2 citations · 2 across the 7 of their papers we have counts for
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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…
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…
The necessity of the sausage-string structure for mode-locking regions of piecewise-linear maps
David J. W. Simpson
Piecewise-smooth maps are used as discrete-time models of dynamical systems whose evolution is governed by different equations under different conditions (e.g.~switched control sys…
How to compute multi-dimensional stable and unstable manifolds of piecewise-linear maps
D. J. W. Simpson
For piecewise-linear maps the stable and unstable manifolds of hyperbolic periodic solutions are themselves piecewise-linear. Hence compact subsets of these manifolds can be repres…
The structure of mode-locking regions of piecewise-linear continuous maps: II. Skew sawtooth maps
David J. W. Simpson
In two-parameter bifurcation diagrams of piecewise-linear continuous maps on , mode-locking regions typically have points of zero width known as shrinking points. Nea…
The structure of mode-locking regions of piecewise-linear continuous maps
David J. W. Simpson
The mode-locking regions of a dynamical system are the subsets of the parameter space of the system within which there exists an attracting periodic solution. For piecewise-linear…