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Dynamical Integrity and Its Background

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Lectures on Nonlinear Dynamics

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Abstract

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.

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Acknowledgements

This paper summarizes the work I have done on dynamical integrity in the last decades together with Giuseppe Rega and other collaborators, among which my former PhD students Pierpaolo Belardinelli, Laura Ruzziconi and Nemanja Andonovski. I am extremely in debt to all of them.

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  1. Polytechnic University of Marche, via Brecce Bianche 12, Ancona, Italy

    Stefano Lenci

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  1. Stefano Lenci
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Correspondence to Stefano Lenci .

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  1. Escola Politécnica da Universidade de São Paulo, São Paulo, Brazil

    José Roberto Castilho Piqueira

  2. Escola Politécnica da Universidade de São Paulo, São Paulo, Brazil

    Carlos Eduardo Nigro Mazzilli

  3. Escola Politécnica da Universidade de São Paulo, São Paulo, Brazil

    Celso Pupo Pesce

  4. Escola Politécnica da Universidade de São Paulo, São Paulo, Brazil

    Guilherme Rosa Franzini

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Lenci, S. (2024). Dynamical Integrity and Its Background. In: Castilho Piqueira, J.R., Nigro Mazzilli, C.E., Pesce, C.P., Franzini, G.R. (eds) Lectures on Nonlinear Dynamics. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-031-45101-0_11

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