden amidst chaos. Chaos generating order.

Chaos as described in mathematics, measuring fluctuations in the intensity of forces, stresses, pressures plotted into graphs or even plain mathematical functions, self-referential, iterated loosing themselves into chaos. Translated into all kinds of forces exhibited in all systems possible. Physical, mental, emotional, psychological, social, spiritual systems analysed in the very same way as mathematical chaos.

"... however their iteration map is one-dimensional and in close agreement with that arising from Sil'nikov theory."

The map of the iterated points is one-dimensional? All falling onto a single line? Or the map, being a function and the one-dimensional referred to, its mathematical property. And all this, .. 'in close agreement with that arising from Sil'nikov theory'.

"Thus, the iteration map of the time intervals becomes the most appropriate indicator of this chaos."

An appropriate indicator of this chaos? Sil'nikov chaos. The one-dimensional iteration map of the time intervals?

"The residual width of the experimentally measured maps is due to a transient fluctuation enhancement peculiar to macroscopic systems, which is absent in low-dimensional chaotic dynamics."

Macroscopic systems distinct from low-dimensional chaotic dynamics? Underlying systems different than systems above? Chaotic dynamics of underlying systems, systems embedded within systems above, low-dimensional as embedded systems are bound to have less dimensions, than the system they are embedded in, embedding system which includes more than one low-dimensional systems.

The transient fluctuation enhancement peculiar to macroscopic systems? Due to the influence of additional parametres a low-dimensional system is exposed to, by the other low-dimensional system(s) that together comprise the macroscopic system they are embedded in?

Each low-dimensional system compensating for the introduced parameters, tweaked. Embedded low-dimensional systems integrated and together now determine the properties, attributes, the behaviour of the macroscopic system.

My very own thoughts developing in a similar process. All systems possible, physical psychological, social developing likewise.

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