An Introduction to Chaos

Chaos theory, perhaps more correctly if less interestingly called dynamical-systems theory, deals with systems of equations (that might correspond to real world systems). These systems are usually in the form of differential equations or iterative maps and the ones we are interesting in are,

• Apparently random
• Deterministic
• Sensitive to initial conditions

Or more pithily,

Chaos: When the present determines the future, but the approximate present does not approximately determine the future.
Edward Lorenz¹

In this article, we will look at a few classic examples of chaotic systems. Later, we will try to understand them more deeply and explain their relevance to astrophysics.

Newton Raphson

The Newton-Raphson method² is a numerical algorithm to find the roots (the value for $x$ ) of an equation of the form $f(x) = 0$ . It is iterative and so requires an initial guess of the root ($x_0$ ). Subsequent estimates are computed using,

$$x_{n+1} = x_{n} - \frac{f(x_n)}{f'(x_n)}$$

and we continue until $f(x_n)$ is close enough to $0$ , $|f(x_n)| < \epsilon$ .

How does this work? This iteration fits a line around the local value ($x_n$ ) of the function. We then move a distance that solves the equation, assuming that linear fit is correct. If $f(x)$ is well approximated by a line, we immediately find the solution. For example,

$$\begin{align} f(x) &= 4 + 2x \quad \text{with} \quad x_0 = 5 \\ x_1 &= 5 - \frac{14}{2} = -2 \end{align}$$

If $f(x_n)$ is not well approximated by a line we approach the solution more slowly,

$$\begin{align} f(x) &= x^2 \quad \text{with} \quad x_0 = 5 \\ x_1 &= 5 - \frac{25}{10} = 2.5 \\ x_2 &= 2.5 - \frac{25}{20} = 1.25 \\ & \ldots{} \\ x_n &= 0 \end{align}$$

As this function flattens out as it approaches the root, our linear approximation does not move far enough. As the guess gets close to the true root, the linear approximation improves, and we eventually find the solution.

This method does not always work. If we run into a point where the derivative is $0$ we move to $\infty$ . We can also oscillate around the root without drawing near to it. However, what we are interested in here is the behavior of this method when there are multiple roots. Consider $x^2 = 4$ which has two solutions, $x = \pm 2$ . Which solution is found for each $x_0$ ? The general rule is that the Newton-Raphson method will move to the closest solution. However, for some functions the behavior around the equidistant points is interesting. We can see small changes in initial conditions resulting in large, hard to understand changes in the result. The method is chaotic in these regions.

Below we show the results of $f(x) = x^3 - 1$ . This has only one real solution, $x = 1$ , but two complex solutions, $x = -0.5 \pm \frac{\sqrt{3}}{2} i$ . We plot this in the complex plane where the color shows which solution is found for that $x_0$ . The shade shows the number of iterations required to reach a solution (light $\rightarrow$ few, dark $\rightarrow$ many). Click on a point to see the steps taken to the solution and scroll to zoom.

Lorenz System

Perhaps the most well known chaos theorist was Edward Lorenz (not to be confused with Hendrik Lorentz of the relativistic transforms). Lorenz was a mathematician and meteorologist who worked on modeling and forcasting weather. We will show the main result from his most well known paper, Deterministic Nonperiodic Flows (Lorenz 1963).

This paper studied the following systems of differential equations.

$$\begin{align} \frac{dx}{dt} &= \sigma(y - x) \\ \frac{dy}{dt} &= x(\rho - z) - y \\ \frac{dz}{dt} &= xy - \beta z \end{align}$$

where $x$ , $y$ , and $z$ vary with time and describe temperature variations and heat flow in the system, and $\sigma$ , $\rho$ , and $\beta$ are constants describing physical properties of the system. Lorenz, quite reasonably, thought that two systems that started in similar states would remain similar over time. However, even small changes in initial conditions (for some values of $\sigma$ , $\rho$ , and $\beta$ ) resulted in large changes after some time.

I’m leaving a lot of detail out here, both about what exactly these variables are and how they relate to fluid flow. I’m doing this because they are quite complicated, I don’t fully understand them, and all we really want to show here is that a simple set of equations can lead to complicated, chaotic results.

Below we evaluate these equations up to $t = 30$ for varying values of the constants. Click on the image to peturb the initial conditions slightly and see how this can significantly change the final state (the black dot) of the system.

References