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.

Part IV is about strange attractors. If you allow something to wander chaotically for a while, what kinds of patterns does it make? Surprisingly, the answer is often a fractal.

---

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. I reference my post of Fractal Dimensions, but I don’t think it’s necessary to understand this. If you don’t understand anything, please ask – in the comments or by emailing thechaostician@gmail.com.

Originally Written: January 2017.

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

---

---

Attractors

One of the most important (and sometimes most difficult) problems when dealing with something that might be chaotic is to figure out what it will do after you’ve let it run for a long time. This long time behavior becomes more tractable if the system you are studying has an attractor.

If you start close to an attractor, you will come closer and closer as you watch the system for a long time.

There is one other requirement which guarantees that you have identified a single attractor, instead of multiple attractors. If you start on the attractor, you must eventually get close to every other location on the attractor.

We have already seen several examples of attractors when we studied the logistic map.

• When $r = 2.9$, the initial conditions approached a single fixed point. This is an example of an attractor, specifically an attracting fixed point.
• When $r = 3.2$, the initial conditions approached a periodic orbit of period two. When $r = 3.5$, the initial conditions approached a periodic orbit of period four. Each of these is an example of an attractor, specifically an attracting periodic orbit.
• For the other values of $r$, it is not as clear whether or not there is an attractor.
• For $r = 3.835$, there is an attracting periodic orbit of period three. However, it takes a while to settle down onto this periodic orbit. This suggests that there might be some other hidden structure.
• For $r = 3.65$, you could say that there is an attractor which covers most of the interval between 0 and 1. If you start outside of this region, you will eventually end up there. It is difficult to figure out if the second condition holds. If you start in this region, will you eventually get close to every other point in this region? We typically don’t use the word ‘attractor’ to refer to something with the same dimension as the space (in this case, the space, [0,1], is one-dimensional). Instead, we might call it a trapping region.
• When $r = 4$, there definitely isn’t an attractor. If you start at a random location in between 0 and 1, the motion doesn’t attract to anything. Instead, it chaotically wanders throughout the entire region of [0,1].

There are many examples of attractors in physical systems as well. If you start a pendulum with any initial position (except exactly balanced upside-down) and let it go, it will eventually settle down and stop at the bottom. The lowest point of a pendulum is an attracting equilibrium point.

---

Attractors of Chaotic Systems

An attractor in a one-dimensional chaotic system can only be part of the line segment between zero and one. This is still useful information: the Bifurcation Diagram in Figure 8 of Part II can be thought of as a plot that shows how the attractor changes with $r$.

The Hénon map

To see an attractor with more interesting structure, we will have to go to higher dimensions. The Hénon map is a two dimensional extension of the logistic map. Instead of having a point moving around on a line in discrete time, the Hénon map has a point moving around in two-dimensional space in discrete time. The $x$ and $y$ coordinates of the point at any time are given in terms of the $x$ and $y$ coordinates at the previous time: $$ x_t = 1 + y_{t-1} – a \ {x_{t-1}}^2 $$ $$ y_t = b \ x_{t-1} $$ $a$ and $b$ are numbers that we can change – just like $r$ in the logistic map.

Let’s see how the position changes in time for the Hénon map with $a = 0.9, b = 0.3$ and with $a = 1.4, b = 0.3$. In both cases, we will start with two initial conditions: $(0.1,0.1)$, shown in red, and $(0.101,0.101)$, shown in blue.

When $a = 0.9$, the motion eventually settles down onto an attracting periodic orbit of period two. When $a = 1.4$, the motion is chaotic. While I have not shown the entire thing here, there is a period doubling cascade as $a$ increases from $0.9$ to $1.4$, like the logistic map.

It is clear that the chaotic motion does not go everywhere. It is confined to certain regions. There is an attractor here. Let’s see what it looks like. To see it, we will need to focus on the long time behavior only. At earlier times, the motion is still moving towards the attractor, but hasn’t reached it yet.

Visualizing the Attractor

Here is how we will visualize the attractor:

• Choose a value of $a$ and $b$.
• Choose an initial condition, e.g. $x_1 = 0.1, y_1 = 0.1$.
• Apply the Hénon map 10,000 times. Since we want to see what it does after a long time, we should ignore what it does initially.
• Record the position for the next 1,000,000 times. This shows the long time behavior. I am recording a lot of points because I want to see any fine details of the shape of the attractor.
• Plot all of the recorded positions in two dimensions.

For $a = 0.9$, we can see an attracting periodic orbit of period two.

For $a = 1.4$, we can see the attractor when the Hénon map is chaotic. It is a strange attractor.

An attractor is strange if its shape is a fractal. I define a fractal to be any object which doesn’t get boring when you zoom in a lot.^[1]See my post about Fractal Dimensions for more details.

Let’s see what happens if you zoom in on the strange attractor of the Hénon map.

• 

• 

• 

• 

• 

• 

• 

• 

We keep on finding structure as we zoom in. There is even some self-similarity. Notice that the top part of the attractor is split into three lines. As we zoom in on the top line, we see that it splits into three lines too. As we zoom in on the top line of these three, we see that they also split into three even smaller lines. Eventually, the picture starts to break up into individual points. This isn’t because the Hénon attractor doesn’t have any more structure there. It’s because this picture ‘only’ plots a million of the points on the attractor.

The attractor of Hénon map is a fractal. Since fractals are sometimes considered ‘strange’ geometric objects, we call it a ‘strange attractor’.

---

Basins of Attraction

In the logistic map, we saw at most one attractor for each value of $r$. Similarly, in the Hénon map, we saw at most one attractor for each combination of $a$ and $b$. Could you have multiple attractors?

Let’s consider an example with a pendulum. Make the ball of the pendulum a magnet, with its north pole facing down. Instead of just letting the pendulum move with the force of gravity alone, place several strong magnets on the ground under the pendulum. Place a magnet with its north pole facing up (so it will repel the pendulum) directly under the lowest point the pendulum could be at. Place magnets with their south pole facing up (so they will attract the pendulum) in a circle around it. The ball of the pendulum will be attracted to each of the magnets in the circle. Instead of having a single attracting equilibrium point at the bottom, we now have multiple attracting equilibrium points: one above each of the magnets in the circle.

If you allow the pendulum to swing, which equilibrium point will it attract to?

The answer depends on where you start the pendulum.

If you start really close to one of the magnets in the circle, you will attract to that magnet. What happens if you start about halfway between two magnets or far away from any of the magnets? You might think that the pendulum will attract to the closest magnet, but this doesn’t always happen. The pendulum can gain enough momentum to overshoot the closest magnet and approach another one. The pendulum will be pushed and pulled by the various magnets as it passes by them. Its motion will look chaotic. Eventually, because of friction, the pendulum will settle down close to one of the magnets in the circle. Videos of this kind of magnetic pendulum can be found on Youtube.

The ‘basin of attraction’ for each magnet is the set of all initial conditions which eventually settle down onto that magnet. Because the pendulum can overshoot the magnets, the basin of attraction is not all of the initial conditions closest to that magnet. Instead, it is something more complicated.

To get an idea of what the basins of attraction might look like, let’s consider a simpler system that also has six evenly-spaced attracting equilibrium points. We will use a two-dimensional map defined according to:^[2]This map is derived by using Newton’s method to find the roots of $z^6 – 1 = 0$ in the complex plane. There are six roots, evenly spaced around the unit circle. $$ x_t = \frac{5}{6} \ x_{t-1} + \frac{1}{6} \ \frac{{x_{t-1}}^5 – 10 \ {x_{t-1}}^3 \ {y_{t-1}}^2 + 5 \ {x_{t-1}} \ {y_{t-1}}^4}{({x_{t-1}}^2 + {y_{t-1}}^2)^5} $$ $$ y_t = \frac{5}{6} \ y_{t-1} + \frac{1}{6} \ \frac{{y_{t-1}}^5 – 10 \ {y_{t-1}}^3 \ {x_{t-1}}^2 + 5 \ y_{t-1} \ {x_{t-1}}^4}{({x_{t-1}}^2 + {y_{t-1}}^2)^5} $$ This may look complicated, but it is just $x_t$ and $y_t$ written as explicit functions of $x_{t-1}$ and $y_{t-1}$, just like the Hénon map.

There are six attracting equilibrium points: $(x = 1, y = 0)$, $(x = -1, y = 0)$, $(x = \frac{1}{2}, y = \frac{\sqrt{3}}{2})$, $(x = -\frac{1}{2}, y = \frac{\sqrt{3}}{2})$, $(x = \frac{1}{2}, y = -\frac{\sqrt{3}}{2}),$ and $(x = -\frac{1}{2}, y = -\frac{\sqrt{3}}{2})$. We can plot all of their basins of attraction.

Notice how interesting the shapes of the boundaries of the basins of attraction are. If we zoom in on them, we would discover that they are also fractals.

This gives us another form of sensitive dependence on initial conditions. Near the boundary of the basins of attraction, all of the basins come extremely close together. Even small changes in the initial conditions could change which attracting equilibrium you eventually end up at and how long it takes to get there.

In this example, each basin of attraction approached an attracting equilibrium point. This is not always the case. Basins of attraction exist for attracting periodic orbits and for strange attractors. Whenever you have multiple distinct attractors in your system, each will have its own basin of attraction.

---

Repellors

If you could balance a pendulum perfectly upside-down, it would stay there. However, if it shifts by even the smallest amount, it would fall and eventually settle down to the attracting equilibrium point.

Similarly, if you started a magnetic pendulum exactly on the boundary between basins of attraction, you would stay on the boundary. It would not have to stay still. Instead, it would wander chaotically around on the fractal boundary. If it ever gets bumped even slightly off of the boundary, then it would be in a basin of attraction and thus would eventually settle down onto one of the attracting equilibrium points.

These two examples are in some ways similar to the attractors discussed above. If you start exactly on them, you will stay on them. They can have various structure, depending on what system you are describing. There is an important difference: Points close to these objects do not come closer as time goes on. Instead, the nearby points get farther away.

These examples are called repellors.

The upside-down pendulum is a repelling equilibrium point. The boundary between the basins of attraction of the magnetic pendulum and the boundaries of Figure 5 are strange repellors. They can be harder to see than strange attractors, but are still important to understand the dynamics.

Strange repellors can explain the phenomenon of transient chaos.

We have seen two examples of transient chaos already. The first example was the logistic map with $r = 3.835$. The motion initially looked chaotic, but eventually settled down onto the periodic orbit of period three. The second example was the magnetic pendulum. Videos of the magnetic pendulum showed it wandering around for a while before eventually settling down onto one of the fixed points.

If you start extremely close to a strange repellor, you will follow the motion of the repellor initially. The motion on the strange repellor is chaotic, so your motion will look chaotic too. The distance between you and the repellor grows exponentially, so after some amount of time, you will get far enough away that you can no longer feel its influence. You then settle down onto an attractor. The chaos only appears initially; it eventually dies down.

Attractors and repellors are some of the most important things governing how a chaotic system will move in time. Attractors govern the long time behavior. If whatever you are studying has an attractor, you can see it just by waiting long enough. Some attractors are simple: attracting equilibrium points and attracting periodic orbits. Strange attractors, which are associated with chaotic motion, are more complicated. They are fractals. Repellors also can be simple equilibrium points / periodic orbits or complicated strange repellors. They govern the short-time behavior of any motion which begins close to them. They can also form the boundaries between basins of attraction if the system has multiple attractors.

Next post.