fact the analogy can be taken one step further by comparing the spring and dashpot to electric components. If one equates the mechanical stress (pressure in the material) with electric pressure (i.e. voltage) and the time derivative of the strain with the flow of electrons (i.e. the electric current), then the spring corresponds to a capacitor that stores energy, while the dashpot corresponds to a resistor that transforms electrical energy to heat. With the aid of analogy we have gone from an extremely complex living cell to a homogeneous viscoelastic material, further on to a construction of springs and dashpots and lastly to an electric circuit.
82 Worked Examples T1
T2
Fig. 11 Schematic image of Rayleigh–Bénard convection. When a thin layer of liquid is contained between two surfaces with a temperature difference ofT =T1−T2convection cells might appear depending on the magnitude ofT
The concept was popularised by the meteorologist Edward Lorenz, who in the 1960s was the first to mathematically describe a hydrodynamic system that exhibits chaos. Based on his observations he also coined the phrase “the butterfly effect”, which refers to the sensitivity to changes in initial conditions, and has its origin in the rhetorical question “Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?”13
Lorenz studied so called Rayleigh–Bénard convection that takes place when a thin layer of liquid is heated from below and cooled from above (see Fig.11). If the difference in temperatureT =T1−T2between the top and bottom is small, then the fluid is stationary and the temperature of the fluid forms a smooth gradient from top to bottom. IfT is increased, so called convection cells appear, in which the fluid alternately rises and sinks in a circular fashion. This well-ordered structure is however destroyed ifT is increased further, and replaced by a turbulent motion without any sort of regularity, and this was the behaviour that interested Lorenz.
This system, at least superficially, resembles the dynamics in the atmosphere, where air is being heated close to the surface of the Earth, rises, is cooled down and falls to the ground. And if the idealised model system exhibits turbulent and chaotic behaviour, why not the entire atmosphere? Lorenz approached this ques- tion by simplifying the Rayleigh–Bérnard system with the prospect that the simpler system would also exhibit deterministic chaos.14 Initially the convecting fluid was described by two coupled partial differential equations, one for the direction of flow (Footnote 12 continued)
great French mathematician Henri Poincaré partially solved the three-body problem in 1887 and in doing so claimed the winning prize in a competition organised by the Swedish king Oscar II.
In his solution Poincaré for the first time described the concept of deterministic chaos and laid the foundations for modern chaos theory. But the road to success was not straight. A mistake was found in Poincaré’s original submission and he had to use all of the prize money to publish a revised version of his article in the journalActa Mathematica, still one of the most prestigious journals in the mathematical sciences. For a detailed history of theN-body problem see for example F. Diacu (1996), The solution of theN-body problem. The Mathematical Intelligencer, 18(3).
13Lorenz, E. (1966). The Essence of Chaos. CRC Press.
14Lorenz, E. (1963). Deterministic non-periodic flow, Journal of the Atmospheric Sciences, 20, 130–141.
Fig. 12 A visualisation of the Lorenz attractor. The initial condition (X(0),Y(0),Z(0))= (0.1,0,0)is shown as a circle at the bottom of the figure
and one for the temperature. These equations are however not easy to handle, but by applying a clever transformation Lorenz was able to reduce the system to three coupled ordinary differential equations of the form:
⎧⎪
⎨
⎪⎩
d X
dt = −σX+σY
dY
dt =r X−X Z−Y
d Z
dt = X Y −b Z.
(10)
Here X is proportional to the intensity of the convective flow,Y is proportional to the difference in temperature between the liquid flowing up and down, and Z is proportional to the deviation from a linear flow. The constants σ,r andb only depend on the physical properties of the liquid, such as the viscosity and thermal conductivity.
This system of equations might seem highly abstract and far from providing a description of the processes taking place in the atmosphere of the Earth, but we have to keep in mind that Lorenz tried to prove that weather systems can exhibit chaotic dynamics. If a simplified toy model does so, then it is very likely that the real system, with its many more degrees of freedom, also does so. And indeed he was right, at least when it comes to the toy model. The state of the Lorenz model (10) at a timetis described by the triple(X(t),Y(t),Z(t)), and the dynamics of the system can there- fore be represented as an orbit through three-dimensional space, where the direction of motion at each point in space is determined by Eq. (10). Figure12shows such an orbit that was started with the initial condition(X(0),Y(0),Z(0))=(0.1,0,0)
84 Worked Examples and reveals the complicated structure of the dynamics. This structure is referred to as the Lorenz attractor since it attracts orbits from all possible initial conditions.
Because of its peculiar properties the attractor has gathered considerable attention from mathematicians. For example it has been proven that the Lorenz attractor is fractal.15The model also exhibits the desired sensitivity to initial conditions and has become a canonical example of chaotic systems.
We started out with weather and wind, and via convection ended up with the relatively simple Lorenz model that exhibits complicated and even fractal dynamics.
The model does not have any predictive power, but instead makes it possible to explain a specific property of the weather, its sometimes unpredictable behaviour.
Because of its properties the model is located at the outer edge of the predictive- explanatory spectrum of models: it cannot help us predict the weather, but instead gives insight into its basic dynamics.