Lorenz Attractor






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The Lorenz attractor is generated by the differential equation written by

dx/dt=-10x+10y
dy/dt=28x-y-xz
dz/dt=-8/3z+xy.

This equation was first proposed by Edward N. Lorenz in his paper titled "Deterministic nonperiodic flow" published in Journal of the atmospheric sciences in 1963, and this system converges to an strange attractor with fractal properties.
This paper is thought to be the first one which treated the deterministic chaos and its mechanism.
Prior to Lorenz's paper, a Japanese engineer Yoshisuke Ueda had found the deterministic chaos in the modified Duffing equation, but its publication was late.

Moreover, it is thought that a mathematician Henri Poincaré (in the late 19th century) and a Balthasar van der Pol (1927) had encountered chaos. For further information, please see "Quotation about chaos" section.

Lorenz's work is important even today because he had an insight of the essence of chaos although no one did not know it in those day and his work settled the today's chaos theory.
He found a unimodal structure in the flow of the solution, and he understood that it was the source of randomness.

The Lorenz equation is derived from the Navier-Stokes equations with the Boussinesq approximation, which described the atmospheric convection. Although the Lorenz equation loses the correspondence to the actual atmosphere in the process of approximation, it is important that chaos appears from the equation which describes the dynamics of the nature.
Namely, chaos might exist ubiquitously around us, e.g., in the blow of wind or in the flow of water.

If the dynamics of atmosphere are chaotic, the long-term weather forecasting would not be reliable any more, although short-term one might be somewhat effective.

Such understanding of nature conflicted with the general knowledge of the physics in those days, namely, "we can predict the future of the system precisely if we know the initial states and the forces"; therefore, it gave a large impact on the society.

The famous phrase "a flap of a butterfly's wing might change the weather in New York (butterfly effect)", assumes that the dynamics of atmosphere are chaotic with the sensitive dependence on the initial condition. This phrase is thought to be based on the subtitle of Lorenz's lecture in 1972, "Does the flap of a butterfly's wings in Brazil wet off a tornado in Texas?".

Are the dynamics of atmosphere actually chaotic? To this question, only some conjectures can be given because the states of atmosphere are not stationary in space nor in time. Moreover, there would be many causes which change the atmospheric states such as stochastic noise. It is very difficult to analyze such a high-dimensional and stochastic system and to judge whether it is chaotic.
However, in the restricted area where the atmosphere is stationary, I think that it might be possible that the short-term behaviors of atmosphere are chaotic because we empirically know that the weather forecasting is not reliable.

Lorenz has become a professor emeritus of MIT in 1981, and he was invited to Dr. Steven Strogatz's lecture of nonlinear dynamics as a guest lecture every year. Lorenz died from cancer at his home in Cambridge on 16th April, 2008. As written in Wikipedia, he had finished a paper a week ago with a colleague.

In this page, you can observe the Lorenz attractor from various view angles.

The red, green, and blue lines denote the x, y, and z axes, respectively.
You can change the view with the three left bars, and you can change the zoom with the most right bar.

Reference



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Introduction to Chaos and Nonlinear Dynamics