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.
Read more about this topic: Butterfly Effect
Famous quotes containing the words theory, mathematical and/or definition:
“The theory of the Communists may be summed up in the single sentence: Abolition of private property.”
—Karl Marx (1818–1883)
“What is history? Its beginning is that of the centuries of systematic work devoted to the solution of the enigma of death, so that death itself may eventually be overcome. That is why people write symphonies, and why they discover mathematical infinity and electromagnetic waves.”
—Boris Pasternak (1890–1960)
“Perhaps the best definition of progress would be the continuing efforts of men and women to narrow the gap between the convenience of the powers that be and the unwritten charter.”
—Nadine Gordimer (b. 1923)