Chapter 11 Issues in applying toy systems behavior to complex real-world systems

Outline: 11.1 Objective11.2 Analogies for parts and forces11.3 Hierarchy of systems11.4 Importance of dramatic events11.5 lack of suitable models with over three parts.11.6 Models with six parts 11.7 Different Source of chaotic looking waves11.8 Prospects for success

11.1 Objective

The objective of this chapter is to briefly describe some issues in trying to draw lessons from observing toy systems that might apply to large natural and man-made systems. These include climate, ecological systems, economic, political and perhaps even systems of belief. Strogatz, a noted expert on chaos theory, says this is pretty much unexplored territory, although some work has been done by economists trying to explain randomly occurring stock market crashes. This involves building a bridge from small isolated systems with about three interacting parts to these larger systems which have many parts and are not isolated. There are a variety of challenges in doing so.

I haven’t time to get into this subject nearly far enough to do more than discuss some of the issues and difficulties and to delve briefly into a number of specific real-world systems where variables appear to change or oscillate randomly, and probably chaotically in a way that is intriguingly similar to the waveforms produced by the chaotic double pendulum. This similarity suggests, not proves, that there is some common underlying mechanism. Thus there are sections on molecules, el Nino, mantle convection, the business cycle, social systems, ecosystems, etc.

To begin I discuss some of the issues in trying to relate the behavior of toy systems to more complex real-world systems. Toward the end I’ve simply included a number of links that might be useful to the reader.

11.2 Analogies for parts and forces

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In the double pendulum and Lorenz waterwheel there are discrete physical parts with mass and the force that governs their behavior is gravity. In some natural systems like the solar system there are also discrete well defined parts and gravity also governs so its relatively easy to see how lessons from toy systems might apply. The same is also true for molecules, which are comprised of atoms bound by electromagnetic forces. In other systems the parts and the system of hidden forces connecting them are harder to identify.

Slide 115 is how I tend to visualize the situation. Here we have a factory interacting with headquarters, a supplier factory, and a group of customers. In looking at any complex system I try to identify what the major parts might be and visualize them as balls or masses connected by spring-like forces. Move one part and it affects the others. Over time all the parts oscillate. A video of the real world parts morphing into balls and vice versa would be helpful in making this idea even more tangible.

I think this visualization is key to understanding real-world systems which otherwise are too complex. One needs to reduce them to their fundamentals.

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Earth systems like the atmosphere, ocean currents, lithosphere magma convection and the like are governed by gravity but their “parts” are gases and fluids and are not easy to visualize as discrete parts except perhaps at a high abstract level like saying for instance that the atmosphere is a part that interacts with the oceans, suns radiation, biosphere, and ice sheets. The dynamics of gases and fluids have received extensive study. Oscillation and chaos have been observed, however –as in the Rayleigh-Benard cells introduced in Chapter 4- chaos takes place in 3-D space as well as in time. In other words something like the temperature measured along a

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line thru the gas or fluid will vary chaotically. Its certainly true in the atmosphere or a cross-section of water in the north Atlantic with all its ever changing eddies. It appears that energy occasionally concentrates momentarily and randomly in some locations just as it does in the double pendulum so maybe that much is a phenomenon in common. The result is rogue waves, hurricanes, and perhaps large volcanic eruptions such as caused the Deccan and Siberian traps and mass extinctions. https://en.wikipedia.org/wiki/Rogue_wavehttps://en.wikipedia.org/wiki/Deccan_Traps, https://en.wikipedia.org/wiki/Siberian_Traps

At the ecosystem level plants and animals might be considered as mega-parts. At a more detailed level the parts might be individual species. One system that has been studied consisted of wolves and deer.

In economic systems its easy to think of individual companies as discrete parts that interact with each other. Its obvious they do interact as competitors or as suppliers to each other. At a higher level we might consider all consumers as a part, all companies as another, all savers or investors as another, and so forth. Its hard to know what parts to include. Probably government as a regulator should be included. But how do they influence each other? Its tempting to think that money is the equivalent of gravity or electromagnetic force as the mechanism that causes the behavior of one part to affect other parts. Money in a corporation’s hands is like potential energy. It has the potential to cause something to occur. When its spent it converts to kinetic energy as someone actually builds something.

Going a bit further we might suppose the economic system has a limited amount of total energy (ie: money) and it simply sloshes back and forth within the system just like energy sloshes back and forth between the arms of a double pendulum.

At a simplistic level this all seems reasonable. In their attempt to understand why market crashes occasionally occur economists have postulated the equivalent of parts, forces and built computer models using same. Sometimes the models produce spikes that are felt to represent market crashes. I’ve only looked briefly as several examples but it appears this is a work in progress so to speak.

The most obvious parts in political systems are things like nation states, which obviously interact with each other with war and trade. We might also consider the major political parties, major funders, the government, the press, and of course the voting public as constituting a system. I have no idea yet as to what forces mediate the interaction between these parts.

Finally, I think there are systems of belief. Some religious, others political or social in nature. Mind share might be the equivalent of energy in such systems. A given population only has so much mental energy or attention to devote to believing one thing or another. Where their attention is devoted seems to ebb and flow over time.

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Thus we see the public’s attention focused on political affairs at times while at other times it shifts to war, disease, climate, national debt, clean air, global warming, celebrity fashions, etc. On a more fundamental level there is probably a widespread mindset associated with any period in history of a given people. It was manifest destiny with the US west as the land of opportunity at one time. During WW2 that was the overriding focus. After WW2 there was another wave of optimism, only to crash in the 1960’s focus on Vietnam and race relations. Now it’s a complex mix typified I think by uncertainty.

To reiterate, it seems plausible that the total amount of mental energy in a group of people is equivalent to the total energy in a toy system. Its finite. If so some of the same basics physics and behaviors may apply. For instance the energy in any given person or sub-group could oscillate between the different things it might be focused on. Maybe terrorism for a while, then sports, then family or health issues, then political issues. It’s a very interesting topic to explore. And it all does come down to energy, ultimately because mental energy requires the brain to work and we know that the brain and head dissipate considerable energy in relation to other parts of the body. I also suggest it oscillates between things at different levels on the Maslow hierarchy of needs.

In sum, in order to relate the findings from the study of toy systems to real world systems it seems we must at least try to imagine those systems as comprised of parts that interact. Its tempting to say that interaction is chaotic, and there is some evidence for that in natural systems.

It seem even more difficult to identify the forces that cause one part in these societal systems to effect other parts. And there may be several forces involved. For example one company might effect another in terms of both money flow and say pay or management philosophy. The topic simply needs more thought. And it might get impossibly complex and intangible.

We note that a system won’t oscillate unless the parts have some equivalent to inertia so that an applied force doesn’t cause instant change in the other parts but rather it takes time to get them moving. Its not entirely hard to imagine that corporations, government, and even customers have inertia.

Finally, the laws of physics cause PE to convert to KE and vice versa thus setting up an oscillation within toy and natural systems. An equivalent to the laws of physics is needed to explain societal systems. For instance if a corporation has PE in the form of funds on hand what causes them to begin spending them and thus affecting other entities?

Energy analogy: I’ve oft noted that energy sloshes around in multi-part systems going from one place or part to another, in orderly fashion if the system is periodic and in random fashion if its chaotic. In order to extend the behavior of toy systems

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to living and societal systems either energy or some equivalent to energy must slosh around. In economic systems that something might be money. It boils down to this. Money must come in potential and kinetic forms for the analogy with toy systems to hold. If a corporation or body of consumers has money in hand they have potential energy. If they spend it causes someone else (that other part) to move and do something like manufacture or consume a product and maybe that’s like kinetic energy. But having done that the manufacturer or consumer must rise up like a pendulum bob and convert that kinetic energy back to potential energy. And then they must spend it again. And so forth so as to create an oscillation. If the economic system doesn’t have a mechanism like this, and doesn’t oscillate then it seems the dynamics of toy systems is irrelevant to understanding their behavior. I haven’t time to attempt to determine if money is the equivalent of energy and if they oscillate in this manner. It doesn’t seem implausible and is probably worth investigating. There is a like challenge in regard to political and social systems of all sorts.

Spatial/temporal chaos: The individual factories, stores, banks, offices, etc. that constitute parts with economic systems are obviously distributed geographically and so any force applies to them will hit some parts before others and those parts will move or reach before others. In other words all factories won’t react simultaneously to a force applied. Nor would or do all people act as a monolithic block in response to changing pressures. In short these domains are more like fluids or gases than solid masses. Thus anything we learn by observing toy systems like the double pendulum with its point masses may not be applicable to systems comprised of many, many parts. Or perhaps they are applicable only to the extent these multi-part communities act more or less simultaneously. Otherwise they act like fluids or gases and we should use simple fluid and gas toy systems like the Raleigh-Benard cell to better understand them. The Raleigh-Benard –if enough energy is applied- exhibits what’s called spatial/temporal chaos. This means that variables, like temperature or fluid velocity, are different from place to place and how they vary across space is disorderly and random. And this pattern is changing randomly or chaotically over time as well.

11.3 Hierarchy of systems

No real-world system exists in isolation, the parts of real world systems range widely in mass or the equivalent of mass, and there are often many more parts in real-world systems than in the toy systems. These facts makes it much more difficult to apply toy system lessons to real world systems. Here are some thoughts on that issue.

Fig 126 illustrates the hierarchy of systems that exists in the real world. All interact by exerting forces on each other. The largest or most massive system consists of “A”

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parts. The next smaller systems contain “B” size parts. The A parts are closely bonded and interact strongly with each other. In other words they oscillate periodically, quasi-periodically, or chaotically. However whatever they do effects the smaller B parts whereas the B parts have little effect on the more massive A parts. This is indicated by the bold arrows where one head is larger than the other. The same idea applies to how the B parts interact with the smaller C parts.

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To make this more relevant lets assume the A parts are planets in our solar system. Over long timeframes they subtly alter each other’s orbits, but clearly noting happening on earth will alter the interaction between planets. The fact that earths tilt gradually changes as does its distance from the sun might be variables at this level. Lets assume the B systems are major on-earth system like the atmosphere, the oceans, the earth’s molten mantle. Another B level system might consist of the

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plant kingdom interacting with the animal kingdom and solar radiation. Average global temperature is a variable at this level. The earths tilt will effect all them but they can’t effect earths tilt. Finally economic and political systems may be examples of C levels systems. Things like agricultural production will be effected by climate but with the exception of global warming and deforestation what the economy does will little effect the biosphere.

These examples are not necessary accurate but only serve to make the point that systems at all levels are interconnected and that more massive systems effect smaller ones more than the reverse. It seems likely that large systems oscillate at a higher frequency than do small systems but that’s not necessarily true because small masses will oscillate slowly if the spring-like forces between them are weak. Also large systems can change rapidly if something powerful happens to them. The meteor that killed the dinosaurs 67 million years ago comes to mind as do the massive volcanic eruptions that created the Deccan and Siberian traps. https://en.wikipedia.org/wiki/Deccan_Traps, “This massive eruptive event spanned the Permian-Triassic boundary, about 250 million years ago, and is cited as a possible cause of the Permian-Triassic extinction event.” https://en.wikipedia.org/wiki/Siberian_Traps

If we look at the waveforms of the major variables in any one of these systems it will represent not only the internal oscillations within that system but also the imprinted effects of all the other systems. If the A level system is changing very slowly, as things like earths tilt does, then it will produce very long low frequency wave forms and the effects of more rapidly oscillating B and C level systems will show up as minor deviations in those waves.

If on the other hand we look at B level waveforms we may see that they ride atop a long-term change or trend. For instance daily or yearly temperature fluctuations ride atop a long-term increase in earths temperature due to global warming.

This was another brief attempt to identify possible levels of systems. Clearly the systems at each level are strongly influence by those at prior levels.

a) Level A systems: solar system dynamics and mantle convection/plate tectonics (Massive underlying systems. Clearly not effected by anything that happens on earths surface.)

b) Level B systems: Major natural non-living systems like carbon cycle, pole to equator atmospheric and oceanic circulation, solar radiation, earths reflectivity or albedo, GHG, biomass, frozen methane, ice. Note: Some of these are fluidic or gaseous systems that can exhibit temporal/spatial chaos thus further complicating any analysis.

c) Level C systems: Living systems at the macro level like the interaction between total biomass in plant and animal form.

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d) Level D: technology, knowledge (Can these be “parts”? Whatever they are both short-term scientific breakthroughs and long-term increases in education clearly effect economic, political and societal systems.

e) Level E: security, nations, borders, (Perhaps the first and most basic human or societal systems were established to meet security needs. Reference the Maslow hierarchy of needs. https://en.wikipedia.org/wiki/Maslow's_hierarchy_of_needs

f) Level F: commerce, trade , communications, transport, energy

g) Level G: attitudes, passions, world views, ideologies, religions

Again these are only suggestive starting point that invite further thought. One can nevertheless see that the “parts” within each of these levels interact and influence each other. Even at Level G there is a historic ebb and flow of where societies collective mind is most focused. Each area of interest or concern has a certain mind-share, and that mind share changes over time. An analysis of word count in the mass media would identify these areas of interest or concern and show how they change. Speculatively speaking, total mental attention (within a population) may be equivalent to total energy within some physical system. It moves from topic to topic, perhaps in the way energy moves within the double pendulum or other simple toy systems. We and the media focus on the national debt, then get concerned about the mid-east, then attention shifts to some virus epidemic, and then we’re all focused on some election. Over a longer time frame the mind-share of different religions has changed as has the general attitude toward manifest destiny, the American dream, WW2, the cold war, and so forth. Are any of the same basic laws of physics or their equivalent driving this? I find this an exciting area. All these changes may have some underlying pattern or driver. We might detect that a social focus on some area is likely to change because such things do as metal attention oscillates between all the things it can focus on. In the 1950’s our focus was on economic security and our attitude was positive. In the late 1960’s attention shifted to self-actualization as youth said a good 9 to 5 gray flannel suit job with a big company was not the end-all. When we have achieved satisfaction at one level in the Maslow hierarchy of needs social attention may, and arguably will, shift to a higher level. We’ll be dissatisfied there until it is also satisfied.

Coupled systems: I’ve said that couple systems imprint their behavior on each other. Here are some simple spring mass runs that show such behavior. Unfortunately each system in this case is a very simple two-part system, namely a mass connected to a wall by a spring. There is one system on the left and another on the right in these screenshots.

I made the left mass ten times greater than the right mass so obviously the big body will effect the small body more than the converse. In addition I adjusted the spring

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stiffness so left would oscillate fairly slowly and right would oscillate at a higher frequency.

This is the waveform of the left mass when acting alone as an isolated system.

This is the waveform of the right mass when its acting alone:

This is the waveform of the heavy left mass when the two systems are coupled by a spring in the middle. Obviously the faster oscillations of right imprint themselves on lefts lower frequency oscillations. If the middle spring is weakened to simulate a weaker coupling then lefts waveform just shows minor ripples.

Lets imagine that left part is a large company like GM and the right part is some small supplier to GM. Whatever this small part does will affect GM but only slightly. However what GM does could affect that small supplier far more. Likewise global

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climate is little effected by mans short-term changes to local forest cover for instance, but global climate might well strongly effect forests worldwide.

This plots right’s velocity and shows that lefts behavior effects it quite a bit.

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Mega systems and mega trends: The complex hierarchy of systems described above presents a huge challenge in trying to apply lessons learned about the dynamics of simple toy systems to complex systems. However we do know that our higher level social and economic systems are effected by long term trends in massive underlying systems. In other words the mega systems produce mega trends. Maybe the parts in the mega systems actually oscillate over time, or maybe they interact in a one-way fashion that never oscillates. Not sure.

I think it could be fruitful to try to better identify mega systems, identify their “parts” and think about how those parts may interact over the next decade or so. I haven’t thought this thru but just notionally perhaps there is some interaction between mega trends in climate change, educational level, and technical change that could be identified and studied. It might lead to some general predictions as to what might happen or become possible. In other words take a systems look at things.

11.4 Importance of dramatic events

When the double pendulum reaches a certain energy level the outer arm is able to swing over the top (OTT) as opposed to falling back. It’s a tipping point of sort, and marks a qualitative change in behavior which I call a dramatic event. Just a tiny energy increase can send the system over it. Such dramatic events happen when a significant portion of the total system energy gets momentarily concentrated in one part or one place. Dramatic OTT events occur often in the double pendulum simulations, and they also happen when the Lorenz waterwheel reverses.

I feel this kind of dramatic change in behavior is important but its been hard to see what the connection is between how it manifests in the double pendulum and how it might manifest, and be important, in real-world systems if their internal energy level was increased. Global warming is the obvious situation that comes to mind and one wonders if some pattern of ocean circulation, or ecosystem behavior might have a tipping point that increase in energy could push it beyond with dire consequences.

I don’t know of any important real world systems that are near a tipping point of this sort and its probably hard to determine. Nevertheless I feel this is something others may want to investigate. Meantime here are some thoughts.

Its hard to express this but the system that is oscillating and then goes over the tipping point may not “care” that it is doing so. The double pendulum simply goes over the top but it accelerates down on the far side just like it would have accelerated falling back. The waveforms in either case are much the same. And the Lorenz waterwheel carries on just as well falling back as it does after going over. The only thing that seems to care is the observer who is amused by its behavior. But can it matter in some more meaningful way?

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One concept as to how it might matter is to consider something outside the oscillating system but coupled to and affected by it in some way. For example I think of how the circulation of the Gulf Stream affects the climate of northern Europe. The Gulf Stream happily does its thing obeying the laws of physics caring not about how it affects anything else. But in fact it affects the climate of northern Europe making it much warmer than it would be otherwise. I don’t understand that system well enough to comment further except to say it MAY be one system with a tipping point global warming could affects in some way. Meanwhile I need some tangible way to express the way dramatic events might have outside impacts.

I turn to the Lorenz waterwheel and to a lab demo by Dr. Steven Strogatz. Its visible on uTube at https://www.youtube.com/watch?v=7iNCfNBEJHo. The reader needs to bring it up now and pause at the 3:40 time point. The idea is to watch how much green water is on the right side of the wheel. Between about 3:40 and 4:50 the waterwheel rocks back and forth and the bulk of green water never goes over the top. Instead it approaches but then the wheel reverses and it falls back. This is also what the double pendulum does when it doesn’t have enough energy to go over the top. What we note it that the amount of green water on the right side oscillates at a more or less regular frequency like the progression of day and night or winter to summer. However at about 4:50 Strogatz increases the energy in the system (by reducing the brake friction) and now the green bulk of water goes over the top on random occasions. This obviously changes the predictable cyclic way the amount of water at right used to change. You can see how it changes unpredictably and randomly. Now make the conceptual jump and imagine this green water represents the amount of warm Gulf Stream water west of Great Britain. If so it would make the climate vary irregularly in that area and affect everything depending on it, like crops. This analogy between the toy waterwheel and a real-world system is hopefully more tangible now. Behavior of the large core system affects the behavior of things riding atop it so to speak. Only a small increase in system energy caused the big system to change its behavior.

Again I’m not suggesting the Gulf Stream behaves this way. Perhaps its average temperature does not oscillate so in some years there is more warm water up north than in other years. Maybe those oscillations are not bringing the warm water near some tipping point where it could go over the top to somewhere else. Given all the turbulent eddies in the Gulf Stream in the north Atlantic its not easy to picture what might happen. I suspect we could profitably look at ocean currents with these thoughts in mind, and look at other natural systems as well.

So the challenging question remains. What real-world system could experience dramatic changes in behavior when their internal energy level exceeds some tipping point?

I don’t think this would be easy. If there were a historical record of prior dramatic events it would prove they are possible. If we had a simulation model of the system and determined it was chaotic then dramatic events would be expected to occur at

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random intervals providing there was sufficient energy in the system to enable them. I’ve not time to follow this conjecture further.

11.5 lack of suitable models with over three parts.

In order to build a bridge between the dynamics of toy systems with just a few parts to large real world systems that exist within a hierarchy of other systems per Figure 126 it seems that an additional simulation model would be useful. What I have in mind is a spring/mass model somewhat like the “molecule model” described much earlier. It would consist of masses connected by forces and looks like Figure 126. It would allow the user to insert up to say 10 or 20 masses and make each a different weight. It would allow the forces between them to be of different types. Simple linear springs being one. But one should be able to insert non-linear springs. Better yet would be an attracting force that decreases exponentially with distance as gravity does, and a repelling force that does likewise. This would allow the system to settle into a realistic molecule-like equilibrium after being disturbed. It would need adjustable damping. It would allow these forces to be inserted between any two bodies. Thus there might be tens of such “springs, again like what I show in Figure 126. That’s the model. The plots and readout should allow the forces, displacements, and speeds of each mass to be plotted over time. And for one variable to be plotted against another or another two so as to get partial phase space plots. To test for SDIC and chaos the model should either support two nearly identical systems running in parallel and show if their waveforms diverge, or it should allow waves from successive runs to be superimposed and compared.

There are four key things I would like to learn from such a model. One is what the waveform of any given variable, like the force on one part, looks like over time. I suspect it’s a very complex waveform, perhaps thus explaining why the waveforms of things in the real world like temperature, snowfall, and stock market vary in such a complex manner. To explore this the user might increase the number of parts in the system from say 4 to 6 to 10 to 20 and see what difference that makes.

Second I would like to know if energy spikes or concentrates in or on one part randomly thus disturbing it greatly. I do believe this should happen in the model as it seem to happen in the ocean with rogue waves, in the climate with hurricanes, and in the stock market with crashes. The alternative is that energy becomes distributed and stays that way. Its like all the waves in some container eventually having about the same height and never forming high rogue waves. Either it will become an equi-energy situation, or not.

Third, I would like to know if the presence or absence of chaos plays and significant role in the systems behavior, and especially whether it effects the occasional concentration of energy in one part or place. Thus I would run the model after

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displacing one part just a little, which is equivalent to releasing the double pendulum at a low angel thus inserting minimal energy. Then I would displace one part quite a lot, or maybe several parts a modest amount so as to insert a fairly high level of energy in the system to see if that makes it chaotic.

Forth, I would like to know if we add a small amount on energy to every part -by displacing it somewhat from equilibrium before starting the sim- would all or most that energy eventually concentrate in one place. I’m trying to explore the notion that adding say 1 unit of energy to each of 100 parts in a system –al la global warming- might result in 100 units randomly, occasionally, and momentarily concentrating on one part or in one place thus causing something extreme to happen there.

So I haven’t located a suitable simulation model but the section below describes some results with models with just a few more parts than the double pendulum.

11.6 Models with six parts

Objective: In this section we will look at the behavior of several systems with more than three parts in an attempt to build a bridge between the behavior of the three part double pendulum and larger systems. Unfortunately all of these are spring/mass systems with linear springs. Most real-world systems presumably have non-linear forces connecting their parts so the applicability of these models may be limited.

What’s significant it that all show that energy moves within these systems. Any given part will oscillate calmly at times and then begin to oscillate violently.

6-mass/spring model: The following screenshots were taken from a simulation model developed by Dr. Dooling at UNCP. https://sites.google.com/site/uncpdooling/jar The system consists of six masses connected by linear springs as shown in the screenshot below. The user is able to pull one or more masses from their relaxed equilibrium position to insert potential energy before starting the run. To start this oscillation mass one was pulled aside 0.4 units of distance. In this system the PE is actually in the stretched or compressed spring(s) while the KE is in the masses as they move.

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The intent of this run was to see if all or at least most the total potential energy initially inserted into the system by moving the leftmost mass left by 0.4 units of distance–thus stretching the spring to its right while compressing the one to its left- would eventually move and become concentrated in some other mass’s KE or some other springs PE. In this case the total energy inserted into the system was 0.16 units and it stayed constant per the hard to see yellow line. I chose then to follow mass#6 (the red one) and see how much of the total energy ever concentrated there. Would it ever get all 0.16 units of energy?

The red waveform showed that red’s KE was high at times and low at others. High peak values were rare but did occur at t=183 and 218. At t=183 almost all the energy in the entire system had momentarily concentrated in red’s KE. There were also periods where red didn’t move very fast, especially between t=245 and 250.

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There seems to be a repetitive beat to this pattern of oscillation but there also is some randomness.

The plot below shows how the potential energy in spring#6 (the rightmost one) varied over time. It never got 100% of system energy but it did once peak at 78% of it. That energy that started at the left of the system when mass#1 was displaced had migrated all the way across the system to mass#6. A plot of any other variable such as the kinetic energy or speed in mass #3 would have looked generally the same.

We note several other things of import. First, the spikes in energy occur relatively far apart. In other words the energy does not concentrate in PE6 or any other place very often. I haven’t shown that this behavior applies to other systems like climate but if it did, and the spikes represented strong storms, this plot suggests they wouldn’t happen very often in a particular location. This and the plot below shows they occur on random intervals, which suggests the system is chaotic even thought systems with linear springs are not supposed to be capable of behaving chaotically

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because the forces are linear. Because intense spikes are rare in this particular location (spring) it suggests they are happening elsewhere in this system. There are also noticeably quite periods, especially around t=70 in this run. We note how rapidly a quiet period can escalate into a violent one. There is little ramp-up to intense events that might provide warning they are imminent.

The screenshot below plots the velocity of mass#6, which is of course proportional to its KE. The relatively quiet areas are underlined in green.

So what’s the point here? This simulation shows that energy moves from part to part in systems with linear forces as opposed to the non-linear forces in the double pendulum. It also shows that particular parts are violently disturbed at seeming random intervals even in this system with linear forces. With its linear forces this system is not supposed to be capable of becoming chaotic, according to the experts. I haven’t time to test it for SDIC so have no opinion on that.

The fact that this system DOES exhibit random spikes and calms suggests to me that its not that important to know whether the forces in economic and societal systems are linear or non-linear. It seems random spikes and calms would occur in either case. Clearly this needs further investigation as we try to determine how the behavior of toy systems relates to important real-world systems.

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Non-linear spring/mass model: This sub-section probably bellows somewhere else but it conveniently fits here.

Dr. Dooling at UNCP was kind enough to develop a two mass two spring model for me that allowed for non-linear springs. The model called “ejs-model-nonlinear(1).jar” is available at his web site: https://sites.google.com/site/uncpdooling/jar

The criteria for a system to operate chaotically are not clearly stated in the technical literature, except that all experts agree the system must be non-linear. This probably means the equations describing the dynamics must be non-linear, but its not clear to the author whether just having non-linear forces is sufficient to make the equations non-linear. I suspect it would but am no longer into the math.

To help answer this question the runs below were made to see if this system would operate chaotically if the springs were non-linear. This was a quick and imprecise test since I didn’t know how to test for SDIC when I did it and now I haven’t time.

The first screenshot shows UI settings for the first run using non-linear springs where F=kx^3, (x is the stretch or compression from relaxed length). To insert energy before starting the run the left mass was pulled 0.3 meters to the left.

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This setup yielded the following plot. Because the trace followed approximately the same pattern time after time this run was judged to be quasi-periodic, not chaotic.

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The screenshot below came from a run with roughly the same total energy and it too appeared quasi-periodic.

In the run below the non-linear springs were stiffened so the system had about one hundred times as much total energy as above. It was thought this might drive the system into chaos but in fact it turned out to be almost perfectly periodic.

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In one last attempt to see if the non-linear spring system could operate chaotically the left spring was stretched still further before release. Again the same general pattern appears leading to the tentative conclusion that this particular system cannot operate chaotically. In other words, just because the forces that connect parts in a system like this are non-linear that apparently isn’t sufficient to allow chaos. Perhaps the governing equations are not non-linear, or perhaps the fact the masses can only slide back and forth does not provide enough degrees of freedom for this system to become chaotic. The literature seems to say that a technical term called “principle of linear superposition” applies here. The waveforms apparently reflect that principle, not chaos.

To most non-technical readers its sufficient to note that some, perhaps all, of these simple mechanical systems where the parts are connected by springs may be incapable of operating chaotically. Still they are legitimate systems where the

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behavior of one part affects the behavior of other parts. Thus things can be learned from their behavior.

Wolfram spring mass cube: I think this 9-mass system is more realistic of systems in general than the 6 in-line mass system described above. Unfortunately its not well instrumented and the user can’t control the initial conditions accurately. Its hard to tell what variable each line represents but that’s not really important. The fact that so many waves are superimposed, and the short time the waves are plotted make it near impossible to see if the energy in one spring or in one mass oscillate periodiczlly or have those random spikes and calms. http://demonstrations.wolfram.com/DynamicsOfACubeShapedMassSpringNetwork/

Only one thing is really clear from the plots. It is that energy transfers within this system. Several masses were pulled aside to start this run but its clear from the plot all the others soon began to oscillate. Actually watching this system in motion makes this obvious and is the recommended way to use this model. Just watch it.

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Fabric models: Spring/mass odels -probably for animated movies- have been made to simulate the behavior of fabrics. This simulation video shows how a disturbance to one part of a 21 by 21 matrix of spring-connected masses propagates throughout the system. https://www.youtube.com/watch?v=ib1vmRDs8VwIts compelling to watch and is perhaps the most useful tool I’ve found to visualize how disturbing one part in a large system radiates waves that end up disturbing all the other parts. IF this model had appropriate controls and plots it

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would be most useful in bridging the gap between simple toy systems like the three-part double pendulum and large systems in general. A screenshot is shown below.

Other material: A Wolfram simulation of a quadruple pendulum (5 body system) is at: http://demonstrations.wolfram.com/MotionOfQuadruplePendulum/This does not produce enough plots to be useful, only a trace of bob movement.

NOTE: I went thru the first 45 pages of physics sims on wolfaram at: http://demonstrations.wolfram.com/topic.html?topic=Physics&start=421&limit=20&sortmethod=recent but didn’t find mass/spring network sim. But there may be one somewhere since there are 114 pages total

Other multi-mass videos: masses coupled with springs sim model: http://www.ffn.ub.es/oscar/Mecanica/Applets/Coupled%20Oscillators/CoupledOsc.html not useful

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http://physics-animations.com/Physics/English/thermo.htm (shows apparently random movement of gas molecules)

https://search.yahoo.com/yhs/search?p=+entropy+thermodynamics+animation&ei=UTF-8&hspart=mozilla&hsimp=yhs-001 (35 min. highly animated but somewhat confusing video about energy and entropy)

http://farside.ph.utexas.edu/teaching/315/Waves/node18.htmlhttp://scienceworld.wolfram.com/physics/SpringsThreeSpringsandTwoMasses.html the math only

I generated the table below -called Excel1- to illustrate the difficulty of generalizing the behaviors of simple systems like a double pendulum to the more complex systems that effect our health and welfare. Lots of detailed things can be said about the double pendulum or Lorenz waterwheel. But there are very few generalizations this author feels comfortable about making that apply across the broad range of systems. That they all can oscillate seems the one safe generalization, followed by the fact most oscillate irregularly to some degree.

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11.7 Different Source of chaotic looking waves

Here is still another complication in trying to apply what we’ve learned about toy systems to real-world systems.

If we see a chaotic wave or variation in some natural system it wasn’t necessarily produced by a few parts that oscillate chaotically relative to each other. Before trying to find such discrete parts consider this alternate explanation. What we may be watching is how conditions change at one particular place within some large volume of liquid or gas that is behaving chaotically. Rather than having nice even flows like convection currents in a pot that volume of fluid might be experiencing what’s called spatial/temporial chaos. This occurs when fluid in any container is overheated enough so simple orderly convection currents break down and the fluid becomes turbulent.

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This also means that what we’ve learned about chaos in the double pendulum and other small systems comprised of a few discrete parts may, or may not apply. Put another way, systems with discrete oscillating parts can produce chaotic waveforms, AND systems that exhibit spatial/temporial chaos can do so as well. Refer back to the Raleigh-Benard cells in Chapter 4 for more detail on this type of chaos.

This is best explained by a example. Just below we see several chaotic looking weather related waveforms. These records were from some specific weather station whose name I can’t recall. They represent conditions at just one location in the global atmosphere. It may be tempting to ask what three, four or five “parts” might be oscillating so as to produce these waveforms. That approach won’t work here.

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Instead those waveforms recorded changing conditions at one place within a turbulent global weather system like this:

The screenshot below –a satellite image of winds over the US - explains the situation. Suppose the waveforms above were taken at the location I marked with a red cross. The complex chaotic pattern of winds and other weather conditions generally drift from west to east across the US and pass over that spot changing temperature, wind speed, rainfall and pressure at that spot so as to create chaotic waveforms as they pass. This video from a satellite shows real clouds like this in motion: https://www.youtube.com/watch?v=VFsWU2j5bPc

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These complex flows originated with the simple behavior of a simple system. Air was heated near the equator, rose and headed toward the poles. This creating a simple convection cell (called the Hadley cell) much like the Raleigh-Benard convection in the small containers and donut shaped tubes shown in Chapter 4. That Hadley cell flow may or may not be chaotic, but I suspect it is since landmasses and different ocean temperatures would have distorted its path in a complex manner. In any case the coriolis force wrapped the NS flow into EW trade winds as seen in the diagram below.

This diagram is further explained at: https://www.youtube.com/watch?v=Ye45DGkqUkE and https://www.youtube.com/watch?v=ONjrSFvdKjQ The best explanation and set of actually satellite videos relating to global weather and ocean currents I’ve seen was made by NASA and is found at: https://www.youtube.com/watch?v=6vgvTeuoDWY

Then the trade winds became turbulent and chaotic. In sum, weather waveform plots are chaotic because they record conditions at one point within a turbulent flow, not because a few discrete parts interact chaotically.

Extras:

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This image was created using data from SeaWinds onboard QuikSCAT (satellite). It shows ocean winds on September 20, 1999. Orange areas show where winds are blowing the hardest and blue show relatively light winds. Image credit: NASA, From: https://directory.eoportal.org/web/eoportal/satellite-missions/q/quikscat

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Weather plots above are from Saint George, Bermuda during Oct 2914. From: https://www.wunderground.com/history/airport/TXKF/2014/10/17/MonthlyHistory.html?&reqdb.zip=&reqdb.magic=&reqdb.wmo=

Snow and trees: Both the waveforms of snowfall in specific locations or the waveform of growth rate in some tree (as derived from tree ring analysis) look quite chaotic as the following images show.

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From: https://www.allianz.com/en/about_us/open-knowledge/topics/environment/articles/111118-natures-climate-archives-corals-wood-and-pollen.html/

Another chaotic waveform probably from a tree or trees in one particular location.

From: http://www.rsc.org/Education/Teachers/Resources/jesei/treering/home.htm

Historic rainfall and river flow can be deduced using tree ring data.

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11.8 Prospects for success

Given all the complexities described above I’m not optimistic that understanding the behavior of toy systems will provide useful and practical insights into how to better cope with or manage large natural or societal systems. Its disappointing after three years studying this topic to come to such a null conclusion. There are however three possible exceptions: 1)the trend toward equilibrium, 2) expecting and preparing for random spike events, and 3) trying to understand the consequences of several mega-trends acting in combination. In any case even if there are no practical insights we can act upon its nice to better understand at a fundamental level why dynamic systems behave as they do.

1) Trend toward equilibrium: Virtually all my analysis has dealt with frictionless systems, however all real-world systems have friction or some other way to shed energy. As noted a system at equilibrium will oscillate if some part is disturbed, but with friction that oscillation will die down and the system will return to a stable motionless equilibrium. All competing forces will be balanced. This is an important topic that merits more exploration.

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One can imagine economic and political systems that were in stable equilibriums for hundreds of years during the dark and Middle Ages. No significant technical advances or foreign invasions occurred to disturb them. Peasants had their role and nobles theirs. Sons did as their fathers and grandfathers did. Now our world with all its economic, political, and belief systems is being constantly disturbed by advances in telecom, transport, medicine, education, and a host of other things. It must therefore constantly change, perhaps always seeking a new equilibrium but never allowed to settle into one. This stressful situation is, I think, what we are experiencing. Its what the system is doing, almost per the laws of physics.

2) Spikes: I feel that the most interesting thing I’ve learned is that energy can move from part to part or place to place and occasionally spike randomly to high values. This could happen in large complex system and provide us with some theoretical understanding of why it happens. But we already know that spikes happen in real-world natural, economic and societal systems so this isn’t new knowledge. Its simply a reminder to be prepared for those random events.

3) Mega-trend interaction: The most promising area to explore in my view is trying to identify mega systems and see how the interaction between their parts –assuming we can identify them- might produce mega-trends we haven’t thought about. Some might cause these unanticipated consequences. We could look at mega trends like global warming, resource depletion, increasing knowledge, increased global dissemination of information, automation, and things like that to see if considering how they might interact or holistically might provide some idea of how the world will be changing. What difficult situations might result from the combination of these mega trends. Maybe there is a mega system that connects these mega trends and it could be modeled to some degree. Or maybe we just need to speculate on the interaction or combination of mega trends.

There has been some academic research looking for techniques to take advantage of SDIC to control chaos. The idea is that applying a small corrective force on a system early allows the effect of that force to be magnified and significantly alter downstream conditions per the butterfly effect. Clearly applying such a force would alter downstream conditions as all my SDIC runs have clearly proven. However, I don’t see much promise for this idea for several reasons. First, any real-world system will also be affected by random forces being applied from outside the system. These may well counteract the deliberately applied force, possibly causing it to make the situation worse not better. Second, the system must oscillate or cycle multiple times before such a corrective force would take effect. Its likely we don’t know exactly what target value we want that far into the future. Third, it assumes we have a good simulation model of the subject system so we can accurately predict what some variables value will be at some future date without the corrective force, and then with it. For economic, where we might want to prevent crashes or unemployment, there are apparently no models capable of making long term predictions so we can’t know what butter-fly weight interventions to apply. Short-

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term interventions are common governmental tools, so we can add nothing helpful there, but they take massive effort and $, not a light touch because they don’t have time to magnify.

Other thoughts: What does the behavior of a simple two or three part toy systems suggest about the behavior of more complex systems with many more moving parts? For instance climatic, oceanic, ecological and economic systems. It suggests their behavior will be far more complex in terms of the patterns of oscillation than it is for a double pendulum. When operating at sub chaotic levels of energy, periods of approximately periodic behavior, if they exist, may be much longer. Prediction of any sort may be far more difficult. Perhaps we can only hope that there are a few relatively deep and powerful dynamics at work in complex systems that dominate their behavior. I’ve discussed this in terms of mega-systems that may boil down to having a few large parts and exhibit mega-trends. One thing to keep in mind is the possibility that several, or many, small parts may somehow get in phase and have the power of a large part. Acting together they would become another mega part.

I suspect the notion that a system can become chaotic if energy is increased beyond a certain level still applies to complex systems. For this to matter in a practical sense we would need to find real-world systems that operates periodically or quasi-periodically today and thus stands a chance of becoming chaotic in future due perhaps to something that adds energy like climate change. It’s a compelling topic to explore since these large systems are so important to humanities health and welfare.

Perhaps its been done but it might be interesting to see if any economic models that oscillate, as they arguably should. If so it may be possible to see if there is any energy threshold that sends them into chaos.

***end of Chapter 11 ****

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