caa a22 4500 605541302 CHVBK 20210128100915.0 cr unu---uuuuu 210128e20150201xx s 000 0 eng 10.1007/s00371-014-1036-0 doi (NATIONALLICENCE)springer-10.1007/s00371-014-1036-0 Differential geometric methods for examining the dynamics of slow-fast vector fields [Elektronische Daten] [Martin Gutschke, Alexander Vais, Franz-Erich Wolter] In this work we present computational methods for examining dynamical systems. We focus on those systems being characterized by slow-fast vector fields or corresponding differential algebraic equations that commonly occur in physical applications. In the latter ones scientists usually consider a manifold of admissible physical states and a vector field describing the time evolution of the physical system. The manifold is typically implicitly defined within a higher-dimensional space by a system of equations. Certain physical systems, such as relaxation oscillators, perform sudden jumps in their state evolution when they are forced into an unstable state. The main contribution of the present work is to model the dynamical evolution incorporating the jumping behavior from a perspective of computational geometry which not only provides a qualitative analysis but also produces quantitative results. We use geodesic polar coordinates (GPC) to numerically obtain explicit parametrizations of the implicitly defined manifold and of the relevant jump and hit sets. Moreover, to deal with the possibly high co-dimension of the considered implicitly defined manifold we sketch how GPC in combination with the cut locus concept can be used to numerically obtain an essentially injective global parametrization. This allows us to parametrize and visualize the dynamical evolution of the system including the aforementioned jump phenomena. As main tools we use homotopy approaches in conjunction with concepts from differential geometry. We discuss how to numerically realize and how to apply them to several examples from mechanics, electrical engineering and biology. Springer-Verlag Berlin Heidelberg, 2014 Differential geometry nationallicence Dynamical system nationallicence Slow-fast vector field nationallicence Jump set nationallicence Hit set nationallicence DAE system nationallicence Geodesic polar coordinates nationallicence Cut locus nationallicence Gutschke Martin Welfenlab, Leibniz University Hannover, Hannover, Germany aut Vais Alexander Welfenlab, Leibniz University Hannover, Hannover, Germany aut Wolter Franz-Erich Welfenlab, Leibniz University Hannover, Hannover, Germany aut The Visual Computer Springer Berlin Heidelberg 31/2(2015-02-01), 169-186 0178-2789 31:2<169 2015 31 371 https://doi.org/10.1007/s00371-014-1036-0 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s00371-014-1036-0 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Gutschke Martin Welfenlab, Leibniz University Hannover, Hannover, Germany aut NATIONALLICENCE 700 1- Vais Alexander Welfenlab, Leibniz University Hannover, Hannover, Germany aut NATIONALLICENCE 700 1- Wolter Franz-Erich Welfenlab, Leibniz University Hannover, Hannover, Germany aut NATIONALLICENCE 773 0- The Visual Computer Springer Berlin Heidelberg 31/2(2015-02-01), 169-186 0178-2789 31:2<169 2015 31 371