5 papers
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…
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…
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…
From two-dimensional continuous maps to one-dimensional discontinuous maps: a novel reduction explaining complex bifurcation structures in piecewise-linear families of maps
D. J. W. Simpson, V. Avrutin
Piecewise-linear maps describe dynamical phenomena that switch between distinct states and readily generate complex bifurcation structures due to their strong nonlinearity. We show…
A piecewise-linear fixed point theorem
David J. W. Simpson
We prove that if a continuous piecewise-smooth map on is comprised of two linear functions, has a bounded orbit, and satisfies a certain non-degeneracy condition, th…