This book is primarily concerned with the computational aspects of predictability of dynamical systems - in particular those where observations, modeling and computation are strongly interdependent. Unlike with physical systems under control in laboratories, in astronomy it is uncommon to have the possibility of altering the key parameters of the studied objects. Therefore, the numerical simulations offer an essential tool for analysing these systems, and their reliability is of ever-increasing interest and importance. In this interdisciplinary scenario, the underlying physics provide the simulated models, nonlinear dynamics provides their chaoticity and instability properties, and the computer sciences provide the actual numerical implementation.

This book introduces and explores precisely this link between the models and their predictability characterization based on concepts derived from the field of nonlinear dynamics, with a focus on the strong sensitivity to initial conditions and the use of Lyapunov exponents to characterize this sensitivity. This method is illustrated using several well-known continuous dynamical systems, such as the Contopoulos, Hénon-Heiles and Rössler systems. This second edition revises and significantly enlarges the material of the first edition by providing new entry points for discussing new predictability issues on a variety of areas such as machine decision-making, partial differential equations or the analysis of attractors and basins. Finally, the parts of the book devoted to the application of these ideas to astronomy have been greatly enlarged, by first presenting some basics aspects of predictability in astronomy and then by expanding these ideas to a detailed analysis of a galactic potential.

Miguel A.F. Sanjuán is full professor of Physics at the Universidad Rey Juan Carlos in Madrid, Spain, where he is the Director of the Research Group in Nonlinear Dynamics, Chaos and Complex Systems. He has been a Visiting Research Professor at the University of Tokyo, funded by the Japan Society for the Promotion of Science; a Fulbright Visiting Research Scholar at the Institute for Physical Science and Technology of the University of Maryland at College Park, Visiting Research Professor at Beijing Jiaotong University, and Visiting Professor at the Kaunas Technological University. He is Honorary Professor of Sichuan University of Science and Technology (Zigong, China), and Honorary Professor of Huaqiao University (Xiamen, China). He also serves as the Editor General of the Spanish Royal Physics Society. He is a Corresponding Member of the Spanish Royal Academy of Sciences, a Foreign Member of the Lithuanian Academy of Sciences, and a regular member of the Academia Europaea. He has published the monograph Nonlinear Resonances (Springer, 2015). 

Juan C. Vallejo is an astrophysicist in the Nonlinear Dynamics, Chaos and Complex Systems Research Group at the University Rey Juan Carlos since 1999. His research has focused on analyzing the impact of chaotic dynamics in computer simulations for astronomy. He worked for twenty years at the European Space Astronomy Centre in Madrid, and is also working in the Joint Center of Ultraviolet Astronomy at the Universidad Complutense of Madrid.

Preface.- Forecasting and chaos.- Lyapunov exponents.- Dynamical regimes and timescales.- Predictability.- Chaos, predictability and astronomy.- A detailed example: galactic dynamics.- Appendix.