I chose the name The Chaostician for this blog. To live up to this name, this blog should be about chaos, in particular, what a physicist means when referring to ‘chaos theory’.
This seven eight part series is my explanation of chaos theory to a popular audience. Chaos is a mechanism that allows deterministic objects to behave unpredictably. I will explain why this happens and what kinds of predictions we can make when something is chaotic.
Those of you who are already experts in nonlinear dynamics know that there are many ways to approach chaos. In this series, I will focus on behavior on strange attractors in dissipative chaos. In particular, the goal is to explain the ideas underlying periodic orbit theory and trace formulas you can use to calculate expectation values. Other series will address other approaches to chaos.
So far, we have discussed mostly phenomenology. I would present some model – either physical or mathematical – and then describe its behavior. Now, instead of just describing the behavior, it’s time to understand how this behavior arises.
What is the basic mechanism creating the strange behavior of chaotic systems? Why do some systems exhibit sensitive dependence on initial conditions? How can similar initial conditions rapidly become dramatically different, without everything flying apart? By now you should have seen that these sorts of behavior do happen. But how do they happen?
The underlying mechanism is the title of this section: stretch and fold.
Part I: Introduction.
Part II: The Simplest Chaotic System.
Part III: Lyapunov Exponents.
Part IV: Strange Attractors.
Part V: Continuous Time.
Part VI: Stretch and Fold.
Part VII: Partitions and Symbols.
Part VIII: Periodic Orbit Theory.
Prerequisites: I hope this is understandable using high school level math. If you don’t understand anything, please ask – in the comments or by emailing thechaostician@gmail.com. I reference the previous Parts even more than usual because we will be returning to and explaining the examples in them.
Originally Written: April – July 2017.
Confidence Level: Established science since the mid twentieth century. Occasional philosophy mixed in.
Kneading Dough
Think about kneading dough. To knead, you take a lump of dough, stretch it out in one direction, then fold it back onto itself and smush it back into a single lump. Repeat this over and over until the dough is the right consistency.
Kneading dough exhibits sensitive dependence on initial conditions. Start with two tiny chocolate chips in the dough which are extremely close together. Every time you stretch the dough, the two chocolate chips get farther apart. And yet, there is a maximum distance between the two chocolate chips given by the size of the lump of dough. The distance between the chips increases exponentially until the distance between them is close to the size of the lump of dough. After that, their motion is completely independent. If you start with a blob of food coloring, kneading the dough will eventually spread it out so it is evenly distributed throughout the entire lump of dough. During the process, you can see intricate marbled patterns of different colors, all formed by repeating a simple motion.
This is paradigmatic of motion on a strange attractor.
Examples
We can see this behavior in the examples we’ve used previously.
Logistic Map
Let’s start with the logistic map with $r = 4$. This was the example we used back in Part II. Recall that the motion of the logistic map is defined by the parabola: $$ y = r \ x \ (1 – x) $$ We can use this parabola to see how the logistic map stretches and folds the line between zero and one, resulting in chaotic motion for any point along that line.
- Begin with a bunch of random points between 0 and 1. We will use these as our initial conditions. Draw them along the $x$ axis.
- For each point along the $x$ axis, calculate the corresponding $y$ value according to the logistic map. Plot the $(x,y)$ Cartesian coordinates in two dimensions. The initial line along the $x$ axis between 0 and 1 has stretched into the (longer) parabola. Points that were close together are now farther apart.
- For the next time, we will only need the $y$ values. Drop the $x$ coordinate of each point. We are left with a bunch of points along the $y$ axis. The parabola has been folded. Some points that were previously far apart are now smashed close together.
- To iterate the logistic map, we take the $y$ values of the previous time and make them the $x$ values of the next time.
- Stretch the points along the $x$ axis up to the parabola again.
- Fold the parabola onto the $y$ axis again.
- The $x$ values of the next time are the $y$ values of the previous time.
Repeat this over and over again to see the chaotic motion of the logistic map.
Rössler Attractor
Something similar happens with the Rössler attractor.
The motion on the attractor in continuous time consists of two parts: circular motion around the vertical axis and chaotic motion perpendicular to the circular motion. We can separate out these two parts of the motion using a Poincaré section. In Figure 4 of Part V, we chose multiple Poincaré sections at different locations along the circular motion to see what happens in the perpendicular directions.
We will now look at this again to see if we can see it stretch and fold.
- The attractor begins as a narrow horizontal line.
- The attractor stretches horizontally a bit.
- The attractor stretches horizontally a bit more.
- The attractor begins to bend up and stretches more.
- The attractor stretches to its maximum length and is mostly bent upwards.
- The bend of the attractor becomes sharper.
- The attractor starts to fold down onto itself.
- The folding is complete. The attractor is once more smashed down to a narrow horizontal line.
This then repeats every time you go around the attractor.
Tearing
Not all of the chaotic systems we’ve seen can be understood by stretching and folding dough. For some of them, another kneading technique is needed: tearing. Instead of just stretching and folding the dough, you can also stretch the dough, tear it, then squish the pieces back together. If you follow a chocolate chip embedded in the dough, its motion will be chaotic.
Pinball
When discussing Lyapunov exponents in Part III, we introduced a game of pinball that consisted of a ball bouncing between three circular disks. We showed that this game has a positive Lyapunov exponent – if you consider two games which are initially only different by the diameter of an atom, within 10 bounces, the two games become entirely different.
The pinball game’s attractor is split into three parts – one for each of the disks. Consider a ball which has just bounced off of the first disk. It can do one of three things: it can hit the second disk, it can hit the third disk, or it can miss both disks entirely and leave the game. We could say something similar if the ball had just bounced off of the second or third disk.
We can think of the pinball game in terms of kneading dough. Consider three piles of dough, corresponding to the three disks. Bouncing off of one of the disks corresponds to stretching the dough of each pile. Then tear the dough of each pile. Some of the dough of the first pile is added to the second pile, some is added to the third pile, and some is discarded. Some of the dough of the second pile is added to the third pile, some is added to the first pile, and some is discarded. Some of the dough of the third pile is added to the first pile, some is added to the second pile, and some is discarded. None of the dough remains in the same pile it started in. Repeat this over and over to model the motion of the pinball. The dough that moves from one pile to another corresponds to pinballs that bounce between these two disks. The dough that is discarded corresponds to the pinballs that leave the game. No pinballs immediately return to the disk that they bounced off of.
This may not be a particularly effective way of kneading dough, especially because you discard some of the dough each time. Nevertheless, it should eventually mix the dough thoroughly. Any initial lumps in the dough will be broken up and spread evenly throughout the three piles of dough.
Lorenz Attractor
The most famous example of chaos is the Lorenz attractor, or the “Lorenz butterfly”. It is very similar to, and slightly more complicated than, the Rössler attractor, so we have focused on the Rössler attractor. The Lorenz attractor is famous both because it was one of the first chaotic systems studied and because it has a memorable shape.
Just like the Rössler system, the Lorenz system is defined by a differential equation in three dimensions. The velocity is determined by the position according to: $$ \begin{array}{rcl} v_x &=& 10 \ (y – x) \\ v_y &=& x \ (24.5 – z) – y \\ v_z &=& x \ y – \tfrac{8}{3} \ z \end{array} $$ The strange attractor that results is shown in Figure 4. The main difference between it and the Rössler attractor is that the Lorenz attractor chaotically goes around two circles, while the Rössler attractor only goes chaotically around a single circle.
Depending on where the moving object starts, it could either continue going around the same circle or it could switch and start moving around the other circle. The Rössler attractor just stretches and folds the dough each time the object goes around. The Lorenz attractor tears the dough, keeping some of the dough around the original circle and sending some of the dough to the other circle.
Describing the behavior of the Lorenz system is slightly more complicated than the Rössler system. You need two Poincaré sections, one for each circle. Because the motion is clearly still chaotic, we can use similar techniques to analyze both the Rössler and Lorenz attractors.
Why Fractals?
When we first introduced strange attractors in Part IV, we found that the Hénon attractor continues to have interesting, self-similar structure when we zoomed in on it. It is a fractal. We could also zoom in on the Rössler and Lorenz attractors and we would find that they continue to have interesting, self-similar structures even when you zoom in a lot.[1]The Lorenz attractor is not just a butterfly. It is a fractal butterfly. Why do strange attractors often have fractal structure?
I should emphasize that a fractal is a shape, while chaos is a description of motion. I could make an analogy between their relationship and the relationship between a circle and rolling. If you were to study rolling in detail, you would find yourself regularly encountering circles and conclude that there is some intrinsic connection between this shape and this type of motion. When we study chaos in detail, we find ourselves regularly encountering fractals and conclude that there is some intrinsic connection between these shapes and these types of motion.
Fractals are often created iteratively. You start with some shape and apply a simple rule to that shape over and over again. Each time you apply the rule, it creates structures of smaller and smaller sizes. As you zoom in, you keep on seeing these smaller structures. The shape never becomes boring, regardless of how much you zoom in on it.
Strange attractors are what you see in a chaotic system after a long time. They are formed by repeated stretching and folding. Each time it stretches and folds, it makes smaller and smaller structures. After a long time, you get a shape that continues to look interesting as you zoom in. When the dynamics stretches and folds, the attractor will be a fractal.
References
| ↑1 | The Lorenz attractor is not just a butterfly. It is a fractal butterfly. |
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