Higher-Dimensional Models
Philosophy is written in this grand book, the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the letters in which it is composed. It is written in the language of mathematics.Galileo Galilei, on the Book of Nature waiting to be decoded

One of the earliest indications of chaotic behavior was in an atmospheric model developed by Edward Norton Lorenz. His goal was to describe the motion of a fluid layer that is heated from below. The result is convection, where warm fluid at the bottom rises, then cools at the top, and finally descends as a result. Lorenz simplified the description by restricting the motion to two spatial dimensions. This situation has been observed in the laboratory and is known as a Rayleigh-Bénard cell.

In the equations below, the dimensionless parameters $\sigma$, $\rho$, and $\beta$ are determined by various fluid properties, the size of the Rayleigh-Bénard cell, as well as the temperature difference in the cell. (In the program, $\rho$ = r and $\beta$ = b). Note that the variables $x$, $y$, and $z$ have nothing to do with actual spatial coordinates, but are instead simply convenient measures of the state of the system. This exercise is meant to reinforce the idea that even simple sets of equations can exhibit chaotic behavior.

## Rayleigh-Bénard Convection

This program obtains a numerical solution to the Lorenz equations via a common fourth-order Runge-Kutta method.
$\displaystyle{ \frac{\mathrm{d} x}{\mathrm{d} t} = \sigma (y - x) }$ $\displaystyle{ \frac{\mathrm{d} y}{\mathrm{d} t} = x (\rho - z) - y }$ $\displaystyle{ \frac{\mathrm{d} z}{\mathrm{d} t} = x y - \beta z }$
 Parameterized fluid velocity $x$ circulating around the cell: Parameterized temperature difference $y$ between the rising and falling fluid regions: Parameterized difference in the temperature profile $z$ between the top and bottom of the cell: Dimensionless parameter $\sigma$ of the system (determined by various fluid properties): Dimensionless parameter $\rho$ of the system (determined by the size of the Rayleigh-Bénard cell): Dimensionless parameter $\beta$ of the system (determined by the temperature difference in the cell): Time-step $\mathrm{d} t$ of the system: sec0.001 ≤ $\mathrm{d} t$ ≤ 0.1 sec
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