Changes to one part of a complex system can lead to dramatically different changes in other parts.

A minor change to the state of an object or event (such as the flap of a butterfly’s wings or a slightly different starting position of a double pendulum’s arms) can eventually, through countless compounding effects, cause a dramatic change later on (like the metaphorial Texan tornado or a different double pendulum pattern).

The double-rod pendulum is one of the simplest dynamical systems to demonstrate nonlinearity. Compared to a linear system of one arm swinging from a fixed position, two swinging arms attached to each other produce behaviours that are close to impossible to predict.

Let’s Complicate Things

Chaos Theory

A system’s sensitive dependence on initial conditions is often referred to as the butterfly effect. The idea is that a butterfly flapping its wings in an Amazon rainforest can eventually, through countless compounding effects, cause a tornado in another country weeks later. This metaphor refers to the title of a talk called “Predictability: Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?” given in 1972 by Edward Lorenz, a meteorologist and the founder of modern chaos theory.


Causal determinism

Causal determinism (or physical determinism) is the philosophical view that any state (of an object or event) is completely determined by prior states.


In 1814, Pierre Simon Laplace published A Philosophical Essay on Probabilities where he articulated causal determinism:

“We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.”