On a sort of on-topic note, all the phenomena described below may be applied to any fluctuating system such as the stoch market. To explain why gold price is high and junior gold shares prices are low, we may write: $IAPOJGS = (( POG*(NFfs)M1) - APOOC*(NFfs)M1 - NOIIIOS*(NFf2)M1) )*PMMP )*T~K(ch)m
Where IAPOJGS = Index of Average Price of Junior Gold Shares
POG = price of gold
Nffs = normalizing fudge factor
Nff2= second normalizing fudge factor
M1 = money to buy stock
APOOC = average price of other commodities
NOIIIOS = number of interesting investments in other sectors
PMMP = precious metal market phase factor. or insert random number
T~K(ch)m = Time (frequency) Derived Market Chaos Constant which assigns a value due to turbulence. This is derived from a physical model which imitates the real world by having the same inputs. Due to trouble with setting the amplitude and frequency of starting points of the manifold inputs and the system dynamics per se, its necessary run time cannot be sped up beyond real world. The phase space and amplitude of its outputs may not correspond exactly. The randomness of its phase dynamics may correspond roughly.
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It depends on whether you consider weather and climate chaotic or random. Weather may not be chaotic in the sense that you get 40 below in the summer, or hurricanes in the winter, but it may be chaotic, or semi-chaotic in that the timing, duration or frequency of these events is not predictable. Some would argue this is randomness not chaos. Climate is subtlely diffent. Climate swings over time may be chaotic within certain bounds.
For a dynamical system to be classified as chaotic, it must have the following properties:
* it must be sensitive to initial conditions,
* it must be topologically mixing, and
* its periodic orbits must be dense.
Sensitivity to initial conditions means that each point in such a system is arbitrarily closely approximated by other points with significantly different future trajectories. Thus, an arbitrarily small perturbation of the current trajectory may lead to significantly different future behaviour.
Sensitivity to initial conditions is popularly known as the "butterfly effect", so called because of the title of a paper given by Edward Lorenz in 1972 to the American Association for the Advancement of Science in Washington, D.C. entitled Predictability: Does the Flap of a Butterfly’s Wings in Brazil set off a Tornado in Texas? The flapping wing represents a small change in the initial condition of the system, which causes a chain of events leading to large-scale phenomena. Had the butterfly not flapped its wings, the trajectory of the system might have been vastly different.
Sensitivity to initial conditions is often confused with chaos in popular accounts. It can also be a subtle property, since it depends on a choice of metric, or the notion of distance in the phase space of the system. For example, consider the simple dynamical system produced by repeatedly doubling an initial value (defined by the mapping on the real line from x to 2x). This system has sensitive dependence on initial conditions everywhere, since any pair of nearby points will eventually become widely separated. However, it has extremely simple behaviour, as all points except 0 tend to infinity. If instead we use the bounded metric on the line obtained by adding the point at infinity and viewing the result as a circle, the system no longer is sensitive to initial conditions. For this reason, in defining chaos, attention is normally restricted to systems with bounded metrics, or closed, bounded invariant subsets of unbounded systems.
Even for bounded systems, sensitivity to initial conditions is not identical with chaos. For example, consider the two-dimensional torus described by a pair of angles (x,y), each ranging between zero and 2p. Define a mapping that takes any point (x,y) to (2x, y + a), where a is any number such that a/2p is irrational. Because of the doubling in the first coordinate, the mapping exhibits sensitive dependence on initial conditions. However, because of the irrational rotation in the second coordinate, there are no periodic orbits, and hence the mapping is not chaotic according to the definition above.
Topologically mixing means that the system will evolve over time so that any given region or open set of its phase space will eventually overlap with any other given region. Here, "mixing" is really meant to correspond to the standard intuition: the mixing of colored dyes or fluids is an example of a chaotic system.
Linear systems are never chaotic; for a dynamical system to display chaotic behaviour it has to be nonlinear. Also, by the Poincaré–Bendixson theorem, a continuous dynamical system on the plane cannot be chaotic; among continuous systems only those whose phase space is non-planar (having dimension at least three, or with a non-Euclidean geometry) can exhibit chaotic behaviour. However, a discrete dynamical system (such as the logistic map) can exhibit chaotic behaviour in a one-dimensional or two-dimensional phase space.
Chaos in the Atmosphere
Before they could understand how climates change, scientists would have to understand the basic principles for how any complicated system can change. Early studies, using highly simplified models, could see nothing but simple and predictable behavior, either stable or cyclical. But in the 1950s, work with slightly more complex physical and computer models turned up hints that even quite simple systems could lurch in unexpected ways. During the 1960s, computer experts working on weather prediction realized that such surprises were common in systems with realistic feedbacks.
The climate system in particular might wobble all on its own without any external push, in a "chaotic" fashion that by its very nature was unforeseeable.
By the mid 1970s, most experts found it plausible that at some indeterminate point a small push (such as adding pollution to the atmosphere) could trigger severe climate change. While the largest effects could be predicted, important details might lie forever beyond calculation.
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Do climate theorists want their indeterminate cake and to probably eat it too? Do they want climate to be indeterminate and chaotic, but to PREDICT that a small push of CO2 will definitely make it go inexorably one way? Why not up and down? Why not down? Why, if Climate is chaotic, can we point to any one single thing as definitely causational in any particular way?
My thesis is if a butterfly's wings can make a climate change, then you cannot just go around stomping on butterflies as you don't know which ways the wings beat, positif ou negatif.
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Another thesis is to assume large systems are little effected by individual small inputs but more affected by accumulation of these inputs. Most study of chaotic natural systems has been by analysis with relatively impoverished collection of inputs and perhaps unrelated outputs. Most study of inputs of chaotic systems in artificial environments has been with rather macro or fuzzy examination of inputs on rather small and bounded but rather finely and complexly interacting systems. If they measured the pendulum swing of the chaotic system with a micrometer and transit and its release was made to the microsecond without any input artifact perhaps the predictive nature of the supposed chaotic systems would be not so chaotic. In other words if as Laplace thought that all inputs could be accounted for finely enough, any system is predictable. At a certain level of input and bounds analysis some complexly interacting systems are chaotic it can safely be said.
But people fear that large complex systems are not steady state with random perturbations but in their very complexity given to mass changes in state. Fermi with his Maniac computer was one of the first to realize this quite accidentally.
For example, a pair of scientists wrote a simple system of equations for the exchanges of carbon dioxide gas among the Earth's atmosphere, oceans, and biosphere, and ran the equations through a computer. The computations tended to run away into self-sustaining oscillations. In the real world that would mean climate instability — or even fluctuations with no regularities at all. Nothing specific came of these and other peculiar results. It is not uncommon for scientists to turn up mildly anomalous calculations. They stick them away in the back of their minds until someone can explain what, if anything, it all has to do with the real world.
That did not necessarily apply to the climate system, which averaged over many states of weather. So Lorenz next constructed a simulacrum of climate in a simple mathematical model with some feedbacks, and ran it repeatedly through a computer with minor changes in the initial conditions. His initial plan was simply to compile statistics for the various ways his model climate diverged from its normal state. He wanted to check the validity of the procedures some meteorologists were promoting for long-range "statistical forecasting," along the lines of the traditional idea that climate was an average over temporary variations. But he could not find any valid way to statistically combine the different computer results to predict a future state. It was impossible to prove that a "climate" existed at all, in the traditional sense of a stable long-term average. Like the fluid circulation in some of the dishpan experiments, it seemed that climate could shift in a completely arbitrary way.
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