The starting hypothesis postulated that it was possible to integrate random variations into a determinist model and bring to light certain regularities within it. The first equations used were relatively simple. They concerned linear functions, accepting the existence of a proportional link, a cause and effect relation between several points situated at equal distance over time. ‘These equations allow regular and exponential notions, such as the theory of compound interest, to be modelled,’ explains the researcher. ‘But they do not allow other much more unpredictable financial realties to be envisaged, such as the evolution of a market index or the formation of a product’s price.’

To take up a citation by Ian Stewart, quoted by Professor Bair, ‘the science of today shows us that nature is ruthlessly non linear.’ The idea which followed was to start from a non linear discrete dynamic model, in other words a model presented (in the most simple cases) in the form of Y_t+1 = f(Y_t). According to this equality one can calculate the value of Y_t+1 at time t+1 on the basis of this Y_t at a previous time t, thanks to the f function being considered, which is not linear but makes a parameter intervene. Such a model enables a notion of chaos to be embraced within a determinist system, and thus allows predictable and unpredictable magnitudes to be modelled.

Chaos can be characterised thanks to properties of which the main ones are the density, which defines the ensemble of the variations of the function used, the sensitiveness to the initial conditions, or the butterfly effect, which allows us to understand that a slight modification of the initial condition can generate considerable changes, and the order. This latter notion enables us to see that ‘ordered structures can appear within fluctuations which seem random.’

This equation has enabled several phenomena to be modelled, such as the evolution of a market index or the establishment of prices. In the latter case, a simply linear equation allowed a monopoly market to be brought to light, in which a business company can raise prices at its leisure. By integrating an extra parameter the reality becomes more unpredictable. This parameter, competition, given in the example offered by the article, has a weakening effect, a negative feedback, which thus introduces a non-linearity, and regulates the evolution of a product’s price. ‘We noticed that, depending on the variation of this new parameter, the predictions were either purely and simply chaotic, or that a regularity was established within the chaos.’