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 focused on systems that change in discrete time.
To describe how something changes in time, we apply some function over and over again. Applying the function moves you one step forward in time. It doesn’t make sense to halfway apply the function, so we cannot use this to determine the motion of the object between time steps.
In a few cases, it is clear how to interpolate between time steps. For example, the pinball moved along a straight line with constant velocity between bounces. Most of the time, it is not clear how to interpolate.
Part V is about how to describe something that moves chaotically in continuous time.
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, but I might be pushing the limits[1]Pun intended. again. I do refer to something known as a ‘differential equation’, but I don’t assume that you know what it is and don’t try to explain it beyond an intuitive level. 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.
An Example: The Rössler Attractor
The tool we use to describe something that moves in continuous time is called a differential equation.
A differential equation is a relationship between something’s location and its velocity (speed and direction of motion). The object’s velocity depends only on its current position. Given the object’s current position, the differential equation tells you how fast that object is moving in each direction.
The simplest differential equation which exhibits chaos is the Rössler system: $$ \begin{array}{rcl} v_x &=& – y – z \\ v_y &=& x + 0.2 \ y \\ v_z &=& 0.2 + z \ (x – 5.7) \end{array} $$ The position of the object in three dimensions is given by $(x,y,z)$. The velocity of the object in three dimensions, i.e. how fast it is going in each direction, is given by $(v_x, v_y, v_z)$.
A solution to a differential equation is a description of how an object moves in time, $\big(x(t), y(t), z(t)\big)$, from some starting location.
For most differential equations, including this one, it is impossible to write down explicitly how the object will move in time. One alternative option is to use a computer to find a close approximation of the solution.
To get a feel for the long time behavior of the Rössler system, we can plot the attractor. To find the attractor, use a similar technique as before. Choose an initial condition. Move the system forward in time for a while without recording anything so you can get rid of any transient behavior. Plot the path that the object follows for a long time. This path shows the shape of the attractor.
Any initial condition will eventually get close to this attractor. Once it is close to the attractor, it will wander around chaotically on the attractor.
The Connection to Discrete Time
The attractor looks somewhat like a warped circle, centered at the vertical axis. As the object traverses the attractor, its motion is mostly going counterclockwise around the circle. The motion around the circle isn’t very complicated – it isn’t where the chaos is happening. Instead, you can see the chaos by looking at how far away the object is from the vertical axis.
It would be nice if we didn’t have to keep track of the simple part of the motion. The circular motion doesn’t reveal any of the essential features of the Rössler attractor’s chaos. Instead, it just makes all of the calculations more tedious. There is a technique, called a Poincaré section, that can be used to separate the tedious simpler aspects from the more interesting chaotic motion. It also illustrates the connection between a continuous time system and the discrete time systems we have studied previously.
Consider the plane which starts at the vertical axis and extends in the positive $x$ direction.[2]This is really only half of a plane, but it is simpler to refer to it as a plane. The plane could be described as the set of all points in three dimensions with $y = 0$ and $x > 0$. This plane will be called the Poincaré section.
Every time the object goes around the circle, it crosses through this plane. We can mark the location where it crosses the plane by a dot on the plane. Instead of recording the entire orbit, we could instead just record the dots where the orbit crosses this plane. The result is a sequence of dots in the plane.
These dots can be thought of as a dynamical system in two dimensions with discrete time. The dot’s location at time $= 1$ is the location where the object crossed the Poincaré section for the first time. The dot’s location at time $= 2$ is the location where the object crossed the Poincaré section the second time, etc. Although you need to calculate the motion in continuous time to figure out where the object will cross the Poincaré section next, once you’ve found the next point, you can forget about the continuous time motion and just keep track of the dots which appear / move in discrete time.
We can consider the strange attractor for the discrete time system in addition to the original strange attractor for the continuous time Rössler system.
This Poincaré section was defined using the plane $y = 0$ and $x > 0$. The choice of this particular plane was arbitrary. Since the purpose of the plane is to separate the simpler circular motion from the chaotic motion, any plane which cuts across the disk can be used to make a Poincaré section. In particular, I could take any plane which starts at the vertical axis and extends in any direction. Eight Poincaré sections formed in this way are shown in Figure 3. We can look at these Poincaré sections to give us a better understanding of how an object which obeys the Rössler system moves.
Poincaré sections are a powerful tool for unraveling the long time motion of a chaotic system. They separate out simpler, often circular, motion from the chaotic motion that you are hoping to understand. By choosing multiple Poincaré sections along the path, you can get ideas of what happens to the attractor as it moves. Most importantly, Poincaré sections allow us to extend everything that we’ve learned about discrete time chaotic systems to continuous time chaotic systems. It is often easier to work with and understand motion in discrete time. The results can then be lifted to motion in continuous time, which is more immediately applicable to almost all experiments.