With the advent of electronic computers, numerical simulations of dynamical models have become an increasingly appreciated way to study complex and nonlinear systems. This has been accompanied by an evolution of theoretical tools and concepts: some of them, more suitable for a pure mathematical analysis, happened to be less practical for applications; other techniques proved instead very powerful in numerical studies, and their popularity exploded. Lyapunov exponents is a perfect example of a tool that has flourished in the modern computer era, despite having been introduced at the end of the nineteenth century.

The rigorous proof of the existence of well-defined Lyapunov exponents requires subtle assumptions that are often impossible to verify in realistic contexts (analogously to other properties, e.g., ergodicity). On the other hand, the numerical evaluation of the Lyapunov exponents happens to be a relatively simple task; therefore they are widely used in many setups. Moreover, on the basis of the Lyapunov exponent analysis, one can develop novel approaches to explore concepts such as hyperbolicity that previously appeared to be of purely mathematical nature.

In this book we attempt to give a panoramic view of the world of Lyapunov exponents, from their very definition and numerical methods to the details of applications to various complex systems and phenomena. We adopt a pragmatic, physical point of view, avoiding the fine mathematical details. Readers interested in more formal mathematical aspects are encouraged to consult publications such as the recent books by Barreira and Pesin (2007) and Viana (2014).

An important goal for us was to assess the reliability of numerical estimates and to enable a proper interpretation of the results. In particular, it is not advisable to underestimate the numerical difficulties and thereby use the various subroutines as black boxes; it is important to be aware of the existing limits, especially in the application to complex systems.

Although there are very few cases where the Lyapunov exponents can be exactly determined, methods to derive analytic approximate expressions are always welcome, as they help to predict the degree of stability, without the need of actually performing possibly long simulations. That is why, throughout the book, we discuss analytic approaches as well as heuristic methods based more on direct numerical evidence, rather than on rigorous theoretical arguments.