64 citations · 387 across the 46 of their papers we have counts for
40 papers · 1 filter
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…
Extended normal forms for one-dimensional border-collision bifurcations
P. A. Glendinning, D. J. W. Simpson
The border-collision normal form describes the local dynamics in continuous systems with switches when a fixed point intersects a switching surface. For one-dimensional cases where…
Nonsmooth folds as tipping points
D. J. W. Simpson
A nonsmooth fold is where an equilibrium or limit cycle of a nonsmooth dynamical system hits a switching manifold and collides and annihilates with another solution of the same typ…
Differentiable conjugacies for one-dimensional maps
P. A. Glendinning, D. J. W. Simpson
Differentiable conjugacies link dynamical systems that share properties such as the stability multipliers of corresponding orbits. It provides a stronger classification than topolo…
Border-collision bifurcations from stable fixed points to any number of coexisting chaotic attractors
David J. W. Simpson
In diverse physical systems stable oscillatory solutions devolve into more complicated dynamical behaviour through border-collision bifurcations. Mathematically these occur when a…
Normal forms, differentiable conjugacies and elementary bifurcations of maps
Paul A. Glendinning, David J. W. Simpson
We strengthen the standard bifurcation theorems for saddle-node, transcritical, pitchfork, and period-doubling bifurcations of maps. Our new formulation involves adding one or two…