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Solutions leaving a codimension-\(\varvec{2}\) sliding

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Abstract

Piece-wise smooth differential equations (their regularization, numerical integration, and classification of solutions) is the topic of the present work. The behaviour close to one discontinuity surface and also the entering into the intersection of two discontinuity surfaces is well understood. Here, we study the solutions that exit a codimension-2 sliding mode. Some results are expected, others come as a surprise. We are able to explain situations, where difficulties in numerical computations are reported in the recent literature. The analysis is based on asymptotic expansions for singularly perturbed problems and on the study of a time-parameterized two-dimensional dynamical system (hidden dynamics). Various situations are illustrated by examples.

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Notes

  1. There is an interesting connection to the dynamical system (6) of the hidden dynamics. Considering t as a fixed parameter, the values \(u_0(t)\) and \(v_0(t)\) can be interpreted as an equilibrium of (6) corresponding to \(y^*=y_0(t)\).

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Acknowledgements

We are grateful to both anonymous referees for valuable suggestions and remarks. Part of the work was carried out during stays of the first author at the University of Geneva and of the second author at GSSI in L’Aquila. The research of this article was supported by the Fonds National Suisse, Project No. 200020_159856, and the Italian INdAM GNCS (Gruppo Nazionale di calcolo Scientifico).

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Authors and Affiliations

  1. Dipartimento di Matematica Pura e Applicata, Università dell’Aquila, via Vetoio (Coppito), I-67010, L’Aquila, Italy

    Nicola Guglielmi

  2. Dept. de Mathématiques, Univ. de Genève, CH-1211, Genève 24, Switzerland

    Ernst Hairer

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  1. Nicola Guglielmi
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  2. Ernst Hairer
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Correspondence to Ernst Hairer.

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Guglielmi, N., Hairer, E. Solutions leaving a codimension-\(\varvec{2}\) sliding. Nonlinear Dyn 88, 1427–1439 (2017). https://doi.org/10.1007/s11071-016-3320-1

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