Thursday, 14 June 2007

Measure Lyapunov exponents

28052007 MON

Note in 12052007 SAT 1105 brings the idea to use vs. in title for "Our linear vs. non-linear lives". Use the idea of dissipation of energy from the large scales down the small scales in James Gleick Chaos book...

0802 Chaos page 115. Becoming familiar with the concept of self-similarity. Early sense, an organising principle, limitations on the human experience of scale, how to imagine the very great and very small, the very fast with the very slow, but as extensions of the known? Human vision extended by telescopes and microscopes. New discoveries were realisations that change of scale brought new phenomena and new kinds of behaviour. This statement should go along in a discussion of hierarchical system organisation that each system within another has its own phenomena and behaviour which are revealed as we change scale but are self-similar as the organising principle.
0813 Continued: the process for particle physics has never ended. Accelerators as they increase in energy and speed extend science's field of view to tinier particles and briefer time scale and every extension bring new information.
0817 Continued: the parallel trend in science of reductionism. Break things apart and look at them one at a time. The power of self-similarity begins at much greater levels of complexity. It is a matter of looking at the whole. Avoid reducing the complexity when you examine a system, leave the complexity intact and apply self-similarity on the whole and not the parts.

0823 Always when we examine a factor be that a condition or a feature we should make certain we examine from the perspective of the part and its connections with the whole, leave its connections with the whole intact and examining applying self-similarity.
Transfer knowledge the information from other disciplines other systems and applies it to explain the phenomena from the known perspective.

0830 James Gleick page 117. Statements for art. How we perceive art.

0833 Gleick page 260. Struck gold, Shaw’s view that chaotic and near-chaotic systems bridged the gap between the macroscales and microscales. Chaos was the creation of information.
0844 Developing a basic argument to justify learning. Displaying the connections with chaos going along my personal experience, my
practice followed as a father and a powerful argument against the uneducated prevailing attitudes about the importance of school education. Quoting Packard, which I have to find his whereabouts, talking about thought processes, complicated systems that generate information. Information created and stored in our structure. In the development of one person's mind from childhood.

0854 a full circle and back to the task set for the day. Inserting a deeper concept in the chaosandorder project. I have to assimilate in the text.

0856 Lyapunov: page 253, a way of measuring the conflicting effects of stretching, contracting and folding in the phase space of an attractor. Finding regions that lead to stability and instability in the phase space of an attractor. Talking about phase space it
means the whole of the space a system unfolds its processes, the encompassing environment? Not the physical environment the process space that in human life systems is the ideas thoughts contained in cultures and its manifestations the events driven by culture. This is the phase space of the attractor and there the Lyapunov exponents are applied to provide a way of measuring the conflicting effects of stretching, contracting, and folding to determine what properties of a system would lead to stability or instability.

0916 Lyapunov continued: Exponents values above zero positive nearby points separate. Stretching. Instability? Values below zero negative nearby points converge. Contracting. Stability? A fixed-point attractor all Lyapunov exponents negative. Contraction. The direction of the pull inward towards the final steady state. Stability then. Negative values, contraction stability. Positive values, stretching, points diverge, instability. A strange attractor had to have at least one positive Lyapunov exponent.

0928 The Lyapunov idea developed in the most practical ways. Measure Lyapunov exponents and relate them to other important properties. How some systems can create disorder (instability) in one direction while remaining trim and methodical (stability) in another direction. Direction provided as a result of changing values in the variables defining features and conditions. Using Lyapunov analysis to determine contraction or stretching, convergence or divergence, stability or instability.

0937 Lyapunov exponents: (Oh where will I stop). Page 255. What happened to a tiny cluster of nearby points-representing initial conditions-on a strange attractor as the system evolved in time. Nearby points on a strange attractor? What is the significance of this? Has the strange attractor need to be defined first? Does it only apply to points on the strange attractor or the whole of its phase space? It should apply to all the points the main prerogative being to be nearby. Points anywhere in any given neighbourhood and applying Lyapunov analysis to determine whether they will diverge or converge. It is expected that if they are found outside the attractor they would certainly diverge. The most interesting objective to look for is when while on the attractor, therefore stable whether they will diverge or converge. We can accept for systems like life systems any given points to be lying on the strange attractor for that system and with that assumption to continue Lyapunov analysis to see whether the systems would lead to stability or instability.

0954 experiment described: The cluster began to spread out and loose focus, Lyapunov exponents positive separation, divergence. Instability. It turned into a dot then a blob. For certain kinds of attractors? What is that? The blob would quickly spread all over. Such attractors were efficient at mixing. Mixing? Bring notions of integration in a multicultural society, a social event where mixing between its guests is the goal of the host . In societies of nations, in world scene the mixing of cultures, there is quite a lot of mixing needed in the world. Has this anything to do with that? Attractors that are efficient mixers? Are attractors classified taken off the self for use in any occasion? A desired occasion? This kind of attractor leads to instability, separation divergence, however that effect is desirable in that particular case. Does that have significance in human life systems?

1007 the consequent effect of these tendencies is on the predictability factor. Divergence leads to unpredictability (spreading, positive), convergence to predictability (converging, negative). Predictability (representing order) is desired, unpredictability undesired (representing disorder). Attractors whether the spreading would only occur in one direction. Chaotic in one direction and orderly along another. The system had an orderly impulse and a disorderly together. And they were decoupling. Decoupling? One impulse led to unpredictability and the other kept time like a precise clock. Defined and measured. 1019 copy them in a notepad.

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