Differentiating the Discontinuous
Consider these statements from this article:
"Differential equations normally involve continuous functions because small changes in a system's current state generally cause equally small changes in its dynamics."
"Smooth fluid flows routinely develop shock waves, for example, where the flow becomes discontinuous — velocity or pressure changing sharply over a microscopic distance."
In Buchanan's example of shock waves in a fluid, if we assume that the pressure would increase in a 'continuous manner' over time, i.e. it would pass through all possible numeric states, theoretical or states that a measuring instrument can measure, as it increases - this assumption would be wrong.
For example, if the pressure jumps from ≈1 PSIG to ≈25 PSIG suddenly, the assumption that it would pass through all possible numeric states, theoretical or actual pressure states that a measuring instrument can measure, between ≈1 and ≈25 - this assumption would be wrong.
| To be more specific, if we conclude that at any random co-ordinate the pressure would have been ≈15 PSIG sometime, if it has increased from ≈1 to ≈25 PSIG at that co-ordinate, this conclusion would be wrong. Unless we see evidence that the pressure measuring instrument registered ≈15 PSIG at some point. |
Please refer to my previous blog entry for an elaboration. Link: Discontinuity
In fact, no matter how large or small the time frame is, fluid pressure (or any variable or relationship under the sun) cannot be properly mapped by any mathematical function.
If we talk about a very simple system for example current drawn by a simple circuit with fixed resistance when the voltage is varied (V=I.R); even in this case, the system states that we calculate using the formula will never be in very good agreement with the practical results and the actual plot between voltage and current will never be a straight line, and in fact, it will not obey any mathematics.
The actual data plot of the above mentioned system should not be looked upon as a noisy linear system because it's not a linear system. Ohm's law is an imaginary reference, and no more.
| Ohm's Law is an example of 'Empirical Adequacy'. It's not a 'truth' of electricity. |
Please also refer to the James Gleick book passage at the end of this blog entry.
Every system, regardless of what kind of a system it is, is TURBULENCE. If we focus on the quieter, smoother stages of a system, we will see turbulence in them. We will see discontinuity. We will see all the erratic behaviour that is generally associated with periods of high turbulence.
Living at the Discontinuity:
Mark Buchanan Quote:
"...Filippov suggested that discontinuities would have their most interesting consequences in situations where a system's dynamics (away from the discontinuity) act automatically to bring the discontinuity into play. Take the superconductor example again. If the equations for T > TC drive the temperature down towards the discontinuity at T = TC, whereas the equations operating for T < TC drive the temperature upwards, then the discontinuity acts as a kind of trapping surface."
An interesting graph on Discontinuity from Snarketing2dot0.Com:
And finally, here's a passage on Discontinuity from "Chaos: Making a New Science" by James Gleick:
| Discontinuity, bursts of noise, Cantor dusts — phenomena like these had no place in the geometries of the past two thousand years. The shapes of classical geometry are lines and planes, circles and spheres, triangles and cones. They represent a powerful abstraction of reality, and they inspired a powerful philosophy of Platonic harmony. Euclid made of them a geometry that lasted two millennia, the only geometry still that most people ever learn. Artists found an ideal beauty in them, Ptolemaic astronomers built a theory of the universe out of them. But for understanding complexity, they turn out to be the wrong kind of abstraction. Clouds are not spheres, Mandelbrot is fond of saying. Mountains are not cones. Lightning does not travel in a straight line. The new geometry mirrors a universe that is rough, not rounded, scabrous, not smooth. It is a geometry of the pitted, pocked, and broken up, the twisted, tangled, and intertwined. The understanding of nature’s complexity awaited a suspicion that the complexity was not just random, not just accident. It required a faith that the interesting feature of a lightning bolt’s path, for example, was not its direction, but rather the distribution of zigs and zags. Mandelbrot’s work made a claim about the world, and the claim was that such odd shapes carry meaning. The pits and tangles are more than blemishes distorting the classic shapes of Euclidian geometry. They are often the keys to the essence of a thing. |
Here is a graph of the Indian Bombay Stock Exchange Index, the SENSEX.
Its natural movement is filled with gaps. It is a discontinuous flow. Very easily visible.
In the case of equity prices, the prices very often come back, to 'fill the gaps'. That is another aspect of Discontinuity, that I will explore, in my next discontinuity post, Discontinuity-III.
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