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Lorenz Equations: Sensitive Dependence on Initial Conditions

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The essence of chaotic dynamics lies in its sensitive dependence on initial conditions, which leads to unpredictable behavior of the system. In this Demonstration, 10 balls are placed at coordinates at , with small differences . As time progresses, the system begins to exhibit chaotic behavior.

Contributed by: Shinnosuke Wada and Reiho Sakamoto (August 25)
Open content licensed under CC BY-NC-SA

Details

In this Demonstration, we consider the celebrated Lorenz equations:

.

Here, , and .

At , 10 balls are placed at .

Let be the time after which the system begins to exhibit chaotic behavior. Although it is not precisely defined, the following values are observed with respect to :

Δyt00.01'19.8'0.001'21.5'0.0001'24.4'0.00001'26.1'ColumnsRowsColumnsRowsGrid

If is considered as noise in numerical computation, represents the duration within which the numerical solution remains reliable. Attempting to improve the computation by decreasing does not significantly increase . This observation explains the immense difficulty of performing numerical computations for chaotic equations.

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