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 great tragedy of science—the slaying of a beautiful theory by an ugly fact.”
—Thomas Henry Huxley (1825–1895)
“As we speak of poetical beauty, so ought we to speak of mathematical beauty and medical beauty. But we do not do so; and that reason is that we know well what is the object of mathematics, and that it consists in proofs, and what is the object of medicine, and that it consists in healing. But we do not know in what grace consists, which is the object of poetry.”
—Blaise Pascal (1623–1662)
“The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
—Jean Baudrillard (b. 1929)