Lorenz attractor

To study the possibly complicated behavior of three-dimensional systems, there is no better place to begin than with the famous model proposed by Lorenz in 1963. Before this model appeared, the only types of stable attractors known in differential equations were fixed points and closed trajectories. This model illustrates in particular the sensitive dependence on intial conditions, also known by the large public as the `butterfly effect’ (an expression coined by Lorenz himself).

The Lorenz system is given by the equations

$$ \begin{cases} \dot{x} = \sigma (y-x)\\ \dot{y}=\rho x-y -xz\\ \dot{z}=xy-\beta z \end{cases} $$

where $\sigma,\rho$ and $\beta$ are positive parameters. The `historical values’ (those used by Lorenz in his paper) are $\sigma=10$, $\beta=8/3$ et $\rho=28$.

Explaining how Lorenz got his equations would lead us away. We content ourselves with a few words. Lorenz was interested in setting up a simple model that would explain some of the unpredictable behavior of the weather.

Physical sensible models of atmospheric convection involve partial differential equations, and are extremely complicated to analyze. Lorenz sought a much simpler system. He considered a two-dimensional fluid cell that was heated from below and cooled from above. In Fourier modes, the fluid motion can be described by a system of differential equations involving infinitely many variables. Lorenz made a tremendous simplification by keeping only three variables !

Very roughly speaking, $x$ represents the rate of convective `overturning’, whereas $y$ and $z$ can be thought as the horizontal and vertical temperature, respectively. Notice that $x,y,z$ are thus not representing the position of a point in the ambient space, but instead an abstract three-dimensional phase space.

Regarding the three parameters, $\sigma$ is the Prandtl number (related to the fluid viscosity), $r$ is the Rayleigh number (related to the temperature difference between the top and bottom of the cell, and $b$ is a scaling factor (related to the aspect ratio of the rolls).

In the following digital experiment, you can move the orange bullet which represent the initial condition and then press the `start’ button. You can observe that solutions that start out very differently seem to have the same fate, if we forget the `transient behavior’. They both eventually wind around the symmetric pair of fixed points, alternating at times which point they encircle. This forms a complicated set, the so-called Lorenz attractor, on which solutions stay for ever.

The previous experiment can be misleading because it can leave the impression that if you start with two very close initial conditions, the resulting solutions travel very close to each other, before they get on the attractor but also once they are on it. The following experiment shows that this is false ! This time you can see how to initially close points evolve on the attractor.

You can observe that the two solutions move quite far apart during their journey around the attractor. Moreover, you can see that the trajectories are nearly identical for a certain time period, but then they differ significantly as one solution winds around one of the symmetric fixed points, while the other solution winds around the other one. No matter how close two solutions start, they always move apart in this manner when they are on the attractor. This is sensitive dependence on initial conditions, one of the main features of a chaotic system.

Moreover, we observe that solutions pass from one `lobe’ of the attractor to the other in an apparently unpredictable manner, leading to an irregular oscillation that never repeats : we have an aperiodic motion. This is called deterministic chaos because the equations are deterministic but the solutions can behave in a seemingly random way. Recall that `deterministic’ means, given the present state, the future (and the past) are completely determined. Mathematically, this means that, given an initial condition $(x_0,y_0,z_0)$, there is a unique solution $(x(t),y(t),z(t))$ passing through $(x_0,y_0,z_0)$ at time $t=0$. But, to predict the future evolution, we need to know exactly the initial condition, which is impossible in practice. In a chaotic system, this leads to unpredictability. To illustrate this, consider a tiny blob if initial conditions around $(x_0,y_0,z_0)$. We observe that, rapidly, this blob smears out over the entire attractor ! This means that a tiny error on the initial condition is amplified quickly in such a way that we only know that we are on the attractor, but we don’t know precisely where.