8.4.1 The Lorenz equations

The Lorenz system of equations, first proposed by Edward Lorenz in 1963 [1], are a set of coupled first-order differential equations relating three variables $x(t)$, $y(t)$ and $z(t)$:

$$\dot{x}=\sigma (y-x) \tag{1}$$

$$\dot{y}=x(\rho-z)-y \tag{2}$$

$$\dot{z}=xy-\beta z \tag{3}$$

where $\sigma$, $\rho$ and $\beta$ are constants.

In Lorenz’s original model, $x$ is proportional to the rate of convection in the atmosphere when heated from below by a vertical temperature gradient proportional to $z$, with horizontal temperature variation characterised by $y$. This type of convection is known as “Rayleigh-Bénard convection”: the same Rayleigh we met earlier in “Rayleigh’s principle”.

The equations appear simple: linear in each separate variable, and including nothing more extreme than the products $xy$ and $xz$. For some ranges of values of the constants $\sigma$, $\rho$ and $\beta$, the predicted behaviour is indeed quite simple. However, for certain ranges of values complicated chaotic behaviour is found. For all the cases plotted in section 8.4, the values $\sigma =10$, $\beta=8/3$ and $\rho=28$ are used: these are the original set of values used by Lorenz.

[1] Lorenz, Edward Norton: “Deterministic nonperiodic flow”, Journal of the Atmospheric Sciences, 20, 130–141 (1963).