Chaos theory in the context of "Nonlinear system"

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.

A pattern is a regularity in the world, in human-made design, or in abstract ideas. As such, the elements of a pattern repeat in a predictable and logical manner. There exists countless kinds of unclassified patterns, present in everyday nature, fashion, many artistic areas, as well as a connection with mathematics. A geometric pattern is a type of pattern formed of repeating geometric shapes and typically repeated like a wallpaper design.

Any of the senses may directly observe patterns. Conversely, abstract patterns in science, mathematics, or language may be observable only by analysis. Direct observation in practice means seeing visual patterns, which are widespread in nature and in art. Visual patterns in nature are often chaotic, rarely exactly repeating, and often involve fractals. Natural patterns include spirals, meanders, waves, foams, tilings, cracks, and those created by symmetries of rotation and reflection. Patterns have an underlying mathematical structure; indeed, mathematics can be seen as the search for regularities, and the output of any function is a mathematical pattern. Similarly in the sciences, theories explain and predict regularities in the world.

Self-organization, also called spontaneous order in the social sciences, is a process where some form of overall order arises from local interactions between parts of an initially disordered system. The process can be spontaneous when sufficient energy is available, not needing control by any external agent. It is often triggered by seemingly random fluctuations, amplified by positive feedback. The resulting organization is wholly decentralized, distributed over all the components of the system. As such, the organization is typically robust and able to survive or self-repair substantial perturbation. Chaos theory discusses self-organization in terms of islands of predictability in a sea of chaotic unpredictability.

Self-organization occurs in many physical, chemical, biological, robotic, and cognitive systems. Examples of self-organization include crystallization, thermal convection of fluids, chemical oscillation, animal swarming, neural circuits, and black markets.

Negative feedback (or balancing feedback) occurs when some function of the output of a system, process, or mechanism is fed back in a manner that tends to reduce the fluctuations in the output, whether caused by changes in the input or by other disturbances.

Whereas positive feedback tends to instability via exponential growth, oscillation or chaotic behavior, negative feedback generally promotes stability. Negative feedback tends to promote a settling to equilibrium, and reduces the effects of perturbations. Negative feedback loops in which just the right amount of correction is applied with optimum timing, can be very stable, accurate, and responsive.

In physics, specifically classical mechanics, the three-body problem is to take the initial positions and velocities (or momenta) of three point masses orbiting each other in space and then to calculate their subsequent trajectories using Newton's laws of motion and Newton's law of universal gravitation.

Unlike the two-body problem, the three-body problem has no general closed-form solution, meaning there is no explicit formula for the positions of the bodies. When three bodies orbit each other, the resulting dynamical system is chaotic for most initial conditions, and the only way to predict the motions of the bodies is to estimate them using numerical methods.

The balance of nature, also known as ecological balance, is a theory that proposes that ecological systems are usually in a stable equilibrium or homeostasis, which is to say that a small change (the size of a particular population, for example) will be corrected by some negative feedback that will bring the parameter back to its original "point of balance" with the rest of the system. The balance is sometimes depicted as easily disturbed and delicate, while other times it is inversely portrayed as powerful enough to correct any imbalances by itself. The concept has been described as "normative", as well as teleological, as it makes a claim about how nature should be: nature is balanced because "it is supposed to be balanced". The theory has been employed to describe how populations depend on each other, for example in predator-prey systems, or relationships between herbivores and their food source. It is also sometimes applied to the relationship between the Earth's ecosystem, the composition of the atmosphere, and weather.

The theory has been discredited by scientists working in ecology, as it has been found that constant disturbances leading to chaotic and dynamic changes are the norm in nature. During the later half of the 20th century, it was superseded by catastrophe theory, chaos theory, and thermodynamics. Nevertheless, the idea maintains popularity amongst conservationists, environmentalists and the general public.

In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to laminar flow, which occurs when a fluid flows in parallel layers with no disruption between those layers.

Turbulence is commonly observed in everyday phenomena such as surf, fast flowing rivers, billowing storm clouds, or smoke from a chimney, and most fluid flows occurring in nature or created in engineering applications are turbulent. Turbulence is caused by excessive kinetic energy in parts of a fluid flow, which overcomes the damping effect of the fluid's viscosity. For this reason, turbulence is commonly realized in low viscosity fluids. In general terms, in turbulent flow, unsteady vortices appear of many sizes which interact with each other, consequently drag due to friction effects increases.