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In my last post, I ended by suggesting that

Many systems in the world display truly chaotic behavior, wherein small changes to the initial conditions creating vast differences in the outcomes. On the other hand, many have very stable attractors like those described above, where one can discard most of the external factors affecting them in favor of a much smaller number that have the largest effects on the outcome.

Weather is the canonical example of a chaotic system. atmospheric systems are extremely unpredictable, with their behaivior varying greatly depending on small variations of initial conditions. Moreover, weather systems are so immense, and consist of so many component variables, that even if we could accurately simulate them, collecting the right initial conditions to kick off our simulation would be infeasible as well.

Even so, there are features of weather systems that do make them somewhat tractable - we can recognize and predict the behavior of things like storm fronts, high pressure systems, hurricanes, and jet streams. These behave in relatively stable ways for short periods of time, allowing us to make predictions (though over long enough periods of time, the collective weight of small errors in initial conditions build up and render our models useless).

Similar things may be said about other complicated systems. While it is impossible to perfectly predict the behavior of the U.S. electorate during a national election, with enough samples, sites like FiveThirtyEight.com came within an astonishingly small margin of the final tally in the 2008 presidential election. Again, this result was possible because the electorate behaves in a relatively stable way for a short period of time, even if we cannot measure its initial state 100% accurately.

We can even say the same sort of thing about things like the observable world around us. For instance, on my walk to work, I see an incredibly complex world around me, with cars, people, animals, buildings, rain and snow, and thousands of other events happening all the time. If I were to freeze-frame a single moment of my walk, and observe the multitude of objects around me, there are millions of possible "next steps" that might happen. That car on the street might swerve violently to the left and hit a parking meter. That woman in front of me might start spinning in circles in the middle of the sidewalk. The lights might change color at random times, without warning.

But none of those things are likely to happen. In fact, if you showed that snapshot of my walk to a dozen people, and asked them what the same scene would look like three seconds hence, they'd probably unanimously agreee on the major features of the scene. Cars will continue to drive down the road in a predictable way, people will continue to walk straight down the street, lost in their thoughts, and lights will change on a regular timetable. The dozen people might disagree on the details, like what color the next car that comes down the street might be, but the major components would be the same.

So, the world evolves in a chaotic way where the details are difficult to predict, but the major features tend to evolve in a stable and continuous way when observed at a certain scale of time and space. Sound familiar? These relatively stable features are strange attractors in the complex system of the world. For nearby sets of initial conditions, events in the world fall into a the same basin of attraction. A car driving down the street will most likely continue to drive down the street, regardless of whether it is one inch to the right or left. A person walking will continue to put one foot in front of the other, regardless of whether they step on a crack in the sidewalk or not.

This continuity and predictability on a certain scale is a major feature of the world around us, and one that I believe is intimately related to the way our brains perceive the world. If the world really were chaotic, in the sense that outcomes were extremely sensitive to variation in initial conditions, we wouldn't be able to make any sense of it. Unless we had extraordinarily sensitive perceptions, each time we perceived something we would miss subtle details in the initial conditions, and the outcome would be so different that we'd never be able to understand the world.

But this isn't the case. There are millions of possible places that a car that I perceive could be three seconds from now, but my brain can predict with high probability that it won't be on the sidewalk, so I continue walking. Similarly, despite subtle variations in lighting, angle, and distance, I can consistently recognize a friend's face each time I see him. The details are different each time, but the overall impression - "That's my friend" - is the same to my mind.

Our brains are wired in such a way as to mimic these basins of attraction in the world. We do, actually, perceive many subtle variations in our environment, things like lighting, angle, distance, temperature, and smell. To make sense of the world, we look beyond the chaos, and see only the patterns.

As I discussed last time, seemingly discrete events in the macroscopic world (like my possible walks to work) seem to "collapse" themselves into fewer possible outcomes based on intervening events. A large set of initial states result in a single end state, and overall, the total number of initial states gets winnowed down into a much smaller number of final states.

Additionally, at many (if not most) points along the spectrum of starting states, a little variation one way or another doesn't mean much for the outcome - the system is resilient to small changes. However, there are certain points ("inflection" or "tipping" points, one might say), where the outcome changes drastically based on a small change in initial conditions.

All of this might sound familiar if you've read much about chaos theory and strange attractors. Consider that my walk to work is a dynamical system, evolving over time based on a set of constraints, the most significant of which are the red lights. The possible arrival scenarios are attractors - the system tends to evolve towards those points from a large set of initial conditions (the basin of attraction).

Note that there are thousands, if not millions, of other events along my walk to work that affect the timing of my walk to work. This is what makes the system chaotic, and the attractors strange. The sequence of my steps, whether I'm tired or well-rested, the other people walking around me, the prevailing winds, and many other variables affect my speed and my arrival time. However, the red lights have, by far, the most significant effect on my arrival time. That's why I was able to draw the diagrams I did in the previous post to show how the red lights affect my arrival times at work.

In fact, my exact arrival time may vary by fractions of a second one way or another because of these other, more minor effects. But in analyzing my arrival at work, those miliseconds don't matter nearly as much as the multi-second (or even multi-minute) swings caused by the red lights.

(Similarly, one might say that in analyzing my entire day, the difference between arriving at work one minute earlier or later doesn't mean very much, unless, for example, I miss breakfast. Further, one might say that in analyzing my career, my arrival at work on one day doesn't have much effect, unless, for example, I miss an important meeting and get fired as a result. This points to another interesting bit, which is that a similar analysis of the attractors in a system is valid at many different scales, but that's a whole other blog post...)

Many systems in the world display truly chaotic behavior, wherein small changes to the initial conditions creating vast differences in the outcomes. On the other hand, many have very stable attractors like those described above, where one can discard most of the external factors affecting them in favor of a much smaller number that have the largest effects on the outcome.

Next time: rampant speculation on how these attractors relate to how we perceive the world...

My commute to and from work each day consists of a train and subway ride, as well as about 1/2 a mile of walking. This time is when I get most of my reading, writing, and thinking done. Unsurprisingly, given how much time I spend missing trains and held up walking at red lights, my thoughts often turn to the walk or ride itself.

Let's consider one part of my commute: the walk from the PATH station to the office. This walk is about 5 blocks or so on the New York grid, passing through a traffic light at each intersection. Plotted with one dimension of space and one of time, it might look like this:

In this plot, time progresses from left to right, and my walk will take me from the top of the plot (the train station) to the bottom (the office). Each red light along the way is indicated as a red line (green lights are left blank for clarity - they would fill the blank spaces between each red dash).

Let's say I leave the train station at a particular time near the left side of the graph and start walking:

Assuming I walk at a constant speed, my path while walking looks like a diagonal line through time and space. When I get held up at a red light, my path goes horizontal (I'm standing still in space, but still progressing through time). Overall, this gives my path a bit of a zig-zag pattern in the 1-D space + time plot.

Now, on the same morning, let's say I leave the PATH station minute or two later (maybe the train was delayed a bit, or I exited from the back of the train instead of the front):

Once again, my zig-zag path takes me to the office, stopping at the red lights along the way. But interestingly, despite leaving later from the train station, I arrive at the office at exactly the same time:

In both routes, despite stopping at different lights early on my journey, I still get caught at that same last red light. It didn't matter that I started earlier or later - the outcome was the same.

Now, we plot all of the possible starting times and their resulting walks:

Interestingly, despite dozens of possible starting times at the train station, there are only a few possible times of arrival at the office. In effect, the red lights "collapse" the outcomes into a smaller set. For example, the 18 different starting times highlighted in the figure below all have the same result:

Another interesting set of paths to look at are the "tipping points" - any pair of neighboring paths with different arrival times:

Here, a difference of a few seconds in starting time between the blue and yellow paths means many minutes of delay on the arrival side. Also notable is that the blue path is one of the shortest, with respect to total time, whereas the gold path is one of the longest.

Next time: how I think these paths relate to chaos theory and strange attractors...

(Inspired by Ybry's visualizations of train schedules, from Edward Tufte's book Envisioning Information.)