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math.DS2026
One-dimensional first return maps for the two-dimensional border-collision normal form with a zero determinant
David J. W. Simpson
The two-dimensional border-collision normal form is a four-parameter family of continuous, piecewise-linear maps. When this form has a zero determinant, all of its nonlinear dynami…
math.DS2026
Refracting Filippov systems: sliding dynamics without sliding regions
D. J. W. Simpson
This paper develops fundamental mathematical theory for refracting Filippov systems. These are discontinuous ordinary differential equations with solutions defined in the sense of…
math.DS2026
The stability of boundary equilibria of three-dimensional Filippov systems
David J. W. Simpson
For three-dimensional piecewise-smooth systems of ordinary differential equations, this paper characterises the stability of points that belong to a switching surface and are equil…