<Dd> x n = sin 2 ⁡ (2 n θ π) (\ displaystyle x_ (n) = \ sin ^ (2) (2 ^ (n) \ theta \ pi)) </Dd> <P> where the initial condition parameter θ (\ displaystyle \ theta) is given by θ = 1 π sin − 1 ⁡ (x 0 1 / 2) (\ displaystyle \ theta = (\ tfrac (1) (\ pi)) \ sin ^ (- 1) (x_ (0) ^ (1 / 2))). For rational θ (\ displaystyle \ theta), after a finite number of iterations x n (\ displaystyle x_ (n)) maps into a periodic sequence . But almost all θ (\ displaystyle \ theta) are irrational, and, for irrational θ (\ displaystyle \ theta), x n (\ displaystyle x_ (n)) never repeats itself--it is non-periodic . This solution equation clearly demonstrates the two key features of chaos--stretching and folding: the factor 2 shows the exponential growth of stretching, which results in sensitive dependence on initial conditions (the butterfly effect), while the squared sine function keeps x n (\ displaystyle x_ (n)) folded within the range (0, 1). </P> <P> The butterfly effect is most familiar in terms of weather; it can easily be demonstrated in standard weather prediction models, for example . The climate scientists James Annan and William Connolley explain that chaos is important in the development of weather prediction methods; models are sensitive to initial conditions . They add the caveat: "Of course the existence of an unknown butterfly flapping its wings has no direct bearing on weather forecasts, since it will take far too long for such a small perturbation to grow to a significant size, and we have many more immediate uncertainties to worry about . So the direct impact of this phenomenon on weather prediction is often somewhat overstated ." </P> <P> The potential for sensitive dependence on initial conditions (the butterfly effect) has been studied in a number of cases in semiclassical and quantum physics including atoms in strong fields and the anisotropic Kepler problem . Some authors have argued that extreme (exponential) dependence on initial conditions is not expected in pure quantum treatments; however, the sensitive dependence on initial conditions demonstrated in classical motion is included in the semiclassical treatments developed by Martin Gutzwiller and Delos and co-workers . </P>

A set of elements linked together so that a change in one affect the other is known as