Dynamical Integrity and Its Background

This chapter summarizes the main elements of dynamical integrity, the branch of the dynamical systems theory that is aimed at checking the robustness, or practical stability, of solutions of evolutionary systems governed by differential equations (continuous time) or maps (discrete time). Although developed in the field of engineering, the ideas and results apply to any dynamical systems, irrespective of the problem they are aimed to describe. First, the background is illustrated, and the main aspects of dynamical systems are summarized. The classical concept of stability is then recalled, to understand the starting point of dynamical integrity and how this is aimed at adding new and relevant information with respect to classical knowledge. The main elements of a robustness analysis, namely the safe basin, the integrity measure and the integrity profiles, are finally reviewed, and the regularity of the integrity profiles is studied for the first time.