Butterfly Effect - Theory and Mathematical Definition

Theory and Mathematical Definition

Recurrence, the approximate return of a system towards its initial conditions, together with sensitive dependence on initial conditions, are the two main ingredients for chaotic motion. They have the practical consequence of making complex systems, such as the weather, difficult to predict past a certain time range (approximately a week in the case of weather) since it is impossible to measure the starting atmospheric conditions completely accurately.

A dynamical system displays sensitive dependence on initial conditions if points arbitrarily close together separate over time at an exponential rate. The definition is not topological, but essentially metrical.

If M is the state space for the map, then displays sensitive dependence to initial conditions if for any x in M and any δ > 0, there are y in M, with such that

The definition does not require that all points from a neighborhood separate from the base point x, but it requires one positive Lyapunov exponent.