Chaos Theory & The Beautiful Geometry of Double Pendulum

Chaos theory is a field of study in mathematics, with applications in several disciplines including meteorology, sociology, physics, engineering,etc…. In a scientific context, the word chaos has a slightly different meaning than it does in its general usage as a state of confusion, lacking any order. Chaos, with reference to chaos theory, refers to an apparent lack of order in a system that nevertheless obeys particular laws or rules.

Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions. These systems are deterministic, meaning that their future behavior is fully determined by their initial conditions, with no random elements involved. In other words, the deterministic nature of these systems does not make them predictable. This behavior is known as deterministic chaos, or simply chaos. The theory was summarized by Edward Lorenz as follows: “Chaos: When the present determines the future, but the approximate present does not approximately determine the future.” (pg 68).

Chaotic behavior can be observed in many natural systems, such as weather and climate. This behavior can be studied through analysis of a chaotic mathematical model, or through analytical techniques such as recurrence plots and Poincaré maps.

A double rod pendulum animation showing chaotic behavior. Starting the pendulum from a slightly different initial condition would result in a completely different trajectory. The double rod pendulum is one of the simplest dynamical systems that has chaotic solutions.