von neumann

Chaos Theory

4 min read

Overview

Chaos theory is a field within physics and mathematics that investigates phenomena in areas such as weather, traffic, and population dynamics. It describes the boundary of determinism: in a system with completely uniform conditions, even the slightest deviations in the initial conditions can lead to completely different outcomes. The result is unpredictable, which is why it is also referred to as deterministic chaos.

Take the example of a billiards game: even if the balls are arranged seemingly exactly the same way at the start of two games, there are minimal variations in their starting positions. The tiniest gaps between the balls can lead, even with an apparently identical strike, to completely different outcomes.

This is also known as the butterfly effect: the flap of a butterfly’s wings can trigger a tornado. Here, the flap represents the initial condition—a pressure fluctuation in a describable weather model—that leads to an unpredictable result.

Yet, chaos is not random. Chaos has an internal structure. By comparing many sets of data, patterns can emerge. For example, when measuring a person’s heartbeat, deviations from a typical pattern in the naturally non-metronomic rhythm can indicate whether the heart is healthy or not.

Content

Dynamical Systems and Attractors

A dynamical system is a mathematical model that represents a change of state as a function of time. For example, when you drop a ball from a building, you can calculate the position of its trajectory as it falls over time (Galileo Galilei). In the same way, you can calculate the flow behavior of gases and liquids, as well as the movements of celestial bodies.

In dynamical systems, one is often interested in how the system evolves as time approaches infinity. This is referred to as a limit or an attractor. There are different types of attractors.

One of the simplest examples is the damped pendulum. Over time, it always converges to its resting point, which is the lowest point in its swing or trajectory. This point is its attractor; in this case, it is a stable fixed point.

There are also periodic/cyclic attractors. An example of this is the Lotka-Volterra model (predator-prey model). This model oscillates periodically between a peak and a trough. Sometimes there are many predators, causing the prey to disappear. When there is little prey, the predators vanish, allowing the prey to recover, which in turn leads to more predators. A cyclic equilibrium between the two species emerges, with the development being predictable. These are known as stable limit cycles.

Another example of attractors is the chaotic or strange attractors. These are characterized by an evolution over time that is unpredictable and nonlinear. The trajectories of these attractors appear to converge toward a fixed point. However, just before reaching these fixed points, the trajectories change course in a way that is not predictable and, therefore, not periodic. Nevertheless, the trajectories always form a similar pattern—they form fractals. These trajectories are highly sensitive to their initial conditions. If the initial conditions are the same, they will always follow the same path. If the initial conditions are even slightly different, one cannot predict the course; one only knows that they will form a fractal.

Attractors

Lorenz Attractor

dx/dt = sigma(y - x) dy/dt = x(rho - z) - y dz/dt = xy-beta*z

// numerical integration via Euler method
x = x + time * (sigma * (y - x));
y = y + time * (x * (rho - z) - y);
z = z + time * ((x * y) - (beta * z));

Thomas Attractor

dx/dt = sin(y)−bx dy/dt = sin(z)−by dz/dt = sin(x)−bz

// numerical integration via Euler method
x = x + time * ((-b * x) + sin(y));
y = y + time * ((-b * y) + sin(z));
z = z + time * ((-b * z) + sin(x));

Graph View