Butterfly effect in the context of "Polar cells"

Atmospheric circulation is the large-scale movement of air and together with ocean circulation is the means by which thermal energy is redistributed on the surface of Earth. Earth's atmospheric circulation varies from year to year, but the large-scale structure of its circulation remains fairly constant. The smaller-scale weather systems – mid-latitude depressions, or tropical convective cells – occur chaotically, and long-range weather predictions of those cannot be made beyond ten days in practice, or a month in theory (see chaos theory and the butterfly effect).

Earth's weather is a consequence of its illumination by the Sun and the laws of thermodynamics. The atmospheric circulation can be viewed as a heat engine driven by the Sun's energy and whose energy sink, ultimately, is the blackness of space. The work produced by that engine causes the motion of the masses of air, and in that process it redistributes the energy absorbed by Earth's surface near the tropics to the latitudes nearer the poles, and thence to space.

Chaos theory is an interdisciplinary area of scientific study and branch of mathematics. It focuses on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions. These were once thought to have completely random states of disorder and irregularities. Chaos theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnection, constant feedback loops, repetition, self-similarity, fractals and self-organization. The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state (meaning there is sensitive dependence on initial conditions). A metaphor for this behavior is that a butterfly flapping its wings in Brazil can cause or prevent a tornado in Texas.

Small differences in initial conditions, such as those due to errors in measurements or due to rounding errors in numerical computation, can yield widely diverging outcomes for such dynamical systems, rendering long-term prediction of their behavior impossible in general. This can happen even though these systems are deterministic, meaning that their future behavior follows a unique evolution and is fully determined by their initial conditions, with no random elements involved. In other words, despite the deterministic nature of these systems, this does not make them predictable. This behavior is known as deterministic chaos, or simply chaos. The theory was summarized by Edward Lorenz as:

The Lorenz system is a three-dimensional classical dynamic system represented by three ordinary differential equations . It was first developed by the meteorologist Edward Lorenz and describes chaotic behavior of fluid movement when subjected to heating.

Although the Lorenz system is deterministic, its dynamics depend on the choice of initial parameters. For some ranges of parameters, the system is predictable as trajectories settle into fixed points or simple periodic orbits. In contrast, for other parameter ranges, the system becomes chaotic and the solutions never settle down but instead trace out the butterfly-shaped Lorenz attractor, popularly known as butterfly effect. In this regime, small differences in initial conditions grows exponentially making long-term prediction practically impossible.

In physics and mathematics, in the area of dynamical systems, a double pendulum, also known as a chaotic pendulum, is a pendulum with another pendulum attached to its end, forming a complex physical system that exhibits rich dynamic behavior with a strong sensitivity to initial conditions. The motion of a double pendulum is governed by a pair of coupled ordinary differential equations and is chaotic.