Three forms of dimension reduction for border-collision bifurcations

For dynamical systems that switch between different modes of operation, parameter variation can cause periodic solutions to lose or acquire new switching events. When this causes the eigenvalues (stability multipliers) associated with the solution to change discontinuously, we show that if one eigenvalue remains continuous then all local invariant sets of the leading-order approximation to the system occur on a lower dimensional manifold. This allows us to analyse the dynamics with fewer variables, which is particularly helpful when the dynamics is chaotic. We compare this to two other codimension-two scenarios for which dimension reduction can be achieved. • Border-collision bifurcations with linearisations sharing an eigenvalue have dynamics on a lower dimensional manifold. • This is comparable to border-collision bifurcations with a zero eigenvalue or eigenvalue with unit modulus. • Such dimension reduction is particularly helpful when the dynamics is chaotic.