Abstract
Piecewise smooth systems are frequently used as an alternative representation of mena with multiple time scales. One would of course expect that the qualitative behaviour of a model be independent of the choice of a smooth or piecewise smooth representation. We address this issue by building on some classical results from piecewise smooth systems theory.
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Notes
- 1.
In fact, of a slight generalization of the smoothing in [3], where the smoothing function \(\phi \) is allowed to take values in an arbitrary interval \([\lambda _l,\lambda _h]\), as long as it goes to \(\pm 1\) out of an \(\epsilon \)-neighbourhood of the discontinuity.
References
A.F. Filippov, Differential Equations with Discontinuous Righthand Sides (Kluwer Academic Publishers, Dordrecht, 1988)
Yu.A. Kuznetsov, Elements of Applied Bifurcation Theory, 3rd edn. (Springer, Berlin, 2004)
J. Sotomayor, M.A. Teixeira, Regularization of discontinuous vector fields, in Proceedings of the International Conference on Differential Equations (Lisboa, 1996), pp. 207–223
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Colombo, A. (2017). A Choice Between Smooth and Nonsmooth Models. In: Colombo, A., Jeffrey, M., Lázaro, J., Olm, J. (eds) Extended Abstracts Spring 2016. Trends in Mathematics(), vol 8. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-55642-0_8
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DOI: https://doi.org/10.1007/978-3-319-55642-0_8
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