DOUBLE PENDULUM AND BUTTERFLY EFECT

Place the two double pendulums in very similar positions and release them. Note that in some cases they oscillate almost together for a long time, and in others their trajectories diverge almost immediately.

This exhibit summarizes the main characteristic of chaotic systems: sensitivity to initial conditions. Even starting from almost identical starting positions, the small differences are magnified and, in a very short time, each one is moving in a completely different way from the other.

This type of phenomenon became popularly known as the butterfly effect, due to the analogy made with the difference caused in weather conditions that the flapping of a simple butterfly's wings can cause.

This figure shows the famous Lorenz attractor, which, coincidentally, looks like a butterfly. Edward Lorenz discovered, in 1962, that even very simplified meteorological systems would have the property of sensitivity to initial conditions, making it impossible to rapidly increase our ability to predict the weather.

It's not that the butterfly is capable of causing a climate catastrophe. The point is that our inability to pinpoint tiny factors in our atmosphere prevents us from being able to predict weather conditions after a certain number of days.

Chaotic attractors resulting from the iteration of functions from the plane to the plane.

USEFUL LINKS

This Wikipedia page brings an overview of what simple pendulums are, giving their physical definition and the equation that describes their movement. In addition, it brings some information about the so-called physical pendulum, which is ideal, and some of the equations that can be associated with its study.

This Wikipedia page gives some information about the double pendulum, like the one presented in this exhibit. It brings a more formal analysis and interpretation of its movement, in addition to specifying some details about the Lagrangian double pendulum and providing an introduction to the study of chaotic movement.

And who said that mathematical apparatuses don't do art exhibitions? This Exploratorium (a science museum in San Francisco, California) page displays a double pendulum present in the exhibit's collection. It also brings some information about what happens along the movement of the pendulums.

This Dicas do Digão's channel video initially shows two fixed rods that oscillate as if they were one - a simple pendulum -, then the two rods start to oscillate independently, forming a double pendulum and making the difference between the movements of both very clear and the idea of chaotic movement.

This Think Twice's channel video brings a computer simulation of the double pendulum, showing how sensitive this type of system is to initial conditions. Through the simulations, the difference between a periodic behavior (simple) and a non-periodic behavior (the double pendulum) is shown.

This Jelther Gonçalves's article talks about double pendulums and impelled pendulums, bringing, after a long study and formal analysis of the equations that involve their movements, a computer simulation. The article makes some deductions from some known methods, such as Runge-Kutta.                                                                                                                    

Finally, we saw that the study of the movement of the double pendulum is very connected to the idea of chaotic systems. In this Fiocruz's article, we have a general notion of what chaos is, from a scientific point of view. This Wikipedia page complements the study by talking about Chaos Theory, a very broad field that addresses studies in several areas.