Chaos theory is a branch of mathematics which studies the behavior of certain dynamical systems that may be highly sensitive to initial conditions. This sensitivity is popularly referred to as the butterfly effect. As a result of this sensitivity, which manifests itself as an exponential growth of error, the behavior of chaotic systems appears to be random.

That is, tiny differences in the starting state of the system can lead to enormous differences in the final state of the system even over fairly small timescales. This gives the impression that the system is behaving randomly. This happens even though these systems are deterministic, meaning that their future dynamics are fully determined by their initial conditions with no random elements involved.

This behavior is known as deterministic chaos, or simply chaos. Chaotic behavior is also observed in natural systems, such as weather. This may be explained by analysis of a chaotic mathematical model which represents such a system.

Quantum chaos investigates the relationship between chaos and quantum mechanics. Chaotic behavior has been observed in the laboratory in a variety of systems including electrical circuits, lasers, oscillating chemical reactions, fluid dynamics, and mechanical and magneto-mechanical devices. Observations of chaotic behavior in nature include the dynamics of satellites in the solar system, the time evolution of the magnetic field of celestial bodies, population growth in ecology, the dynamics of the action potentials in neurons, and molecular vibrations.

Everyday examples of chaotic systems include weather and climate. There is some controversy over the existence of chaotic dynamics in plate tectonics and in economics. Systems that exhibit mathematical chaos are deterministic and thus orderly in some sense; this technical use of the word chaos is at odds with common parlance, which suggests complete disorder.

However, even though they are deterministic, chaotic systems show a strong kind of unpredictability not shown by other deterministic systems.

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