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.

Let’s begin with an example. Something known as the logistic map is (arguably) the simplest chaotic system. We will describe and analyze it in detail in Part II.

---

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: High school level math, in particular, looking at graphs. If you don’t understand anything, please ask – in the comments or by emailing thechaostician@gmail.com.

Originally Written: December 2016.

Confidence Level: Established science since the mid twentieth century. Occasional philosophy mixed in.

---

---

A Parabola

A chaotic system can be completely described using a parabola: $$ y = r \ x \ (1 – x) \, , $$ where $r$ can be any number between 2 and 4. We will consider the cases $r = 2.9, r = 3.2, r = 3.5, r = 3.65, r = 3.835,$ and $r = 4.$

The parabola goes through $y = 0$ when $x = 0$ and when $x = 1$. If $x$ is between 0 and 1, then $y$ will be between 0 and 1 as well. As we increase the value of $r$, the parabola gets taller. When $r = 4$, the highest point of the parabola touches $y = 1$.

---

Dynamics

To see chaos from this simple parabola, we have to introduce time.

We will use discrete time: measurements are taken once each day, not at every possible intermediate time.

Start at some initial position on the first day, e.g. $x_1 = 0.1$.

To determine the position on the second day, apply the function for the parabola to the position on the first day. $$ x_2 = r \ x_1 \ (1 – x_1) $$

To determine the position on the third day, apply the function for the parabola to the position on the second day. $$ x_3 = r \ x_2 \ (1 – x_2) $$

Repeat this over and over again to determine what the position is on any subsequent day. $$ x_t = r \ x_{t-1} \ (1 – x_{t-1}) $$

Let’s see how the position changes with time. We might get different results for different values of $ r $ and for different initial conditions.

We give a name to this process of applying the function for a parabola over and over again on some initial conditions: the logistic map.^[1]The name was chosen for historical reasons. Don’t worry about how this process is either logistic or a map.

We will consider the behavior starting from two nearby initial conditions: $x_1 = 0.1$ (blue dot) and $x_1 = 0.101$ (red dot).

$r = 2.9$

Both initial conditions approach a fixed point at $x_* = \frac{19}{29} \approx 0.655$ after a few initial bounces.

$r = 3.2$

Both initial conditions approach a periodic orbit of period two. After a few initial bounces, the position alternates back and forth between two locations.

$r = 3.5$

Both initial conditions approach a periodic orbit of period four. After a few initial bounces, the position alternates back and forth between four locations.

$r = 3.65$

The motion does not settle into any regular orbits. They do not eventually become periodic. Although the two initial conditions start close together, they do not stay close together. The motion looks random, but it is completely determined by the parabola and the initial condition. This is our first example of chaos.

$r = 3.835$

Initially, the motion looks chaotic. The two nearby orbits move apart. However, this doesn’t last. The motion eventually settles down onto a periodic orbit of period three. Although both initial conditions end up on the periodic orbit, they are at different locations along the orbit.

$r = 4$

The motion is again chaotic, just like it was when $r = 3.65$. No obvious pattern of the motion can be seen. Instead, it looks random. When $r = 3.65$, the motion was more likely to be in some locations than others. When $r = 4$, the position could be anywhere between 0 and 1 with apparently equal probability.

Finding Structure

Suppose that you encountered something in the physical world that moved chaotically – its trajectory looked like the motion of this system when $r = 3.65$ or $r = 4$. At first glance, you might assume that the motion is random. This is wrong. The motion has much more structure.

The first thing to realize is that this motion is actually governed by a parabola. Each location is not determined randomly – it is determined by the location of the object one day ago. You wouldn’t notice this if you only plot position vs. time for your data. Instead, to find this structure you have to plot the position today vs. the position yesterday. When you do this for every recorded position, you will find that all of the data points lie along a single curve – the parabola of Figure 1.

We can find even more structure.

Something significant is happening when we change $r$. Although the first few bounces look similar for similar values of $r$, the long time behavior can be dramatically different. To find the additional structure, we will consider how the long time behavior changes as we change $r$.

Here is how we will investigate the long time behavior:

• Choose a value of $r$.
• Choose an initial condition, e.g. $x_1 = 0.1$.
• Apply the function 1000 times. Since we want to see what it does after a long time, we should ignore what it does initially.
• Apply the function 1000 more times and record the position for each time.
• Plot all of these positions along a vertical line.
• Repeat the previous steps for 5000 different values of $r$ between 2 and 4.
• Place all of the vertical plots next to each other to form a single plot of $x$ vs. $r$.

The resulting plot can be seen in Figure 8. Clearly, there is a lot more structure here that was not apparent from just looking at the chaotic motion. This kind of plot, which shows how the long time behavior changes as you vary a parameter of the system, is called a bifurcation diagram.

Let’s analyze what is going on with this plot.

Between $r = 2$ and $r = 3$, there is only a single curve on the plot. We saw this sort of behavior in the gif for $r = 2.9$. The motion eventually settles down onto a single point. While point this is can vary with $r$, there is only one final point. On this graph, that looks like a single curve.

Between $r = 3$ and $r = 1 + \sqrt{6} \approx 3.45$, there are two lines on the plot. We saw this sort of behavior in the gif for $r = 3.2$. The motion eventually settles down onto a periodic orbit of period two. After a few initial bounces, the position alternates back and forth between two locations. While these two points can vary with $r$, the long time behavior stays close to only two points. On this graph, that looks like two curves.

Between $r \approx 3.45 $ and $r \approx 3.54$, there are four lines on the plot. We saw this sort of behavior in the video for $r = 3.5$. The motion eventually settles down onto a periodic orbit of period four. After a few initial bounces, the position alternates back and forth between four locations. While these four points can vary with $r$, the long behavior stays close to only four points. On this graph, that looks like four curves.

At $r \approx 3.54$, the four lines split into eight, so the long time behavior is described by a periodic orbit of period eight. Shortly thereafter, the eight lines split into sixteen. Although this is no longer visible in the plot, the lines then split into thirty-two, then sixty-four, and then higher powers of two. The $r$ values where the splitting happens keep on getting closer together. These subsequent splittings and the corresponding creation of periodic orbits with longer and longer periods is called a period doubling cascade.

By $r = 3.57$, the period doubling cascade has reached its completion. After this, the motion is chaotic. When the motion is chaotic, it never settles down onto a periodic orbit. Instead, the motion appears to be randomly scattered across many of the possible values of $x$. We saw this sort of behavior in the video for $r = 3.65$. We can also see this in the bifurcation diagram. Chaotic regions do not have a few lines on the plot. Instead, they have what looks like a shaded region. The shaded region, for each value of $r$, is the region that the chaotic motion explores. For $r = 3.65$, notice that the chaotic motion never gets close to either $x = 0$ or $x = 1$. This can also be seen in the video.

Not all of the values of $r > 3.57$ are chaotic. There are a few regions of $r$ values where the long time behavior is periodic again. In particular, $r = 3.835$ is in a narrow region with three lines embedded in the chaotic regions. There are also narrow regions which are characterized by periodic orbits of any period. Longer periods tend to have narrower widths.

At $r = 4$, the motion is chaotic again. The chaotic region fills the entire interval between $x = 0$ and $x = 1$. This can be seen both on the bifurcation diagram and in the video. Also notice in the graph of the parabola (Figure 1) that the parabola reaches all the way up to $y = 1$. For smaller values of $r$, the parabola never touches $y = 1$, so the long-term behavior can’t cover the entire interval.

Universality

The $r$ values where the long term behavior of the system changes appear to be important. These values of $r$ are called bifurcation points.

Let $r_0 = 3$ be the bifurcation point where the fixed point bifurcates into a period two point.^[2]In case you find this terminology unfamiliar: All I’m say is that I am going to give the name $r_0$ to the number $3$ because that is the value of $r$ where something significant happens. I an … Continue reading Let $r_1 = 1 + \sqrt{6} \approx 3.45$ be the bifurcation point where the period two point bifurcates into a period four point. Let $r_2 \approx 3.54$ be the bifurcation point where the period four point bifurcates into a period eight point. Let $r_n$ be the bifurcation point where the periodic orbit of period $2^n$ bifurcates into a periodic orbit of period $2^{n+1}$.

We can take limits with these bifurcation points. Chaos first appears in this system at $$ r_* = \lim\limits_{n \rightarrow \infty} r_n \approx 3.57 \, . $$

There is an even more interesting limit that we can take. Define the Feigenbaum constant $\delta$ to be $$ \delta = \lim\limits_{n \rightarrow \infty} \frac{r_{n-1} – r_{n-2}}{r_n – r_{n-1}} \approx 4.669 \, . $$ The original paper where Feigenbaum presents this constant is Quantitative Universality for a Class of Nonlinear Transformations.

The Feigenbaum constant is interesting because it is universal.

Any time we observe a period doubling cascade, we can calculate the Feigenbaum constant $\delta$. Surprisingly, the value of this limit is always the same ! We considered an example of a discrete time system built from a parabola. We could have also considered many other discrete time systems – any function with a bump that increases in height as a parameter (like $r$) is increased will have a period doubling cascade. The measurement of $\delta$ for any of these will give the same result, $\delta \approx 4.669$. We could also consider discrete time systems in higher dimensions – instead of just measuring the position along a line on each day, we could measure the position in three dimensional space. Any of these which undergo a period doubling cascade would also have the same value of $\delta$. Period doubling cascades can also occur in continuous time systems described by differential equations. The Feigenbaum constant remains the same. The Feigenbaum constant has even been measured in fluids undergoing a transition to turbulence. For example, an experiment only a few years after Feigenbaum’s original paper measured $\delta \approx 4.4$ in a heated tank of mercury (Period Doubling Cascade in Mercury, a Quantitative Measurement). Mathematical structures from one-dimensional discrete time systems to partial differential equations and experimental systems from heated fluids to populations of predators and prey (Period Doubling Cascades in a Predator-Prey Model with a Scavenger) will all give the same constant if they experience a period doubling cascade.

We have seen that a period doubling cascade is one mechanism by which chaos can form. There are other methods – for example, the breaking of homoclinic or heteroclinic orbits. However, any time a period doubling cascade occurs, you can take the limit for $\delta$ and you will get the same value for the Feigenbaum constant.

There are two main lessons to learn from this example.

The first lesson is that, even if the motion of something looks random, it might be describable by a simple rule. We saw this for most of the values of $r > 3.57$. If we just looked at a video of the motion itself, we would see that it wanders around without any clear pattern. However, we know that there is a simple rule governing the motion. The motion can be completely described using the parabola from Figure 1.

The second lesson is more subtle, but is extremely important in the study of chaos: Universality can appear in unexpected ways.

We have not found a universal law, like gravity, which applies to all objects in the universe. We have not found a fundamental particle, like an electron, which can’t be subdivided by any means we now possess and is common throughout the universe. But we have found something universal.

The universality is a particular number which arises whenever there is a period doubling cascade. It doesn’t matter what is being described or what rules are used to govern the motion. Any time that you have a period doubling cascade, the number that characterizes this structure appears.

This is how universality arises in the study of chaos. We look at what kinds of structures emerge from complicated behavior. Occasionally, we find some pattern which appears commonly among many of the systems that we study. This pattern is universal. The pattern may be in how the system changes in time, or it may be how the system changes when you change some parameter, like $r$. When we recognize this pattern anywhere else, we can apply everything that we learned about it.

The details of what physical or mathematical system you are studying don’t matter very much to a chaostician. What matters is trying to find structure that appears in lots of otherwise unrelated systems.

Next post.