Reflection 1 Sensitivity to initial Conditions
Any system using recursive iteration is dependent on initial conditions. [not really—you can set the
recursion to always crash – recursively multiply by zero for example—and that is just one of many
possible basins of attraction.] These systems can be seen in mathematics and science as well as in
nature. In science it is easy to tell that a recursive iteration method is being used to create a system; such
as in a control system in robotics. In nature, however, it is not quite so obvious. For example: weather
patterns can emerge from an initial state of a certain temperature and pressure. From there, the weather
forms storm systems and currents. Those storms and currents then feed other storms and currents to make
for bigger and more dangerous storms. In addition to natural recursion there is a way to influence the
process as with nuclear fission. The last example is probably the easiest to see the effects of small
perturbations having grand effects on the final result. One small error in exciting an atom could cause
catastrophic disaster. With that in mind, is it possible to predict future outcomes by fully understanding
the initial conditions of a particular system?
Without doing years of experimentation and theorizing this question is quite difficult to tackle.
With the first example of robotics, the initial conditions of a differential equation being integrated to
output a desired direction of motion for the robot. By knowing what we want the robot to do we can
work backwards to find out where exactly we should start. For instance I gave an example in class about
designing a small satellite to traverse from an Earth-Moon Lagrange point to the orbit of a near Earth
asteroid. In the control system, using a MATLAB sub-program known as Simulink we created a forward
dynamic control loop. We wanted the satellite to rotate and face a certain direction, ie at the Earth, to
allow for better communication and data transfer.
By using the kinematics equations from basic physics we were able to analytically determine
what our end position vectors should be. From there it was easy to determine what the initial equations
should be by simply integrating. Those initial equations could then be used to give a new position from
which to restart the integration, forming the iterative loop. The new position became the old position and
so on and so forth until the satellite reached the desired position. The initial positions involved were
selected at random. From our plots of the output position vectors we were able to see the asymptotic
approach to stability where it could then correct its attitude. Based on our selection of the initial
conditions, the display of the asymptotic stabilization looked either more or less overshot. Basically;
wherever we decided to start determined how long it would take to stabilize and then correct its attitude.
This may seem like an elementary concept because you may just assume that the farther away
from the desired position the longer it would take to stabilize and correct. In fact there was a sweet spot
so to speak, but it was not located where we thought. For visualization purposes assume that pointing at
the Earth is 0 degrees as seen in the rough sketch in the figure below. We assumed that starting the
satellite only between +/- 1-2 degrees from 0 would be ideal, since the control system wouldn’t have to
correct for much; however, it seems that anywhere within +/-30 degrees worked equally as well. How is
it that the initial conditions had such a large tolerance for error? This had to do with the type of
calculations we had the computer running. The math involved for correcting the attitude was simple
enough in the +/-30 range that the computer was better able to estimate the next value at each iteration;
however, beyond this range, the computer had to make more and more calculations to estimate the next
position value. Therefore, understanding the range of initial conditions that these simple calculations
worked for, allowed us to predict the approximate amount of time it would take to stabilize and then
correct, somewhere in the span of a few seconds.
Figure 1: Earth-Satellite reference frame
Estimating the time for a computer to make a calculation is great, but where is the
applicability? For us it with our attitude control model it was very specific, but for everyone else it didn’t
create Earth shattering results. However; the second example that I listed, the weather, has much more
dramatic effects on the general public. Admittedly, of the three, this is the subject that I am least
knowledgeable about, but that can’t stop me from wondering about it.
My future brother in-law has a Master’s in meteorology, so whenever we visit I end up learning a
bit about how the weather works, specifically with storm systems. As I have come to understand it, by
looking back to the third grade model of how water is cycled in nature we can see the recursive iteration.
[right, that is a great example] It’s been a while since the third grade, so I’ll quickly explain what I
mean by this. The Sun heats up the water, and the water evaporates. Then the water vapor ends up
condensing in the form of a cloud. When enough vapor has collected the molecules of water start to
combine with each other and form rain droplets. Gravity takes over from here and the rain
precipitates. The precipitation then soaks into the ground or flows back to a body like a lake or ocean
where the whole process can restart. Again it seems that I’m not presenting anything significant to think
about. How would it be possible to understand the initial conditions of rain? Simply put, it wouldn’t be
possible, since at any given moment we can’t calculate the amount of water being evaporated from all the
lakes, rivers, and oceans around the United States let alone the world; however, there are aspects of the
storm clouds that we can measure and predict.
The initial conditions for a storm include low-pressure zones, cold fronts, mountains, and jet
streams. All of these factors have an effect on whether or not rain forms. The low-pressure zone could
form at a higher altitude due to warm air at the Earth’s surface rising. This warm air carries the water
vapor to the cooler altitude where it can then condense into rain. The cold fronts cool the water vapor
where it is and generally forms rain or snow depending on the severity of the front. Mountains can cause
rain or snow by simply forcing the moving air upward to cooler altitudes. The jet stream has more of an
after effect on the storm clouds in that it drives them. A jet stream is basically just a stream of fast
moving air in the upper troposphere created by two different temperature air masses colliding, the warmer
air of the stratosphere colliding with the cooler air of the troposphere. This creates wind that runs along
that boundary of where the two masses collide. By studying all of these factors we can understand how
the storm will form and where it will go once it is formed. With that in mind I think it is possible that
with enough data about how these systems interact we can get an exact prediction of a storm. Right now
we base our predictions on an estimate of what we see happening and run it through a model generator.
The model generator deals with issues like chaos and unpredictability to give a best guess. Now I
know that I am greatly oversimplifying this and perhaps I’m completely wrong in my understanding, but
wouldn’t it be possible to predict the storms movement by fully understanding the causes of the storm
before it starts? Is the system really too chaotic and almost random in nature that we can’t guarantee what
would happen? The answer is that it all comes down to my initial statement of whether or not it was
possible to predict the amount of water vapor being generated. Since that is unpredictable, then perhaps
this is one of the cases where the sensitivity to the initial conditions is extremely high. With my satellite,
the sensitivity was only moderately high since there was a range of positions in which the computer was
able to make the calculations quickly. With the weather it seems that the easy calculation range is just too
small.
Finally I come to my third example of nuclear fission. I see this as the blend between natural and
forced recursion. Uranium 235 is by nature radioactive. This is caused by the excitement of an atom to
the point where it releases a neutron. This neutron then impacts with the nucleus of another atom where
that atom releases more neutrons. Left alone, this is generally a slow process due to the low energy state
of the neutrons, the thermal neutron state. This would be why the element doesn’t just self-destruct and
cause an explosion at random. Currently we use uranium 238 (U238), which is just an isotope of uranium
235 (U235), to induce a fissile reaction. In a nuclear reactor the neutron is excited and ejects from one
atom, which is similar to the natural process and excites other atoms. If left unchecked this fission
reaction would become unstable and go supercritical causing a catastrophic failure ending in a big
bang. For this reason we use moderators such as water, the most common, to slow the fast neutrons
bringing them back to the thermal neutron state.
Here, the initial conditions are type of uranium and the moderator. If U235 was chosen, the
fission reaction would slowly die out after having been introduced to the moderator. This would bring the
reactor to a subcritical state which is useless. Without the water it would have continued to produce
fission, but at a very slow rate, which wouldn’t generate much electricity through the reactor. By using
the U238 without water, we would get failure. By introducing the water we can have a stable critical
fission reaction. There are different types of reactors that use different fuels and moderators. For example
a Molten Salt Reactor uses mixture of molten salt and a fissile material such as U235. This allows the
mixture to act as both the fuel (heating element) and the coolant. The moderator here is solid graphite;
however, it has been discovered that the salt mixture causes the graphite to fracture which in turn limits
the lifespan of the moderator. Certain coatings are being developed to reduce the damage done to the
graphite. Thankfully, all of these initial conditions have been thoroughly studied and understood, and
that’s why we are able to use them in the generation of electricity.
Throughout this paper I have asked if it was possible to determine the outcome simply by
studying the initial state of a system. The short answer is no overall. For simple and straightforward
systems we can have a good idea of what will happen by using a different material or slightly altering the
starting position. But more complicated systems, like the weather, will show that chaos has too large an
effect on the process to say that we can predict the outcome. That being said, I wonder if there will be a
time that we understand the effects of chaos and can accurately predict (rather than give a best guess) as
to the nature of any system by simply looking at the initial conditions. It would be a form of fortune
telling so to speak.
Great essay, well written, with a wonderful application to weather and nuclear reactions.
Regarding the question of prediction, Sensitivity to Initial Conditions (SIC) only occurs for chaotic
systems. For any other basin of attraction, such as periodic orbits, there is no SIC. Grade = A.