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One-parameter bifurcations in planar Filippov systems. (English) Zbl 1079.34029
Filippov systems are considered in the plane. These are piecewise smooth ordinary differential equation systems whose vector fields have jumps at the boundary of two adjacent domains of the plane. If at a boundary point the transversal components of the vector fields corresponding to the two neighbouring domains have the same sign, the trajectory crosses the boundary transversally with a possible brake. If these components have opposite signs on a part of the common boundary, the motion has to stick to the boundary: the state of the system slides on the boundary. There may exist, obviously, singular points on the boundary in which the behaviour is changing. If the system depends on a parameter and the value of the parameter is varied, the system may undergo local and global bifurcations on the boundary (sliding bifurcations) as, e.g., singular points merge, equilibria or limit cycles from inside of the neighbouring domains touch the boundary etc. In this paper, a complete classification of codimension 1 bifurcations of generic planar Filippov systems is presented. Normal forms of the local bifurcations are also provided. The results are illustrated on a predator-prey system in which harvesting of the predator takes place above a certain value of its abundance.
MSC:
34C23Bifurcation (ODE)
34A36Discontinuous equations
34C20Transformation and reduction of ODE and systems, normal forms
92D25Population dynamics (general)
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