Codimension one bifurcations of discontinuous vector fields - the boundary saddle and node cases
DOI:
https://doi.org/10.5902/2179460X83931Keywords:
Bifurcation, Regularization, Discontinuous vector fieldAbstract
We use the method of regularization of discontinuous vector fields, as described by (Sotomayor and Teixeira, 1998), to explain, in terms of the classical smooth bifurcation, two codimension one bifurcations of families of discontinuous vector fields generated by the collision of a saddle with the discontinuity set, and the collision of a node with the discontinuity set. These bifurcations are contained in the list presented both by in (Filippov, 1998), and by in (Kuznetsov et al., 2003).
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Andronov, A. A., Vitt, A. A., & Khaikin, S. E. (1966). Theory of Oscillators. Pergamon Press, Oxford, 1nst edition.
Buzzi, C. A. & Santos, R. A. T. (2019). Regularization of saddle-fold singularity for nonsmooth differential systems. Journal of Mathematical Analysis and Applications, 474(2):1036–1048.
Colombo, A., di Bernardo, M., Hogan, S., & Jeffrey, M. (2012). Bifurcations of piecewise smooth flows: Perspectives, methodologies and open problems. Physica D: Nonlinear Phenomena, 241(22):1845–1860. Dynamics and Bifurcations of Nonsmooth Systems.
Di Bernardo, M., Champneys, A., Budd, C., & Kowalczy, P. (2008). Piecewise-smooth Dynamical Systems. Springer-Verlag, London, 1nst edition.
Filippov, A. F. (1998). Differential Equations with Discontinuous Righthand Sides: Control Systems (Mathematics and its Applications). Kluwer Academic Press, 1nst edition.
Guckenheimer, J. & Holmes, P. (1983). Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields. Springer-Verlag, New York, 1nst edition.
Jeffrey, M. (2018). Hidden Dynamics: The Mathematics of Switches, Decisions and Other Discontinuous Behaviour. Springer Nature, 1nst edition.
Jeffrey, M. (2020). Modeling with Nonsmooth Dynamics. Springer Nature Switzerland, 1nst edition.
Kuznetsov, Y., Gragnani, S., & Rinaldi, A. (2003). One-parameter bifurcations in planar filippov systems. International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 13.
Sotomayor, J. (1973). Generic Bifurcations of Dynamical Systems AMS (MOS) 1970 SUBJECT CLASSIFICATION: 58F99. In PEIXOTO, M., editor, Dynamical Systems, pages 561–582. Academic Press.
Sotomayor, J. & Machado, A. (2002). Structurally stable discontinuous vector fields in the plane. Qualitative Theory of Dynamical Systems, 3:227–250.
Sotomayor, J. & Teixeira, M. (1998). Regularization of discontinuous vector fields. In Equadiff 95 : International Conference on Differential Equations. World Scientific.
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