1997, with Remo Badii*Complexity*.*Lyapunov Exponents*. 2016, with Arkady Pikovsky

This is a comprehensive discussion of complexity as it arises in
physical, chemical, and biological systems, as well as in mathematical models of nature.
Common features of these apparently unrelated fields are emphasised and
incorporated into a uniform mathematical description,
with the support of a large number of detailed examples and illustrations.

The quantitative study of complexity is a rapidly developing subject with special impact in the fields of physics, mathematics, information science, and biology. Because of the variety of the approaches, no comprehensive discussion has previously been attempted. The aim of this book is to illustrate the ways in which complexity manifests itself and to introduce a sequence of increasingly sharp mathematical methods for the classification of complex behaviour. The authors offer a systematic, critical ordering of traditional and novel complexity measures, relating them to well-established physical theories, such as statistical mechanics and ergodic theory, and to mathematical models, such as measure preserving transformations and disccrte automata. A large number of fully worked out examples with new, unpublished results is presented, This study provides a classification of patterns of different origin and specifies the conditions under which various forms of complexity can arise and evolve. An even more important result than the definition of explicit complexity indicators is, however, the establishment of general criteria for the identification of analogies among seemingly unrelated fields and for the inference of effective mathematical models.

This book will be of interest to graduate students and researchers in physics (nonlinear dynamics, fluid dynamics, solid-state, cellular automata, stochastic processes, statistical mechanics and thermodynamics), mathematics (dynamical systems, ergodic and probability theory), information and computer science (coding, information theory and algorithmic complexity), electrical engineering, and theoretical biology.

Lyapunov exponents he at the heart of chaos theory and are widely used in studies of complex dynamics.
Utilising a pragmatic, physical approach, this self-contained book provides a comprehensive description of the concept.
Beginning with the basic properties and numerical methods, it then guides readers through to the most recent advances
in applications to complex systems.
Practical algorithms are thoroughly reviewed and their performance is discussed,
while a broad set of examples illustrate the wide range of potential applications.
The description of various numerical and analytical techniques for the computation of Lyapunov exponents
offers an extensive array of tools for the characterisation of phenomena such as synchronisation,
weak and global chaos in low and high-dimensional setups, and localisation.
This text equips readers with all the investigative expertise needed to explore fully the dynamical properties of complex systems,
making it ideal for both graduate students and experienced researchers.