1 - Introduction

Summary

Early results

The problem of determining the stability of a given regime (e.g. the motion of the solar system) is as old as the concept of the dynamical system itself. As soon as scientists realised that physical processes could be described in terms of mathematical equations, they also understood the importance of assessing the stability of various dynamical regimes. It is thus no surprise that many eminent scientists, such as Euler, Lagrange, Poincare and Lyapunov (to name a few), engaged themselves in properly defining the concept of stability. Lyapunov exponents are one of the major tools used to assess the (in)stability of a given regime. Within hard sciences, where there is a long-standing tradition of quantitative studies, Lyapunov exponents are naturally used in a large number of fields, such as astronomy, fluid dynamics, control theory, laser physics and chemical reactions. More recently, they started to be used also in disciplines, such as biology and sociology, where nowadays processes can be accurately monitored (e.g. the propagation of electric signals in neural cells and population dynamics).

The reader interested in a fairly accurate historical account of how stability has been progressively defined and quantified can refer to Leine (2010). Here, we limit ourselves to the recapitulation of a few basic facts, starting from the Galilean times, when E. Torricelli (1644) investigated the stability of a mechanical system and conjectured (in the modern language) that a point of minimal potential energy is a point of equilibrium.

Besides mechanical systems, floating bodies provide another environment where stability is naturally important, especially to avoid roll instability of vessels. Unsurprisingly, the first results came from a Flemish (S. Stevin) and a Dutch (Ch. Huygens) scientist: at that time, the cutting-edge technology of ship-building had been developed in the Dutch Republic. In particular, Huygens’ approach was quite modern in that he addressed the problem by explicitly comparing two different states. D. Bernoulli too dealt with the problem of roll-stability, emphasising the importance of the restoring forces, which make the body return towards the equilibrium state. L. Euler was the first to distinguish between stable, unstable, and indifferent equilibria and suggested also the possibility of considering infinitely small perturbations.