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.