In March 1963, Edward Lorenz founded chaos theory with his paper “Deterministic Nonperiodic Flow” (Journal of the Atmospheric Sciences, 20, 130–141). In honour of the anniversary, Physics Today has a survey paper, “Chaos at fifty.”
Lorenz’s classic paper described the Lorenz Attractor. The image above (by “XaosBits”) shows two orbits of the attractor. Microscopically different starting points give totally different trajectories. The image below (by “Wikimol”) gives another view of the attractor.
Lorenz’s paper was followed by the discovery of the Rössler attractor in 1976, and then what can only be called an avalanche of further work. Prior to Lorenz, signs of chaos had been seen in the logistic map, but Lorenz showed that the world itself (weather, specifically) was chaotic. Chaos also shows up in orbital motion, in turbulence, and in mixing. The image below (by “Eclipse.sx”) shows the result of simulating a magnetic pendulum.
Lorenz went on to coin the term “butterfly effect.” This effect, more formally known as “sensitivity to initial conditions,” is one of the key marks of chaos. Another is “topological mixing,” which means that the system will evolve over time so that any given region of its phase space will eventually overlap with any other given region. The image below (public domain) shows a circular region (blue) progressively being mapped (purple, pink, red, orange, yellow, and further iterations not shown) to a set of points which leaves no blank areas in the phase space.
Lorenz died on 16 April, 2008, at the age of 90. Physics Today also published an obituary when Lorenz died.