Lyapunov Exponents: A Tool to Explore Complex Dynamics

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).